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Subsections

Quantum Theory

Why do we need Quantum Theory?

Classical (Newtonian) mechanics works perfectly in explaining the world around us, and is accurate enough for even charting the trajectory of probes sent to Jupiter and beyond. So why are we not content with classical physics? Where does the need for quantum theory arise? Quantum theory unveils a new level of reality, the world of intrinsic uncertainty, a world of possibilities, which is totally absent in classical physics. And this bizarre world of quantum physics not only offers us the most compelling explanation of physical phenomena presently known, but is also one of the most prolific source of modern technologies, providing society with a cornucopia of devices and instruments.

Evidence for Quantum Theory

Classical mechanics works very well for large objects that are moving much slower than the velocity of light. Once objects start to move very fast, we need to modify Newton's equations by relativistic equations. On the other hand, for objects that are very small, quantum theory becomes necessary. If one attempts to extend Newton's laws to domains that are far from daily experience, they start to fail and give incorrect results. Historically, at the turn of the nineteenth century, this failure of Newtonian physics became very evident in the studies of the atom. What experimental evidence do we have that classical physics is invalid, and that, to date, quantum theory is the most accurate explanation of how nature behaves? Classical physics is what intuitively follows from our five senses, and we have no reason to naively extend the world apprehended by our five senses to microscopic domains of which we have no direct experience. As it turns out, most of the experimental evidence for quantum theory runs counter to one's day to day experiences. One of the fist achievements of quantum theory was the explanation of the structure and stability of atoms, and of the periodic table of elements. Strange phenomena such as super-conductivity, super-fluidity and so on are more macroscopic manifestations of quantum behaviour. Instead of quoting results from experiments far removed from daily life, suffice it to say that most of what goes under the name of high technology is a direct result of the workings of quantum mechanics, and most modern conveniences that we take for granted today would be virtually impossible without it. Observations of radiation from a blackbody and its radiation (measured by spectroscopic lines) provided the first experimental evidence for quantum theory. Every time one sees a neon or sodium light, one is seeing quantum theory in practice. The light from a neon or sodium source is a spectroscopic line. An electric field excites atoms of the neon or sodium atom to a discrete quantum state; the atom then makes a transition by emitting light that is characteristic of the atom, and yields the particular color of light that one sees. Furthermore, semiconductors and electronic chips in general exist due to quantum theory. Electronic devices, from computers, television, to mobile phones are all based on the semiconductor, and aeroplanes, ships, cars all use semiconductors in an essential manner. More complex technologies such as MRI (Magnetic Resonance Imaging), lasers, physical chemistry, fabrication of new drugs, modern materials science and so on all draw on the principles of quantum theory. It is no exaggeration to predict that twenty first century technology will largely be based on the principles of quantum physics.


Planck's Quantum Hypothesis

A black body which is maintained at a constant temperature T steadily loses energy from its surface in the form of electromagnetic radiation. Since the atoms composing the black body are in contact with a heat bath at temperature $T$, each atom has approximately $kT$ amount of energy, where k = Boltzmann's constant. Since the atoms are jiggling around due to thermal motion, classical electromagnetic theory then predicts that all wavelength's of radiation, in particular upto infinitely short wavelengths, should be emitted by a black body. This classical prediction for the spectrum of radiation that is emitted by such a black-body is contradicted by experiment. Max Planck, a German physicist, correctly explained the experimentally measured black- body spectrum by making the epoch-making conjecture in 1900 that electromagnetic waves are the macroscopic manifestations of packets of wave-energy called photons. Planck further made the quantum hypothesis that the energy of photons is quantized in the sense that the energy of the photons only comes in discrete packets, the smallest packet called a quantum. Photons are massless quantum particles, and all phenomena involving electromagnetic radiation can be fully explained by the quantum theory of photons. The phenomenon of electromagnetic radiation is a classical approximation to the quantum theory of photons. As mentioned in our earlier discussion on radiation in Section 6, classical electro-magnetism is a valid approximation when the typical energy of the photon is less than the characteristic energy of the instrument with which the experiment is being performed. Photons can have wavelength from zero to infinity. For a wave of frequency $f$, or equivalently, of wavelength $\lambda$, the quanta of energy are given by ($c$ is the velocity of light)
$\displaystyle E$ $\textstyle =$ $\displaystyle N\frac{hc}{\lambda}$ (11.1)
  $\textstyle =$ $\displaystyle Nhf$ (11.2)
$\displaystyle \mathrm{with}$ $\textstyle N$ $\displaystyle =1,2,...$ (11.3)

We see from the above that the shorter the wavelength of a photon, the greater is its quantum of energy. The constant $h$ is an empirical constant of nature, required by dimensional analysis, and is called Planck's constant. Its numerical value is given by
$\displaystyle h$ $\textstyle =$ $\displaystyle 6.62618 \times 10^{-34}Js$ (11.4)
  $\textstyle =$ $\displaystyle 4.13570 \times 10^{-15}eVs$ (11.5)

For the sake of convenience, it is customary to work with
$\displaystyle \hbar$ $\textstyle \equiv$ $\displaystyle \frac{h}{2\pi}$ (11.6)
  $\textstyle =$ $\displaystyle 1.054589 \times 10^{-34}Js$ (11.7)
  $\textstyle =$ $\displaystyle 6.58217 \times 10^{-16}eVs$ (11.8)

The quantum postulate immediately solves the problem of the black-body spectrum; radiation with increasingly short (ultra-violet) wavelength is incorrectly predicted by classical physics to make an increasingly large contribution to the energy loss. Due to Planck's quantum postulate, to emit even a single quanta of ultra-violet radiation would require a minimum energy much larger than the typical thermal energy of about kT that is available for emission, and hence would not be present in the radiated spectrum of a black-body. The black-body spectrum obtained from Planck's postulate is confirmed by experiment, and was the first success of quantum theory.


Bohr atom;Electron Wave

Various experiments towards the end of the nineteenth century indicated that atoms are made out of a positively charged nucleus with negatively charged particles moving around it. Atoms must interact with light. A simple manifestation of this interaction is the phenomenon of sight; light is 'reflected' off material bodies composed out of atoms. The reason an object has a color, say green, is similar to our previous analysis of a neon light. The atoms of a green object absorb only the green frequency of light, and then re-radiate green color light, thus making it appear green to our eyes. By analyzing the light emitted by atoms, one reaches the remarkable conclusion that the energy that the atom can absorb or emit only takes certain discrete values. This in turn implies that the atom's energy is quantized, and that the atom can have only a set of discrete energies. Figure 11.1 shows the discrete energy levels of a hydrogen atom. [We will discuss the spectroscopy of atoms in some detail in Chapter 12 on Atoms.]

Figure 11.1: Discrete Energy Levels of Hydrogen
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It was initially thought that the atom was a microscopic version of the solar system. The idea is mistaken for the following reason. An electron moving in a closed orbit has to classically keep on accelerating since (the direction of) its velocity is constantly changing. Since it is known from electromagnetic theory that an accelerating charge always radiates, the electron would continuously lose energy, and consequently would soon spiral into the nucleus, and in effect the atom would collapse. Hence, according to Newtonian physics, atoms are inherently unstable and cannot exist. This classical ``prediction'' contradicted the entire body of knowledge that chemistry had developed based on the idea of atoms, and that there is a different kind of an atom for each of the different elements. The question which confronted physicists was the following: are atoms real, are they made out of material particles? If so, why don't they obey the laws of classical mechanics and of electromagnetism? A complete explanation of the atom only emerged in 1926. In the absence of an alternative to Newton's law, pioneers of quantum theory had to reason intuitively and metaphorically. By the early 1900's pioneers like Niels Bohr, Max Born, Arnold Sommerfeld and Victor deBroglie had developed ad hoc rules to explain the existence of the atom, inspired by the quantum postulate of Planck. Recall Planck's conjecture that light wave consists of photons which cannot have continuous energy - unlike the case for classical radiation - but instead, that the energy of the photon is quantized and can only have discrete quanta of energy. Bohr and deBroglie made a number of postulates regarding the atom, and we only discuss those that were vindicated by later developments. The ideas of Bohr and others were conjectures, since there was no underlying theory from which these could be deduced. An electron inside an atom moves in the Coulomb potential due to the positively charged nucleus, leading to an attractive force and resulting in the electron being in a bound state. Figure 11.2 schematically shows the hydrogen atom, which consists of a nucleus made out a proton ( and zero neutron to be consistent with later discussions), and an electron ``circulating'' it. More precisely, a hydrogen atom consists of an electron of mass $m$ and charge $-e$ which is bound to a nucleus (proton) having charge $+e$. The energy of an electron in a Hydrogen atom, moving with velocity $v$, is then given by
\begin{displaymath}
E=\frac{1}{2}mv^2-k\frac{e^2}{r}
\end{displaymath} (11.9)

where $k$ is given in eq.(3.97). Bohr made the ad hoc conjecture in 1913 that inside an atom, the electron can exist only in certain special allowed states. If the electron is in such a state, it will not radiate even though it is constantly accelerating. This conjecture immediately leads to a discrete set of energies for the atom. Bohr explained the atoms' absorption and emission of radiation by the electron making ``quantum transitions'' from one its allowed states to another one. Bohr further conjectured that there is a lowest energy state for the electron, called the ground state, and once the electron is in this ground state, it will no longer radiate. Bohr conjectured that the size of an atom should be determined by the fundamental constants involved in binding an electron to the nucleus, namely the electron mass and charge $m$ and $e$ respectively, and since the existence of the atom is a quantum phenomenon, Planck's constant $h$ should also appear.
By merely doing dimensional analysis, it can easily be seen that the combination $\displaystyle \frac{h^2}{mke^2}$ has the dimension of length, and has a value of near $10^{-11}$m.
This is a remarkable coincidence, since this is the typical size of an atom, and shows the power of dimensional analysis. A more careful analysis shows that the radius of the hydrogen atom is approximately given by $\displaystyle \frac{2\hbar^2}{mke^2}=0.529\times10^{-10}$m. Question: Why does the speed of light $c$ not appear in the estimate for the size of the atom? What are the special allowed states of the electron? To get the correct energy levels, Bohr was further led to the (correct) conjecture that the angular momentum $L$ of the electron in the atom is quantized such that
\begin{displaymath}
L_{Bohr}=n\hbar
\end{displaymath} (11.10)

Figure 11.2: Pictorial Representation of the Hydrogen Atom
\begin{figure}
\begin{center}
\epsfig{file=core/H.eps}
\end{center}
\end{figure}

Bohr further made the incorrect conjecture that the electron inside the hydrogen atom moves in an exactly circular orbit. It follows from eq.(3.31)that
\begin{displaymath}
L_{Bohr}=n\hbar=mvr_n
\end{displaymath} (11.11)

and if effect the radii $r_n$ for electron motion is quantized.
Furthermore, for a particle moving in an exact circle, we have that the attractive force of the Coulomb potential be exactly balanced by the centrifugal force, and yields
\begin{displaymath}
k\frac{e^2}{r_n}=\frac{mv^2}{r_n}
\end{displaymath} (11.12)

and which yields from eq.(11.9) that the allowed (discrete) energies of the hydrogen atom as given by
$\displaystyle E_n$ $\textstyle =$ $\displaystyle -\frac{1}{n^2}E_{\mathrm{Rydberg}}$ (11.13)
  $\textstyle \mathrm{with}$    
$\displaystyle E_{\mathrm{Rydberg}}$ $\textstyle =$ $\displaystyle \frac{mk^2e^4}{2\hbar^2}=13.6
eV$ (11.14)

The energy of the hydrogen atom comes out to be negative, as expected since the electrons are in a bound state with the nucleus. The energy level of the hydrogen given by the Bohr is correct, although Bohr made a number of correct and incorrect assumptions and conjectures to come up with this result. The structure of the hydrogen atom is shown in Figure 12.2. Why should the electrons inside an atom have only a discrete set of allowed states? Bohr had no explanation for this experimental fact, and it was only later, in 1923, that deBroglie offered the following explanation. Just like the photon is a particle that manifests as electromagnetic waves, one can conversely conjecture that a particle, say an electron, also has a wave associated with it. For a particle of mass $m$ moving at velocity $v$, deBroglie conjectured that there is an associated wave with a wavelength given by
\begin{displaymath}
\lambda=\frac{h}{mv}
\end{displaymath} (11.15)

Figure 11.3: DeBroglie Wave of an Electron
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\input{core/electron.eepic}
\end{center}
\end{figure}

How does the idea of deBroglie explain the behaviour of the electron inside the atom? At the scale of the atom, which is of the order of $10^{-11}$m, deBroglie conjectured that the idea of an electron being a classical particle having a definite position and velocity is no longer valid. Furthermore, reasoning by analogy with the concept of resonance in waves (as discussed in Section 5.4), for which only certain frequencies are allowed, deBrolie conjectured that the only allowed waves for the electron are resonant waves. To see how Bohr's conjectures follow from the idea of an electron wave, for a circular orbit of radius $r_n$, a state of the electron with $n$ complete wavelengths yields, similar to eq.(5.77) for the case of resonance for a circular object, the following.
$\displaystyle 2\pi r_n=n\lambda=n\frac{h}{mv}$     (11.16)
$\displaystyle \Rightarrow L_{Bohr}=mvr_n=n\hbar$     (11.17)

where the last equation reproduces Bohr's conjecture, given in eq.(11.10), that the angular momentum of the electron is quantized.

Figure 11.4: Ground State
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\epsfig{file=core/wave1.eps, height=4cm}
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The restriction of the states of the electron to be only resonant waves in turn provides an explanation of the discrete energy levels observed for atoms. The discrete quantized energies of the atom correspond to the allowed resonant frequencies. The electron can make ``quantum transitions'' from a higher energy state to a lower energy state by radiating photons, and vice versa by absorbing photons. Furthermore, the fact that there is a lowest frequency for a resonant wave explains why an electron can be in a bound state with the nucleus without radiating, since there is no lower state into which it can make a transition.

Figure 11.5: Second Excited State
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Figure 11.6: Third Excited State
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\epsfig{file=core/wave4.eps, height=4cm}
\end{center}
\end{figure}

In summary, an electron can have a definite energy in an atom and be in a stable stationary state, but the price that we must pay is that we no know its exact position. This is the only way we can avoid the classical result that an accelerating charge must radiate. Note the inability to know the position and velocity of an electron in an atom is not like our ignorance in statistical mechanics; rather, the ignorance in quantum phenomena is an inherent limitation that is placed by nature on what can in principle be known, in this case about the electron's position and velocity. This quantum postulates of deBroglie yields a stable atom. But there is the following paradox inherent in deBroglie's postulate of an electron wave. Each and every time an electron is observed in an experiment, it is seen to be a point-like particle; on the other hand, a wave is spread over space. So this is the paradox: how can the electron be a point-particle and at the same time be a ``wave''? This is the famous ``wave-particle'' duality that permeates quantum physics. What is the nature of the wave that deBroglie postulated? Is the electron wave a physical wave, like a sound wave or an electromagnetic wave? For five years, the electron wave that deBroglie postulated was simply an interesting metaphor, without any sound theoretical or mathematical foundation. In 1926 it was finally understood that the electron wave of Bohr is not a physical wave, but instead, is a probability wave. What do we mean by a probability wave, and how does this relate to the behaviour of an electron? We address this question in the section below.

Basic Postulates of Quantum Theory

The modern formulation of quantum theory rests primarily on the ideas of Erwin Schr $\ddot{\mathrm{o}}$dinger, Werner Heisenberg and P.A.M. Dirac. In the period from 1926-29 they laid the mathematical foundations for quantum mechanics, and this theory has successfully stood the test of innumerable experiments over the last seventy years. At present, there is not a single experimental result which cannot be explained by the principles of quantum theory. Unlike Einstein's theory of relativity which reinterpret's the meaning of classical concepts such as time, position, velocity, mass and so on, quantum theory introduces brand new and radical ideas which have no pre-existing counterpart in classical physics. To understand the counter-intuitive and paradoxical ideas that are essential for the understanding of quantum mechanics, we develop it in contrast to what one would expect from classical physics, and from intuition based on our perceptions of the macroscopic world. Recall that a classical system is fully described by Newton's laws. In particular, if we specify the position and velocity of a particle at some instant, its future evolution is fully determined by Newton's second law. In quantum mechanics, the behaviour of a quantum particle is radically different from a classical particle. The essence of the difference lies in the concept of measurement, which results in an observation of the state of the system. A classical particle, whether it is observed or unobserved, is in the same state. By contrast, a quantum particle has two completely different modes of existence, something like Dr Jekyll and Mr Hyde. When a quantum particle is observed it appears to be a classical particle having say a definite position or momentum, and is said to be in a physical state. However, when it is not observed, it exists in a counter-intuitive state, called a virtual, or a probabilistic, state. To illustrate the difference between a classical and quantum particle, let us study the behaviour of a classical and quantum particle confined inside a potential well of infinite depth. Consider a particle of mass $m$, confined to a one-dimensional box, with perfectly reflecting sides due to the infinite potential, of length $L$. Suppose the particle has a velocity $v$, and hence momentum $p=mv$. Let us study what classical and quantum physics have to say about the particle confined to a box.

Figure 11.7: Particle in a Box
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Classical Description The classical (Newtonian) description of a particle is that the particle travels along a well-defined path, with a velocity $v$. Since the box has perfectly reflecting boundaries, every time the particle hits the wall, its velocity is reversed from $v$ to $-v$, and it continues to travel until it hits the other wall and bounces back and so on. We hence have
$\displaystyle p_{\mathrm{classical}}$ $\textstyle =$ $\displaystyle mv$ (11.18)
$\displaystyle E_{\mathrm{classical}}$ $\textstyle =$ $\displaystyle \frac{1}{2}mv^2=\frac{p^2}{2m}$ (11.19)

Figure 11.8: Spacetime diagram of a Classical Particle in a Box
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\epsfig{file=core/figure28.eps, height=6cm}
\end{center}
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The point to note is that the position and velocity of the classical particle are determined at every instant, regardless of whether it is being observed or not. Quantum Description A particle inside a potential well is similar to an electron inside an atom, and hence is be described by a resonant wave. The reason we choose the example is because it has all the features of the $H$-atom, but is much simpler. The specific features of an electron inside an atom discussed earlier reflect the general principle of quantum theory which states that, if the momentum of the particle is fully known, we then have correspondingly no knowledge of its position. The precise relation between the uncertainty in position and momentum is given by the Heisenberg Uncertainty Principle discussed in Section 11.76. Note we can interchange the role of momentum with position, and a similar analysis follows. Hence, similar to the case of the Bohr atom, the electron in the potential well is in a bound state with a definite energy, but at the same time it no longer has a definite position. In summary, the particle inside the potential well has a definite momentum (and hence has definite energy), but its position inside the well is a random variable. When it is not observed, the quantum particle exists in a random state, which in physics is called a virtual state; in particular, the position of the particle can be anywhere within the interval $L$.


\fbox{\fbox{\parbox{12cm}{ {\bf Random Variable}
\ Quantum physics leads us to...
...y a (classical) coin, but is even more subtle, and
which we address later.
}}}


What happens if we perform a measurement to actually ``see'' what is the position occupied by the particle? The measurement will find the electron to be always at some definite point; the act of measurement causes the electron to make a quantum transition from its virtual state to an actual physical state. In summary, the quantum particle has two forms of existence: a virtual state when it is not being observed, and a physical state which is observed when a measurement is performed on the particle. In this sense, the physical state of the quantum particle is Dr Jekyll, and its hidden virtual state is Mr Hyde.


Wave Function

Recall that in our discussion of the atom, we discussed deBroglie's conjecture the electron is described by a (probability) wave.The resolution of the paradoxes posed by Bohr, deBroglie and others is the following: the electron wave that deBroglie postulated is not a classical wave, but rather is a probability wave. The fundamental quantity that specifies quantum probabilities is a wave-like entity called a probability amplitude, also called the Schr $\ddot{\mathrm{o}}$dinger wave function, and is denoted by $\Psi $. It should be emphasized that the probability wave $\Psi(x,t)$ of a quantum particle is not an ordinary classical wave; rather, the only thing it has in common with a classical wave is that it is spread over space. The non-classical nature of $\Psi(x,t)$ can seen from the following fact: the moment the quantum particle is observed (measured), say to be at a definite position $x$, the wavefunction of the quantum particle instantaneously goes to zero ('collapses') everywhere else in space, since once we find the particle at position x there is zero likelihood of finding it at any other point! This process of the collapse of the wavefunction causes an irreversible change in the system and is called decoherence. The wave function $\Psi $ in general is a complex number, and is the central quantity in quantum mechanics, replacing the role played by the position and velocity of a particle in Newtonian mechanics. In quantum mechanics one gives up any attempt to know what the object intrinsically is, what in philosophy is called the object in-itself. The wave function $\Psi $ contains all the possible information that can be extracted from the object by a process of repeated measurements. Quantum probabilities, denoted by $P$, are related to the wave function $\Psi $ by the following relation.
\begin{displaymath}
P(\mbox{\rm {some specific outcome}})\equiv \vert\Psi(\mbox{\rm {some specific
outcome}})\vert^2
\end{displaymath} (11.20)

The equation above is the great discovery of quantum theory, namely, that behind what we observe lies the hidden world of probability, which in turn is fully described by the wave function $\Psi $ The Schr $\ddot{\mathrm{o}}$dinger equation determines how $\Psi(x,t)$ changes with time; given the initial wavefunction of the particle at t=0, i.e. $\Psi(x,0)$, the future behaviour is then fixed by the Schr $\ddot{\mathrm{o}}$dinger equation. In summary, we see that the classical particle which is pointlike has now been replaced by a quantum particle which is described by the wavefunction $\Psi(x,t)$. The quantum particle is inherently random when it is not being observed, and in the sense of probability exists everywhere in space and hence is wave-like; when a measurement is performed to ascertain the position of the quantum particle, it is always found to be pointlike and hence the particle-like behaviour of the quantum particle. This is the wave- particle duality referred to earlier. The non-local collapse of $\Psi(x,t)$ has puzzled physicists since it apparently needs the information that the particle has been observed at position x to be communicated at infinite speed to the rest of space for the instantaneous 'collapse' to take place and would seem to violate the special theory of relativity. However, detailed analysis has shown that quantum measurement theory is consistent with the special theory of relativity; and more importantly, the non-local nature of the wavefunction $\Psi(x,t)$ is consistent with all the experiments that have been devised to test this aspect of quantum mechanics.

*Complex Numbers

A complex number $\alpha$, is represented, for real numbers a and b, by
$\displaystyle \alpha$ $\textstyle =$ $\displaystyle a+ib$ (11.21)
$\displaystyle i^2$ $\textstyle =$ $\displaystyle -1$ (11.22)

Its complex conjugate, denoted by $\alpha^*$, is formed by everywhere changing the sign of $i$ to $-i$, and which yields
\begin{displaymath}
\alpha^*=a-ib
\end{displaymath} (11.23)

Recall the absolute value of a real number is always greater or equal to zero, that is, $\vert\pm a\vert=a \geq 0$. We similarly have for the absolute value of a complex number, multiplying by the rules of (real) arithmetic
\begin{displaymath}
\vert\alpha\vert^2\equiv \alpha \times \alpha^*=a^2+b^2 \geq 0
\end{displaymath} (11.24)

One can verify that adding, subtracting, multiplying and dividing two complex numbers results in yet another complex number. Just like powers of real numbers, a complex number can be raised to a complex power. We have, for real numbers a,x and y
$\displaystyle a^{x+iy}$ $\textstyle =$ $\displaystyle a^x a^{iy}$ (11.25)
  $\textstyle =$ $\displaystyle a^x e^{iy\ln a}$ (11.26)

So we need to understand what does $\displaystyle e^{i\theta}, \theta \equiv y\ln a$ stand for? Note in general we have
$\displaystyle e^{i\theta}$ $\textstyle =$ $\displaystyle a+ib$ (11.27)
$\displaystyle e^{i\theta}\times e^{-i\theta}=(a+ib)(a-ib)$ $\textstyle =$ $\displaystyle a^2+b^2$ (11.28)
$\displaystyle \Rightarrow 1$ $\textstyle =$ $\displaystyle a^2+b^2$ (11.29)

Motivated by the well known trigonometric identity
\begin{displaymath}
1=\sin^2\theta+\cos^2\theta
\end{displaymath} (11.30)

we have the fundamental result
\begin{displaymath}
e^{i\theta}=\cos\theta+i\sin\theta
\end{displaymath} (11.31)


\fbox{\fbox{\parbox{12cm}{In general, from the reasoning given above that, we ca...
...ay}
since the constant is zero by noting that $t=0 \Rightarrow
\theta=0$.
}}}


Complex numbers form a system of arithmetic similar to real numbers. For the more mathematically minded, it can be shown that to solve for the roots of an arbitrary n-th-order polynomial equation, it is necessary and sufficient to extend the real numbers to complex numbers.

Quantum Particle in a Box

We examine more carefully the behaviour of a quantum particle inside a box; in particular, what it means to observe and to not to observe the particle. For the quantum particle confined to a box of length, we know that the probability of finding the particle outside the interval must be zero, and hence we need $\Psi $ to vanish outside the interval. Let $x$ be a point inside the interval; in other words $x$ lies inside the interval $[0,L]$, denoted by $x \in [0,L]$. Following the reasoning of deBroglie, we take $\Psi $ to be a resonant wave; then, from our discussion in Section 5.4 we have the following.
$\displaystyle \Psi(x)$ $\textstyle =$ $\displaystyle C sin(\frac{px}{\hbar}) \mbox{\rm { ;}} x \in [0,L]$ (11.32)
  $\textstyle =$ $\displaystyle 0 \mbox{\rm { ;}}x \ne [0,L]$ (11.33)

[C is a constant which we will fix later.] The constant $\hbar$ that has been introduced in the equation above is purely on dimensional grounds. The argument of a sine function is an angle, which is dimensionless. Since $x$ has dimensions of length, and $p$ has dimensions of mass $\times$ velocity, we have to divide out by a a dimensionful quantity $h$ which has the dimensions of $px$. We will soon see that this constant is none other than the famous Planck's constant, whose numerical value is given in eq.(11.6). We need to smoothly match $\Psi $ for the regions inside and outside the interval $[0,L]$, and hence $\Psi $ must vanish at the boundary points $x=0$ and $x=L$. Note that from above that at $x=0$ we have as expected $\Psi(0)=0$; however, to achieve $\Psi(L)=0$, we need to constrain (quantize) the possible values of $p$. From the properties of the sine function we know that $sin(n\pi)=0$ for any integer $n$. Hence for the momentum $p$ of the quantum particle we have the following.
$\displaystyle \Psi(L)$ $\textstyle =$ $\displaystyle C sin(\frac{Lp}{\hbar})=0$ (11.34)
$\displaystyle \Rightarrow p$ $\textstyle =$ $\displaystyle \hbar\frac{n\pi}{L}$ (11.35)

Figure 11.9: Graph of $\Psi $ with Different Momenta
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\epsfig{file=core/box1.eps, height=8cm}
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We see from eq.(5.33) that the momentum of a quantum particle is not like that of a classical particle. A classical particle inside a box can have any momentum $p$; however for the quantum particle, since the wave function $\Psi $ has to vanish outside $[0,L]$ the momentum of the quantum particle is quantized, and can only have a discrete set of values, and consequently its energy is quantized as well. This is a general feature of a quantum system, and as one can imagine, quantum theory derives its name from this phenomenon. In summary, for a quantum particle, its energy is given by
$\displaystyle p$ $\textstyle =$ $\displaystyle \hbar\frac{n\pi}{L}$ (11.36)
$\displaystyle E$ $\textstyle =$ $\displaystyle \frac{p^2}{2m}=\hbar^2\frac{(n\pi)^2}{2mL^2}$ (11.37)

Note that both momentum $p$ and energy $E$ have been quantized, that is come in discrete amounts measured by $\hbar$ and $\hbar^2$ respectively. Now it is known from classical physics that the energy for example of a string tied at two ends also becomes discrete. But what is unique about the quantization of momentum and energy is that it is measured in terms of a universal constant $\hbar$. For a quantum particle, what can be physically measured is not its position, but rather,the probability of the particle being at different positions inside the box. The probability is given by
$\displaystyle P(x;p)$ $\textstyle =$ $\displaystyle \vert\Psi(x)\vert^2$ (11.38)
  $\textstyle =$ $\displaystyle C^2 sin^2(\frac{px}{\hbar}) \mbox{ ;}x \in [0,L]$ (11.39)

Figure 11.10: First Excited State
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Figure 11.11: Second Excited State
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In Figure's 11.10 and 11.11 the probability of the particle being at various positions in the interval $L$ is plotted. Note the salient point that unlike the classical particle which passes through each point within the interval $L$, the quantum particle, say in the first excited state, will never be found at the midway point of the interval. In general, the probability distribution of the quantum particle shows it to be anything but a point particle when it is in its virtual state.


\fbox{\fbox{\parbox{12cm}{
We fix C. For any system,
the total probability of ...
...}}
\end{eqnarray}
Note the constant $C$ does not depend on momentum $p$.
}}}


The graph of $P(x)$ describes the result of a number of experiments. Since we are looking at a particle, say an electron, with a definite momentum, we prepare such electrons in the following manner. Heat a filament, which causes it to eject electrons with a wide range of momenta. We then subject the electrons to a fixed magnetic field, and the electrons will curve around with different radii depending on their momenta. We position our box at a particular radii, and, let the electrons with a definite momentum allowed by the box to enter into the box, and give it time to 'settle down'.

Figure 11.12: Electrons Going through a B-field and Entering into Box
\begin{figure}
\begin{center}
\epsfig{file=core/figure29.eps, width=8cm}
\end{center}
\end{figure}

We then measure the position of the particle inside the box. The way this is done is to measure the position of the particle by say shining light on it; suppose we find that its position is $x$. So we end up with an electron having a definite momentum $p$ as well,which we had prepared carefully, and with a definite position $x$ due to the measurement that we performed. One may object to the statement that the particle is at a definite position $x$, since did we not assume that the position of the electron inside the box is random? The answer is yes, the position is random. What we have done is to obtain one possible position of the electron. In other words, when the position of the particle in not measured, it is in a (random) virtual state. By the act of measurement, we caused the particle to make a quantum transition from its virtual state to an actual physical state. We have to repeat the experiment again and again, and every time sending in an electron with the same momentum, and then measuring its position. We will soon discover that the position of the (identical) electrons entering into the box varies, and the electron is found at all points inside the box. We repeat the experiment $N$ times, and record the number of times, $n(x)$, that the electron is found at the position $x$. We can then calculate the probability that the electron is at the different positions, given by $P(x;p)$. For the given momentum chosen, say $p$, we have, in accordance with the general formula given in eq.(4.29), the following
$\displaystyle P(x;p)=\frac{n(x)}{N}\pm \frac{\sigma}{\sqrt{N}}$     (11.40)

An an estimate, we have $\sigma^2=<x^2>-n^2(x)$. The result of the experiment will yield, for the second excited state, the distribution of the positions as shown in Figure 11.11. The probabilities computed from quantum theory behave the same way as that of classical probability. What separates classical and quantum probabilities is the existence of the wave function, and we explore these differences in the next section.

Two State System

In quantum theory, a particle is described by specifying all the possible states it can have. To simplify our discussion, consider a particle that can have only two possible states. Such a system is the simplest possible one for a quantum particle, and is also called, for obvious reasons as a two-state system. An example of a two-state system is the spin of an electron. In addition to moving around in space, the electron has an intrinsic angular momentum called spin. The spin of the electron can either point up or down, and hence forms a two-state system. A quantum two-state system is described by determining its amplitude to be in the two states. How should we mathematically describe our two-state system? The two different states should be "orthogonal" to each other, in the sense that being in one state is completely different from being in the other state. The simplest way to realize this expected orthogonality of the two states is to represent them by two-dimensional vectors, and the idea of orthogonality translates exactly into the concept of vectors being perpendicular. Hence, we will represent the wavefunction for a two-state system by two-dimensional vectors. One should note that the two-dimensional vector space has got nothing to do with a physical two-dimensional space, but rather, should be viewed as a mathematical construction for describing the spin of an electron.


\fbox{\fbox{\parbox{12cm}{An aside about notation. In some textbooks, a vector i...
...d{equation}
where, in general, $\alpha$ and $\beta$ are complex numbers.
}}}


Figure 11.13:
\begin{figure}
\begin{center}
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\input{core/figure30.eepic}
\end{center}
\end{figure}

To precisely discuss the spin of an electron, we first have to choose a coordinate system for the electron. Consider an external magnetic field pointing along the z-axis. Consider two special cases for the spin of the electron. Case (a) The spin points along the z-axis, which we denote as the spin is pointing "up". Case (b) The spin points towards the negative z-axis,which we denote as the spin is pointing "down". The wavefunction for these two special cases are the following.
$\displaystyle \vert\Psi(\mathrm {up})>$ $\textstyle =$ $\displaystyle \vert u>$ (11.41)
$\displaystyle \vert\Psi(\mathrm {down})>$ $\textstyle =$ $\displaystyle \vert d>$ (11.42)

diagram spin pointing up, down and aribitrary So far we could have been discussing classical physics, since the spin pointing up or down with 100% certainty is a classical concept. A quantum mechanical spin is more subtle, since we can superpose two states and obtain a state that points up or down along the z-axis with only a certain likelihood. For such a quantum mechanical state, obtained by superposing a quantum spin pointing up with a one pointing down, we have its wavefunction $\Psi $ given by
\begin{displaymath}
\vert\Psi>=\alpha \vert u>+\beta \vert d>
\end{displaymath} (11.43)

with the following physical interpretation
$\displaystyle \vert\alpha\vert^2$ $\textstyle =$ $\displaystyle \mbox{\rm { Probability that the spin is pointing up}}$ (11.44)
$\displaystyle \vert\beta\vert^2$ $\textstyle =$ $\displaystyle \mbox{\rm { Probability that the spin is pointing
down}}$ (11.45)

The fact that a quantum particle can be in two states simultaneously is highly counter-intuitive and paradoxical. Since there is nothing special about spin, one can replace spin up and spin down by any two independent states. To illustrate the paradox, Schr $\ddot{\mathrm{o}}$dinger proposed the following experiment. Suppose a cat is inside a sealed and opaque box, with a radioactive substance inside the box as well. The radioactive material randomly emits alpha particles, and if it emits a strong burst of alpha particles, it will trigger a container to release a poisonous gas, causing the cat to die. The question Schr $\ddot{\mathrm{o}}$dinger asked is the following: As long as we do not open the box (technically speaking: perform a measurement), we do not know what has transpired, and there is some likelihood that the cat is either dead or alive. Hence the cat's wavefunction will be
\begin{displaymath}
\vert\Psi(\mathrm{cat})>=\alpha \vert\mbox{\rm {cat dead}}>+\beta \vert\mbox{\rm {cat alive}}>
\end{displaymath} (11.46)

This famous cat, called Schr $\ddot{\mathrm{o}}$dinger's cat, illustrates the counter-intuitive and bizarre world of quantum mechanics, that the cat can be alive and dead at the time! Schr $\ddot{\mathrm{o}}$dinger felt this was an absurd situation, since - regardless of whether a measurement is performed - the cat should either be dead or alive, since how can the cat be dead and alive at the same time? The paradox that Schr $\ddot{\mathrm{o}}$dinger's cat brings out is the need to understand what is the physical meaning of the entity that we obtain by superposing two, or, for that matter, many states. To understand the superposition principle, we study the famous two-slit experiment.

Quantum Superposition Principle

The most counter-intuitive aspect of quantum mechanics is the essential role that measurement plays in determining the behaviour of physical reality. We already have encountered something strange and bizarre in Schr $\ddot{\mathrm{o}}$dinger's cat, namely, how can a system simultaneously be in two orthogonal states? To fully appreciate the counter-intuitive and paradoxical nature of quantum mechanics we study the two-slit experiment in some detail. The heart of quantum mechanics is tied down to the wave-particle duality of elementary particles. An elementary particle can be localized (captured) as if it were a point like particle; on the other hand it can exist everywhere just like a wave field which has an extended structure. In this section we shall illustrate the wave-attribute and particle-attribute of an elementary particle. In the macro-world, the concept of a particle is easy to comprehend. One starts with a piece of matter and keep on breaking it until one reaches the smallest constituent of matter. This smallest constituent is a particle. For example, the powder of a chalk can approximately be regarded as particles. In this way a particle is a point-like object. In geometry, a point has no size. However in physics one needs to measure 'size', that is, one needs a very powerful microscope in order to determine microscope size. The microscope in this scale is just the high energy accelerator and detector and at present energy the smallest size that we can measure is up to the order of $10^{-19}$ m. A wave is also easy to visualize in the macro-world, it is the motion of a disturbance and a simple wave is an extended entity with a periodic structure. Water wave, sound wave are just energy propagating in continuous media: water and air. Originally it was thought that light wave also needs the presence of a medium, the ether. This ether is ruled out by the constancy of the speed of light. In micro-world we cannot directly 'see' the particles of waves, so how do we extend the concept of particle and wave from macro-world to the micro-world? This is done by examining the behaviour of a physical system under interference experiments. In macro-world, bullets can be taken as particles. In the experiment as illustrated in Figure11.14, a bullet from the firing gun can only go through either slit 1 or slit 2 and it is detected by the movable detector at the backstop. The experiment is first done by covering the slit 2 so that the bullet can go through only slit 1. After firing for a suitable time interval $\Delta T$, the distribution of the bullets detected is plotted along the $x$-direction and the distribution curve $P_1(x)$ is obtained. The experiment is repeated with the same time interval $\Delta T$ with the slit 1 being covered instead of slit 2. The result is the distribution curve $P_2(x)$.

Figure 11.14: Interference Experiment with Bullets
\begin{figure}
\begin{center}
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\input{core/figure41.eepic}
\end{center}
\end{figure}

When both slits 1 and 2 are open, the combined distribution curve $P_1(x)+P_2(x)$ is obtained, and it is found that is in fact a sum if $P_1(x)$ and $P_2(x)$. In this way we identify particle-like behavior for a physical system in the microwrold by the distribution curves $P_1(x)$, $P_2(x)$ and $P_1(x)+P_2(x)$. There is no interference in the sense that the probability that a bullet will reach a point $x$ is by taking a path eitherthrough slit 1 or through slit 2, and this explains the final result being a sum of $P_1(x)$ and $P_2(x)$. A similar experiment is performed for water wave as shown in Figure 11.15. The detector can only measure the intensity $I$ of the wave, which is proportional to the square of the height $h$ of the wave. Let $h_1$ and $h_2$, respectively, be the amplitudes (heights) of the waves arriving at the detector when slit 2 and slit 1 are closed; we then have $I_1 \propto h_1^2(x)$ and $I_2 \propto h_2^2(x)$ are the intensity distributions of the water wave when slit 2 and slit 1 are closed respectively. When slits 1 and 2 are open, the resultant intensity is, from the superposition of waves, given by
$\displaystyle I_{12}$ $\textstyle =$ $\displaystyle (h_1 + h_2)^2$ (11.47)
  $\textstyle \neq$ $\displaystyle I_1+I_2$ (11.48)

Recall the reason that the intensity of the interference pattern, namely $(h_1 + h_2)^2$, is not the sum of the individual amplitude is due to constructive and destructive interference, and which gives the characteristic minima and maxima of interference. Interference, as originally used by Young for light, is the best indication for a phenomenon being wave-like.

Figure 11.15: Waves
\begin{figure}
\begin{center}
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\input{core/figure42.eepic}
\end{center}
\end{figure}

To ascertain whether an elementary particle such as an electron behaves like a wave or particle, we carry out the interference experiment similar to the one we have considered for water waves. What we need to state at the outset that the interference patterns $P_1+P_2$ and $I_{12}$ can both be obtained for the electron depending on how we perform the experiment. The experimental arrangement consists of an electron gun which sends identical electrons through a screen which has two slits to a wall where an apparatus keeps track of the point at which the electron stops. The electron gun produces the electrons one by one, so that at any given time there is only one electron traveling from the electron gun to the wall. We consider two different experiments with this arrangement, namely, one experiment in which a measurement is carried out to determine which slit the electron went through, and a second experiment in which no measurement is made to determine which slit the electron goes through. In both cases a large number, say $10^6$, electrons are sent in, one by one, and the distribution of the positions at which the electrons hit the wall is measured. Experiment with Detection

Figure 11.16: Electron with detectors
\begin{figure}
\begin{center}
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\input{core/figure45.eepic}
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\end{figure}

We perform the experiment as given in Figure 11.16 with both slits 1 and 2, open and with the additional requirement that we determine which slit the electron actually passes through. This can be arranged by fixing two detectors, say a light source, at the back of the slits as shown in Figure 11.16. Since we know which slit the electron goes through we can plot three distribution curves. $P_1$ and $P_2$ are the distribution curves for electrons go through slit 1 and slit 2 respectively. Similar to the result obtained for bullets, the probability of the electron arriving at a point on the wall when both slits are open, denoted by $P_{12}$ is given by
\begin{displaymath}
P_{12}=P_1+P_2
\end{displaymath} (11.49)

$P_{12}$ is the distribution curve for electrons that passes through either slit 1 or 2. We consequently have the result that when the electron's path is measured, it has a particle-like behavior. Experiment without Detection Consider now the same experiment as before, but with the detectors removed, as shown in Figure 11.17. In other words, we do not make any measurement to determine which slit the electron goes through. The result of this experiment is illustrated in Figure 11.17, and shows that a single electron gives rise to an interference. The interference pattern $P_{12}$ is exactly like $I_{12}$ as obtained for water waves. This suggests that electrons behave like waves and we have to introduce a probability amplitude $\Psi_1$ for electrons when slit 2 is closed and an amplitude $\Psi_2$ for electrons when slit 1 is closed. We then have in analogy with waves
$\displaystyle P_1=\vert\Psi_1\vert^2$     (11.50)
$\displaystyle P_2=\vert\Psi_2\vert^2$     (11.51)

When both slits 1 and 2 are open, and no measurement is made, the resultant distribution $P_{12}$ is the square of modulus of the sum of $\Psi_1$ and $\Psi_2$. The probability amplitudes obey the superposition principle when the different paths are not known, and yield
\begin{displaymath}
\Psi_{12}=\Psi_1+\Psi_2
\end{displaymath} (11.52)

It is the superposed amplitude $\Psi_{12}$ that determines the outcome when no measurement is performed. Hence
$\displaystyle P_{12}$ $\textstyle =$ $\displaystyle \vert\Psi_{12}\vert^2=\vert\Psi_1+\Psi_2\vert^2$ (11.53)
  $\textstyle \neq$ $\displaystyle \vert\Psi_1\vert^2+\vert\Psi_2\vert^2=P_1+P_2$ (11.54)

The superposition principle is the unique feature of quantum mechanics, and shows graphically that, under some circumstances, particles behave as probability waves.

Figure 11.17: Electron without Detector
\begin{figure}
\begin{center}
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\input{core/figure43.eepic}
\end{center}
\end{figure}

Note from Figure11.17 that the points of minima, say $x_M$, of the interference pattern indicate that no electrons will be detected at those points. This is a remarkable, since if say only one slit was open there is a finite likelihood of an electron arriving at $x_M$, but with both slits open, unlike the case for bullets, no electron can arrive there. This result is counter-intuitive since one would expect, as in the case of bullets, that for both slits open the electron would have two ways of arriving at point $x_M$. In sum, when we do not observe which path the electron takes, it behaves like a wave. An actual interference experiment for atoms instead of electrons is given in Figure 11.18, and leads to the conclusion that the electron distribution curve can either be wave-like or particle-like depending on whether we require the information as to which slit the electron passes through.

Figure 11.18: Interference Experiment with Atoms
\begin{figure}
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\epsfig{file=core/lwav2_1.eps, width=4cm, angle=90}
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Knowing which slit the electron passes through, the electron exhibits particle-like behaviour and result in eqn.(11.62) is obtained; not knowing which slit the electron goes through, the electron exhibits wave-like behaviour and result in eqn.(11.66) follows. This is the famous wave-particle duality of quantum mechanics. The wave-like character is lost in Figure 11.16 because we ``shine'' light on the electron after it passes through the slit. This ``shining light'' is a ``measurement'' process and it collapses the wave-like distribution of eqn.(11.66) to the particle like distribution of eqn.(11.62). In other words the global wave-like character collapses to a localized particle character and is called decoherence.

Heisenberg's Uncertainty Principle

Planck's quantum postulate has radical and counter-intuitive implications for what can be experimentally measured. This aspect was elucidated only in 1927 by Werner Heinsenburg, another German physicist, and goes by the name of Heisenberg's Uncertainty Principle. Suppose one wants to measure the position of a particle; one can shine light on it and locate it by observing the light that is reflected by the particle. Suppose one wants to know the position of the particle to a very high degree of precision; then, since light of a given wavelength $\lambda$ cannot resolve distances less than $\lambda$ we will have to shine light on the particle with smaller and smaller wavelength to determine more and more precisely what is its position. And this is where we run into the quantum postulate: the minimum amount of light that we can shine on the particle has to have at least one quanta of energy, which is $\displaystyle \frac{hc}{\lambda}$; as we make $\lambda$ smaller and smaller, the energy of the minimum quanta becomes larger and larger.

Figure 11.19: Shining light with large wavelength
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\begin{center}
\input{core/figure53.eepic}
\end{center}
\end{figure}

Hence to make a very precise measurement of the position of the particle, we are forced by the quantum principle to impart a high amount of energy to the particle and results in increasing the kinetic energy to the particle. This kinetic energy imparted to the particle changes the velocity of the particle in an uncontrollable and irreversible manner, and we end up with a final velocity of the particle which is different from the value it had before we made the measurement.

Figure 11.20: Shining light with small wavelength
\begin{figure}
\begin{center}
\input{core/figure52.eepic}
\end{center}
\end{figure}

What is the uncertainty that results from the fact that light only comes in quanta with a minimum energy? For the case shown in Figures 11.19 and 11.20, the uncertainty in position of the particle whose position we are determining approximately equal to the wavelength $\lambda$ of light that we are shining on it, since any distance much smaller than $\lambda$ cannot be resolved. Hence we have
\begin{displaymath}
\Delta x=\lambda
\end{displaymath} (11.55)

In the process of measurement, we have to scatter off the particle, a photon which has at least $\displaystyle E_{\mathrm{photon}}= \frac{hc}{\lambda}$ amount of energy, corresponding to one quantum of light with wavelength $\lambda$. If we knew for certain that the particle would absorb one quantum of light, we could always account for it and there would be no uncertainty. However, and here is where the random and unpredictable aspect of quantum measurement comes in, during the process of measurement, the particle has a finite probability of absorbing any amount of energy from $0$ to $\displaystyle E_{\mathrm{photon}}= \frac{hc}{\lambda}$. One might object that even in a classical measurement, a certain amount of energy has to be imparted to the particle being observed. Although this is true, in classical physics, in principle, the energy imparted in the process of measurement can be made arbitrarily small, whereas in quantum physics, a precision of say $\Delta x$ will necessarily involve a minimum energy of $\displaystyle E_{\mathrm{photon}}= \frac{hc}{\lambda}$ to carry out the measurement. This in essence is the dividing line between classical and quantum measurement theory. Consider measuring the position of a quantum particle which is moving with an initial velocity $v_i$; after the measurement process it will have a final velocity given by $v_f$. The energy of a free quantum particle is not quantized (it is not in a bound state), the particle can absorb any amount of energy upto a maximum of $\displaystyle E_{\mathrm{photon}}$. The way the particle absorbs energy less than $\displaystyle E_{\mathrm{photon}}$ is to first absorb the photon of wavelength $\lambda$, and then with a finite probability spontaneously re-emit another photon of wavelength $\lambda'$, and with energy $E'$; clearly we must have $\displaystyle E'< E_{\mathrm{photon}}$.In effect, the particle absorbs energy equal to $\displaystyle E_{\mathrm{photon}}-E'$. Figure 11.21 symbolically shows a particle absorbing and re-emiting a photons in the process of its position being measured.

Figure 11.21: Particle and Photon denoted by Straight and Wavy Lines, respectively
\begin{figure}
\begin{center}
\input{core/scattering.eepic}
\end{center}
\end{figure}

If one repeats the experiment with initial velocity $v_i$, the particle will have a final velocity which will not have a fixed value, but rather will vary over a range of velocities, denoted by $\Delta v_f$ The variation in the final velocity is due to the varying amounts of energy that the particle absorbs in the process of measurement. The fluctuation in the energy effectively absorbed by the particle is the inherent randomness in the process of a quantum measurement, and which induces a quantum transition from a virtual to a physical state. Hence, no matter what was energy of the particle before the measurement, after the measurement, it has an uncertainty in its energy due to having absorbed energy anywhere between energy $0$ to energy $\displaystyle E_{\mathrm{photon}}$. Hence, the uncertainty in the particle's energy after the measurement process is $\displaystyle \Delta E_{\mathrm{particle}} =E_{\mathrm{photon}}$, and is given by
$\displaystyle \Delta E_{\mathrm{particle}}$ $\textstyle =$ $\displaystyle \frac{hc}{\lambda}$ (11.56)
$\displaystyle \Rightarrow \Delta E_{\mathrm{particle}}$ $\textstyle =$ $\displaystyle \frac{hc}{\Delta x}$ (11.57)

Recall the energy of a free particle after the measurement is given by $\displaystyle \frac{1}{2}mv_f^2$, and hence the uncertainty in the energy of the particle $\displaystyle \Delta E_{\mathrm{particle}}$ translates into uncertainty in the particles final momentum $v_f$. Since all the quantities from now on refer to only the particle that is being observed, we drop the subscript of particle. We hence have
$\displaystyle \Delta E$ $\textstyle =$ $\displaystyle \frac{1}{2}m\Delta (v_f^2)$ (11.58)
$\displaystyle \Rightarrow \Delta E$ $\textstyle \simeq$ $\displaystyle mv_f\Delta v_f=v_f\Delta p_f$ (11.59)

Combining eqs.(11.70)and (11.70) yields
$\displaystyle \frac{hc}{\Delta x}$ $\textstyle =$ $\displaystyle v_f\Delta p_f$ (11.60)
$\displaystyle \Delta x \Delta p_f$ $\textstyle =$ $\displaystyle h\frac{c}{v_f}$ (11.61)

From special relativity we always have that $v_f<c$; hence
\begin{displaymath}
\Rightarrow \Delta x \Delta p_f\geq h
\end{displaymath} (11.62)

Eq.(11.75) states that in making a precise measurement of the position of the particle with initial fixed velocity $v_i$, we introduce uncontrollable uncertainties into the final momentum of the particle $p_f$, the precise amount being given by the Heisenberg Uncertainty Principle.


% latex2html id marker 11565
\fbox{\fbox{\parbox{12cm}{ {\bf Shift of Paradigm}
...
...
of the virtual realm that measurement and uncertainty could be
addressed.
}}}


To recapitulate, we started by trying to precisely measure the position of the particle with no desire to disturb its velocity. But we discovered that, due to the quantum principle, the more precisely we measured the position of the particle the more we uncontrollably disturbed the velocity of the particle. Hence we ended up with a precise measurement of the position of the particle, and due to this very measurement we lost information on the precise value of the particle's velocity. Eq.(11.75) is a special case of the Heisenberg Uncertainty Principle. Heisenberg postulated that a measurement made by any means (not necessarily by using light) will, due to the quantum postulate, introduce uncontrollable disturbances in the object being observed. If say the position of the particle is measured to only to a precision $\Delta x$, then the momentum $p=\mathrm{mass}\times \mathrm{velocity}$ can be known only to a precision of $\Delta p$ which satisfy the celebrated Heisenberg Uncertainty relation.

\begin{displaymath}
\Delta x\Delta p \geq \hbar
\end{displaymath} (11.63)

Heisenberg's Uncertainty Principle states that any measurement made will satisfy the uncertainty relation, and be of only a limited precision; the classical concept of having an arbitrarily precise knowledge of both $x$ and $p$ does not hold in the micro-world. Heisenberg's uncertainty principle has stunning implications. If we fully know the position of a particle, that is $\Delta x=0$, then eq. (11.76) implies that $\Delta p=\infty$, and visa versa! In other words, position and momentum are mutually exclusive, in that complete knowledge of one necessarily means giving up all knowledge of the other. But this is not the end; position and momentum, even though being mutually exclusive, are nevertheless related by eq. (11.76) in that one can have partial knowledge of both. What happens when we don't make any measurement, for example to determine the position or the momentum of the particle? Does it have a definite position or a definite momentum? The answer is no; the particle is in a probabilistic state in which both its position and momentum have a likelihood of having a whole range of values, and has discussed at length in the Section 11.6. The probabilistic state of a quantum particle implies the counter-intuitive result that the outcome of an observation depends on what we decide to measure! For example if we decide to make a very precise measurement of the position of the particle we will end up with a large uncertainty in its momentum, whereas if we decide to make a very precise measurement of the momentum of that same particle we will end up with a large uncertainty in its position! To get a concrete idea of the Uncertainty Principle, consider a hydrogen atom in which an electron is in a bound state with a proton due to their mutual electrical attraction. Since there is an attractive force one may ask why doesn't the electron fall into the nucleus (proton) and by doing this minimize the potential energy of the atom (and which in fact is the incorrect prediction of classical physics)? The reason is the Uncertainty Principle. If the electron were to fall into the nucleus, its position would be determined fairly precisely and this would mean that $\Delta x$ would become very small; the uncertainty principle would then imply that $\Delta p$ is very large, and this would give the electron a very large amount of kinetic energy resulting in the electron flying far away from the proton and in effect breaking-up the atom. The atom reaches a compromise by letting the electron move around in a finite volume whose size is fixed so as to minimize the kinetic energy due to $\Delta p$ and while at the same time lowering as much as possible the electron's potential energy. It is this trade-off between the uncertainty of the position and momentum of the electrons inside an atom which is responsible for the finite size of atoms. The actual size of the atom of course depends on the charge $-e$ and mass $m$ of the electron, together with the extent that quantum effects due to $\hbar$ are operational. This yields the size of an atom to be about $10^{-10}$ m, and is the actual size of a typical atom.


\fbox{\fbox{\parbox{12cm}{ {\bf Quantum Cause and Effect}\\
In classical and d...
...ent, and avoid the paradoxes
which quantum cause and effect could lead to.
}}}



*Path Integral Quantum Mechanics

Recall in classical physics to determine the future evolution of a particle using Newton's second law, we need to specify its exact position as well as its exact velocity at the start of the particle's motion. However, we learnt from Heisenberg's uncertainty relation that we cannot, even in principle, determine simultaneously the exact position and velocity of a particle. The best we can do is to specify the initial coordinate $x$ of the particle, and the future behaviour of a quantum particle is then given by the Schr $\ddot{\mathrm{o}}$dinger equation. We explore the physical implications of the fact that we no longer know the initial velocity of the particle.

Figure 11.22: Quantum Particle Taking All Possible Paths
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\input{core/figure54.eepic}
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To start with, how are we to describe a quantum particle? Recall in the two-slit experiment we saw that if a quantum particle is not observed as going through a particular slit, then, in the sense of probability, a single particle can be thought of as going through both slits simultaneously. Paths taken by the quantum particle in the sense of probability (unobserved) are called virtual paths to distinguish them from experimentally observed paths called physical paths. Now consider making more and more slits with smaller and smaller widths until we have infinitely many slits with zero width as in Figure 11.22 - in other words no slits at all! We now see that the quantum particle, as it evolves from its observed initial position to its final position simultaneously takes all possible virtual paths from its starting to its finishing point. This is the tremendous generalization of quantum mechanics over classical mechanics: in the latter the classical particle takes only one definite physical path in evolving from its initial to its final position, whereas in quantum mechanics the 'particle' propagates probabilistically and takes all possible virtual paths from it's observed initial to its final position. The probability for the quantum particle to take a particular path is given essentially by the potential that is acting on the particle as well as its kinetic energy. Path Integral quantum mechanics was formulated by Feynman in 1949. Path integral quantum mechanics starts directly from the virtual paths that a quantum particle takes, and derives the results of Schr $\ddot{\mathrm{o}}$dinger and Heisenberg. Fundamental to path integral quantum mechanics is the probability amplitude for a particle to go from initial position and time $x_i,t_i$ to the final position and time $x_f,t_f$. To evaluate the probability amplitude we need to define the concept of the Lagrangian. Recall energy is given by
\begin{displaymath}
E=T+V
\end{displaymath} (11.64)

We can also form another quantity, called the Lagrangian $L$, which is defined by
\begin{displaymath}
L=T-V
\end{displaymath} (11.65)

and the action $S$ is defined by
\begin{displaymath}
S=\int_{t_i}^{t_f}L dt
\end{displaymath} (11.66)

The probability amplitude $K$ for the particle is then given by
$\displaystyle K(t_i,x_i;t_f,x_f)$ $\textstyle =$ $\displaystyle \mbox{\rm {Probability amplitude particle
goes}}$  
$\displaystyle \mbox{\rm {from}}$ $\textstyle x_i$ $\displaystyle \mbox{\rm {to $x_f$ in time $t_f-t_i$}}$  
  $\textstyle =$ $\displaystyle \sum_{\mbox{\rm {\{All Possible Paths from $(x_i,t_i)$ to
$(x_f,t_f)$}}\}} e^{iS/\hbar}$  

The above equation states that for every path that goes from $(x_i,t_i)$ to $(x_f,t_f)$, we have a contribution of $e^{iS(\mathrm{Path})/\hbar}$ to $K$; in Figure 11.22, if we were to label the paths as 1,2,3, ..., we would then have
\begin{displaymath}
K(t_i,x_i;t_f,x_f)=e^{iS(\mathrm{Path1})/\hbar}+e^{iS(\mathrm{Path2})/\hbar}+...
\end{displaymath} (11.67)

The discussion on path integral quantum mechanics is reminiscent of our earlier discussion on Fermat's Principle of Least Time. As $\hbar \rightarrow 0$ we see from eq.(11.80) that due to the infinitely rapid oscillations of the exponential, only that path will contribute for which
\begin{displaymath}
\delta S=0 \mbox{\rm { :Principle of Least Action}}
\end{displaymath} (11.68)

From the equation above, we see that Fermat's Principle is actually as special case of the Principle of Least Action, which is a formulation equivalent to Newton's three laws of motion. To relate the path integral formulation of quantum mechanics with the Schr $\ddot{\mathrm{o}}$dinger wavefunction $\Psi(x,t)$ we ask the following question: where is the quantum particle at some intermediate time t between its starting and final time? Since the quantum particle is taking all possible virtual paths, at time t it can be at any point of space with a certain likelihood. Hence, unlike a classical particle whose complete description at time t is given by specifying its position and velocity, to describe the state of a quantum particle we have to specify the likelihood of finding the particle at all the points of space. This probability of finding the quantum particle at the point $x$ of space at time $t$ is given by $\vert\Psi(x,t)\vert^2 $ and is indicated in Figure 11.22. We can now answer our earlier question as to what is the physical significance that if we specify the initial position of the particle, we cannot specify it velocity. The reason that we can see from the above analysis is that the single quantum particle is taking all possible paths simultaneously; consequently it does not have a definite velocity, and this is built into the Schr $\ddot{\mathrm{o}}$dinger equation since the initial velocity of the particle is not specified to determine the way the particle evolves in time.

*Energy in Quantum Mechanics

We return to our two-state system. The two-state system in general changes with time, and at any instance of time $t$, the wavefunction of the system is given by
\begin{displaymath}
\vert\Psi(t)>=\alpha(t)\vert u>+\beta(t)\vert d>
\end{displaymath} (11.69)

For the case when the two state refers to a spin $1/2$ system, we can think of $\vert\Psi(t)>$ as describing the precession of the spin about the z-axis. diagram of spin precessing The change of the wavefunction, for a time interval from $t$ to $t+\Delta t$, is given by
$\displaystyle \Delta\vert\Psi(t)>$ $\textstyle \equiv$ $\displaystyle \vert\Psi(t+\Delta t)>-\vert\Psi(t)>$ (11.70)
  $\textstyle =$ $\displaystyle \Delta\alpha(t)\vert u>+\Delta\beta(t)\vert d>$ (11.71)

Energy in quantum mechanics is the physical quantity that determines how the system will evolve in time, and becomes an operator; for this reason energy is given the special name of the Hamiltonian operator $H$ which acts on the wavefunction, causing it to change. In other words, for a small interval $\Delta t$, the Hamiltonian is defined by
$\displaystyle H\vert\Psi(t)>$ $\textstyle =$ $\displaystyle -i\frac{\Delta\vert\Psi(t)>}{\hbar \Delta t}$ (11.72)
$\displaystyle i \frac{\Delta t H}{\hbar} \vert\Psi(t)>$ $\textstyle =$ $\displaystyle \Delta\vert\Psi(t)>$ (11.73)
  $\textstyle =$ $\displaystyle \vert\Psi(t+\Delta t)>-\vert\Psi(t)>$ (11.74)
$\displaystyle \Rightarrow e^{ \displaystyle{ i \frac{\Delta t H}{\hbar}}}\vert\Psi(t)>$ $\textstyle \approx$ $\displaystyle \vert\Psi(t+\Delta t)>$ (11.75)


\fbox{\fbox{\parbox{12cm}{Taking the continuum limit $\Delta t \rightarrow 0$, w...
... \frac{d\alpha(t)}{dt}\vert u>+\frac{d\beta(t)}{dt}\vert d>
\end{eqnarray}
}}}


*Quantum Mechanics and other Disciplines

It would not be an exaggeration to hold that quantum mechanics has revolutionized our understanding of nature. Our understanding of quantum mechanics is still far from complete and one can be sure there are a lot of surprises awaiting us in the future. On the more practical side, quantum mechanics has led to the creation of most of the modern 20th century technologies that has served society so well. This connection of quantum mechanics with engineering and technology in general has been covered in the Section on Physics and Technology. It can also be safely predicted that 21st century technology will depend even more on quantum mechanics, and those who will study quantum mechanics will be richly rewarded. Quantum theory has had, and continues to have, a far reaching impact on a number of related and not so related fields. About a hundred and fifty years ago, chemistry had almost no connection with physics and concepts of chemistry such as valency, activity, solubility and volatility had more of a qualitative character. The first application of physics to chemistry started in the 19th century with the theory of heat, and was led by the hope of understanding the laws of chemistry in terms of the mechanics of atoms. One of the most successful application of quantum mechanics is the explanation of all the atoms which form the periodic table, and which is the starting point of all chemistry. With the explanation of chemical processes and chemical laws in terms of the quantum mechanics of atoms and molecules, a complete understanding of the laws of chemistry can now be sought in the laws of quantum mechanics. The present relation of biology to physics and chemistry is similar to that of chemistry to physics a hundred years ago. Biological concepts such as life, organ, cell function, perception, adaptation, etc. presently have no explanation in terms of physical and chemical laws. However, with the discovery of the DNA molecule (which contains more than 100,000 atoms), molecular biology now seeks the explanation of terrestrial life in terms of the atoms and molecules which compose the DNA, proteins and other biological macro-molecules, and hence has taken a giant step towards basing itself on the principles of quantum mechanics. The laws of physics and chemistry together with the laws of history (as embodied in Darwin's theory of evolution), have been suggested as forming the conceptual basis for explaining life. Biological evolution has taken place for about 4 billion years, during which nature could try out an almost infinite variety of combinations of atoms and molecules to come up with quasi-stable self-replicating biological macro-molecules which form the basis of living organisms; hence the element of history will probably have to be added to the laws of quantum mechanics to achieve an explanation of the principles of biology. Quantum mechanics has had a profound impact on mathematics and vice versa. The concept of $\vert\Psi(x,t)\vert^2 $ being the probability for occurrence of the different values of the position x has had a major influence on the formal theory of probability. Quantum mechanics has made major contributions to the theory of functional analysis since physically measurable quantities such as position, charge etc. are realized by operators acting on function space and which forms the subject matter of functional analysis. The theory of renormalization, which is an essential feature quantum field theory (the application of the principles of quantum mechanics to the quantization of a classical field), is even today beyond the scope of rigorous mathematics, and find its final justification in the experimental validation of its predictions. More recently, the rapid progress of string theory has opened up new connections between theoretical physics and mathematics, in particular with the more specialized branches of mathematics such as (algebraic and differential) geometry and topology of higher dimensional manifolds, number theory, singularity theorems, knot and link theory, infinite dimensional algebra's and groups and so on. The importance of quantum theory for disciplines more closely related to physics has been even more seminal. Astronomy and astrophysics is concerned with formation and distribution of galaxies, and with the stars which compose them. The processed taking place inside a star as well as its composition are largely determined by quantum theory; in particular, the synthesis via fusion of heavier elements inside a star, the evolution of a star, whether it will become a supernova or a neutron star or a white dwarf or a black hole are all the result of quantum mechanical processes going on inside a star. Cosmology studies the large scale structure of the universe, and one would have thought that quantum mechanics, which apparently is concerned with the micro-world, would not have any relevance to cosmology; in fact, nothing could be further from the truth. The current hot big-bang theory of cosmology relies solely on quantum mechanics to explain the events which occurred within the first 1000 seconds and which are the determinate events which shaped all later evolution of the universe. If one probes even closer to within a few trillionth's of a second after the big-bang, then the universe is seen to be completely dominated by the fundamental quanta's of nature and which find their explanation in the quantum field theory, string theory, gauge fields, quantum gravity etc.

*Quantum Theory and Philosophy

As can be seen from the preceding discussion on the strange and counter-intuitive world of quantum phenomenon, although in practice quantum mechanics has so far been in complete agreement with experimental results, its theoretical underpinning is not well understood; in the words of Bohr, one of the founders of quantum theory, those who are not shocked by quantum mechanics have not understood it. Since the philosophical aspects of quantum theory has many interpretation, we will present the point of view of the Copenhagen school of Bohr and Heisenberg as expressed in the book "Physics and Philosophy" by Werner Heisenberg. The fundamental paradox of quantum mechanics is the following: how can a particle be pointlike when it is observed, and be wave-like when it is not observed? According to Heisenburg, when a quantum particle is not observed it exists as an ensemble of ``possibilities'' (in physics called a virtual state) in which it has a likelihood of existing simultaneously at all points of space; however, when an observation is performed the quantum particle makes a discontinuous jump (called a quantum transition) to a state with some definite position and is said to be in a condition of``actuality'' (in physics called a physical state). The transition from the possible to the actual takes place the moment the quantum particle comes into contact with a measuring device. The act of measurement is not a subjective act and does not need the measurement to be registered in the mind of an observer; the reason being that in the Copenhagen interpretation of quantum theory, the measuring apparatus is supposed to be large enough so that the deterministic laws of classical physics hold for it, and hence once the apparatus registers a reading, the discontinuous ``collapse'' of the wavefunction takes place. To paraphrase Heisenberg, the state of potentiality that the unobserved quantum particle exists in has to assume a form to actualize, and this form is provided by the measuring devise: if the devise measures position, the particle assumes the actuality of being at some definite position, whereas if the momentum of the particle is measured the form assumed by the particle in actualizing is that of being in a state of some definite momentum. In this language, for classical physics all of physical reality consists of only actuality, the interconnection and transition between these two forms of reality being mediated by the act of measurement. The division of physical reality into the ``observer'' and the ``object being observed'' has vexed physicists since the founding of quantum mechanics since there is no hard and fast rule as where does one draws the line between the observer and the observed. One may have noticed that Heisenberg's Uncertainty Principle connects two quantities, in particular position and momentum, in an essential relationship given in eq.(11.76) that was absent in classical physics. In quantum mechanics, position and momentum become mutually exclusive in that completely specifying one means giving up all information about the other. But paradoxically, even though position and momentum are mutually exclusive, they are nevertheless essentially related because $\hbar \neq 0$; in other words, the extent to which position and momentum interpenetrate and make transitions into each other is quantitatively expressed by $\hbar$. A similar relationship exists for the observer and the observed, the wave and particle duality of a quantum particle, possibility and actuality etc. Bohr generalized this feature of quantum physics as the principle of complementarity: nature can be understood in terms of concepts which come in complementary pairs of opposites that are inextricably connected by a Heisenberg-like uncertainty principle; these concepts cannot be applied simultaneously to the problem at hand; rather the two concepts are applicable for describing different aspects of the phenomenon being studied, and taken in its entirety a complete understanding of the phenomenon requires both the concepts. Those familiar with dialectics will recognize that position and momentum and other complementary pairs are examples of a dialectical contradiction, and Bohr's principle of complementarity is a statement about the dialectics of opposites. Virtual and physical state form a unity of opposites, which are mutually exclusive while simultaneously essentially inter-related, and the entire theory of measurement in quantum mechanics is based on the dialectical idea of making a quantum transition from the virtual to the physical state. Quantum physics has performed flawlessly, both in the experimental and theoretical realms, for almost the last hundred years. There are nevertheless questions raised by quantum mechanics, in particular by the interpretation of what constitutes a measurement, which are still in need of clarity and intelligibility. This is the challenge that will be faced by physicists in the 21st century.
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Marakani Srikant 2000-09-11