Next: Atoms and the Periodic
Up: Laws of Physics :
Previous: *Statistical Mechanics
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
technologies, providing society with a cornucopia of devices and instruments.
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
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
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
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 , each atom has approximately 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 , or equivalently,
of wavelength , the quanta of energy are given by ( is the velocity of light)
Planck's Quantum Hypothesis
We see from the above that the shorter the wavelength of a
photon, the greater is its quantum of energy.
The constant is an empirical constant of nature, required by dimensional analysis,
and is called
Planck's constant. Its numerical value is given by
For the sake of convenience, it is customary to work with
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.
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
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.]
Bohr atom;Electron Wave
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
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
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 and charge which is bound
to a nucleus (proton) having charge . The energy of an
electron in a Hydrogen atom, moving with velocity , is then given by
Discrete Energy Levels of Hydrogen
where 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 and
respectively, and since the existence of the atom is a quantum
phenomenon, Planck's constant should also appear.
doing dimensional analysis, it can easily be seen that the combination
has the dimension of length, and has a value
of near 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
Question: Why does the speed of light 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 of the electron in the
atom is quantized such that
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
Pictorial Representation of the Hydrogen Atom
and if effect the radii 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
and which yields from eq.(11.9) that the allowed
(discrete) energies of the hydrogen atom as given by
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.
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 moving at velocity
, deBroglie conjectured that there is an associated wave
with a wavelength given by
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 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 , a state of the
electron with complete wavelengths yields, similar to eq.(5.77)
for the case of resonance for a circular object, the following.
DeBroglie Wave of an Electron
where the last equation reproduces Bohr's conjecture, given in
eq.(11.10), that the angular momentum of the electron is
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.
Second Excited State
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.
The modern formulation of quantum theory rests primarily on the ideas
of Erwin Schr
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
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 , confined to a one-dimensional box, with perfectly
reflecting sides due to the infinite potential, of length . Suppose the
particle has a velocity , and hence momentum
. Let us study what classical and quantum physics have to
say about the particle confined to a box.
Third Excited State
The classical (Newtonian) description of a particle is that the
particle travels along a well-defined path, with a velocity .
Since the box has perfectly reflecting boundaries, every
time the particle hits the wall, its velocity is reversed from
to , and it continues to travel until it hits the other wall
and bounces back and so on.
We hence have
Particle in a Box
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.
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 -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 .
Spacetime diagram of a Classical Particle in a Box
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
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.
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
dinger wave function, and is denoted by
It should be emphasized that the probability wave 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 can seen from the following
fact: the moment the quantum particle is observed (measured), say to be at a definite
position , 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
The wave function 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.
function contains all the possible information that can be extracted from the object
by a process of repeated measurements.
Quantum probabilities, denoted by , are related to the wave
function by the following relation.
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
dinger equation determines how
changes with time; given the initial wavefunction of the particle
at t=0, i.e. ,
the future behaviour is then fixed by the Schr
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
. 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 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 is consistent with all the
experiments that have been devised to test this aspect of quantum mechanics.
A complex number ,
is represented, for real numbers a and b, by
Its complex conjugate, denoted by , is formed by everywhere changing the sign
of to , and which yields
Recall the absolute value of a real number is always greater or equal to zero,
. We similarly have for the absolute
value of a complex number, multiplying by the rules of (real)
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
So we need to understand what does
stand for? Note in general we have
Motivated by the well known trigonometric identity
we have the fundamental result
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 to vanish outside the interval.
Let be a point inside the interval;
in other words lies inside the interval , denoted by
. Following the reasoning of deBroglie, we take
to be a resonant wave; then, from our discussion in Section 5.4 we
have the following.
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.
[C is a constant which we will fix later.]
The constant 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 has dimensions of
length, and has dimensions of mass velocity, we have to
divide out by a a dimensionful quantity which has the
dimensions of . 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 for the regions inside and outside
the interval , and hence must vanish at the boundary points
and . Note that from above that at
we have as expected ; however, to achieve ,
to constrain (quantize) the possible values of . From the properties
of the sine function we know that for any integer .
Hence for the momentum of the quantum particle we have
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 ;
however for the quantum
particle, since the wave function has to vanish outside
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
Graph of with Different Momenta
Note that both momentum and energy have been quantized,
that is come in discrete amounts measured by and
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 .
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
First Excited State
In Figure's 11.10 and 11.11 the probability of the
particle being at various positions in the interval is
plotted. Note the salient point that unlike the classical particle
which passes through each point within the interval , 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.
Second Excited State
The graph of 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'.
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 .
So we end up with an electron having a definite momentum
as well,which we had prepared carefully, and with a definite position
due to the measurement that we performed.
One may object to the statement
that the particle is at a definite position ,
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
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 times, and record the
number of times, , that the electron is found at the position .
We can then calculate the probability that the electron is at the
different positions, given by . For the given momentum
chosen, say , we have, in accordance with the general
formula given in eq.(4.29), the following
Electrons Going through a B-field and Entering into Box
An an estimate, we have
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.
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.
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.
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 given by
with the following physical interpretation
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
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
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
This famous cat, called Schr
dinger's cat, illustrates the
counter-intuitive and bizarre world of quantum mechanics, that the cat can
be alive and dead at the time!
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
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
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
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 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 , the distribution
of the bullets detected is plotted along the -direction and the
distribution curve is obtained. The experiment is repeated with
the same time interval with the slit 1 being covered instead of
slit 2. The result is the distribution curve .
When both slits 1
and 2 are open, the combined distribution curve is obtained, and it is found
that is in fact a sum if and . In this way we identify
particle-like behavior for a physical system in the microwrold by
the distribution curves , and . There is no interference
in the sense that the probability that a bullet will reach a point is by taking
a path eitherthrough slit 1 or through slit 2, and
this explains the final result being a sum of and .
A similar experiment is performed for water wave as shown in Figure 11.15.
The detector can only measure the intensity of the wave, which is proportional
to the square of the
height of the wave. Let and , respectively, be
the amplitudes (heights)
of the waves arriving at the detector when slit 2 and slit 1 are closed; we then have
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
Interference Experiment with Bullets
Recall the reason that the intensity of the interference pattern, namely ,
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.
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 and can both
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 , 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
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.
and 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 is given by
Electron with detectors
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
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 is exactly like as obtained for water waves.
This suggests that electrons
behave like waves and we have to introduce a probability amplitude for
electrons when slit 2 is
closed and an amplitude for electrons when slit 1 is closed.
We then have in analogy with waves
When both slits 1 and 2 are
open, and no measurement is made, the resultant distribution is the
square of modulus of the sum of and . The probability amplitudes
obey the superposition principle when the different paths are not
known, and yield
It is the superposed amplitude that determines the
outcome when no measurement is performed. Hence
The superposition principle is the unique feature of quantum
mechanics, and shows graphically that, under some circumstances, particles behave as
Note from Figure11.17 that the points of minima, say , 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 , 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 . 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.
Electron without Detector
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
other words the global wave-like character collapses to a localized particle character and
is called decoherence.
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
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 cannot resolve distances less than 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
; as we make smaller and smaller, the energy
of the minimum quanta becomes larger and larger.
Interference Experiment with Atoms
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.
Shining light with large wavelength
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 of light that we are
shining on it, since
any distance much smaller than cannot be resolved. Hence we have
Shining light with small wavelength
In the process of measurement, we have to scatter off the particle, a photon
which has at least
amount of energy, corresponding
to one quantum of light with wavelength .
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 to
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 will necessarily involve a minimum
to carry out the measurement. This in essence is the dividing line
between classical and quantum measurement theory.
measuring the position of a quantum particle which is moving with an initial
velocity ; after the measurement process it will have a final velocity given by .
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
. The way the particle
absorbs energy less than
first absorb the photon of wavelength , and
then with a finite probability spontaneously
re-emit another photon of wavelength , and with energy
; clearly we must have
.In effect, the
particle absorbs energy equal to
Figure 11.21 symbolically shows a particle absorbing and re-emiting
a photons in the process of its position being measured.
If one repeats the experiment with initial velocity , 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
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
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 to
. Hence, the uncertainty in
the particle's energy after the measurement process
, and is given by
Particle and Photon denoted by Straight and Wavy Lines, respectively
Recall the energy of a free particle after the measurement
is given by
and hence the uncertainty in the energy of the
into uncertainty in the particles final momentum . 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
Combining eqs.(11.70)and (11.70) yields
From special relativity we always have that ; hence
Eq.(11.75) states that in making a precise measurement of the position of the
particle with initial fixed velocity , we introduce uncontrollable
uncertainties into the final momentum of the particle , the precise amount being given
by the Heisenberg Uncertainty Principle.
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 , then
can be known
only to a precision of which satisfy the celebrated
Heisenberg Uncertainty relation.
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 and 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 , then eq. (11.76)
, 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
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 would become very small; the uncertainty principle would
then imply that is very large, and this would give the electron a very large
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 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 and mass of the electron, together with the
extent that quantum effects due to are operational. This yields
the size of an atom to be
about m, and is the actual size of a typical atom.
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 of the particle,
and the future behaviour of a quantum particle is then given by the
dinger equation. We explore the physical implications of the
fact that we no longer know the initial velocity of the particle.
*Path Integral Quantum Mechanics
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
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
Heisenberg. Fundamental to path integral quantum mechanics is the
probability amplitude for a particle to go from initial position and time to
the final position and time . To evaluate the probability
amplitude we need to define the concept of the Lagrangian. Recall
energy is given by
Quantum Particle Taking All Possible Paths
We can also form another quantity, called the Lagrangian , which
is defined by
and the action is defined by
The probability amplitude for the particle is then given by
The above equation states that for every path that goes from
to , we have a contribution of
to ; in Figure 11.22, if we were to label the paths as
1,2,3, ..., we would then have
The discussion on path integral quantum mechanics is reminiscent
of our earlier discussion on Fermat's Principle of Least Time. As
we see from eq.(11.80) that due to the infinitely rapid
oscillations of the exponential, only that path will contribute
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
wavefunction 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 of space at time is given by
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
equation since the initial velocity of the particle is not
specified to determine the way the particle evolves in time.
We return to our two-state system. The two-state system in general
changes with time, and at any instance of time , the
wavefunction of the system is given by
For the case when the two state refers to a spin system, we
can think of 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 to , is given by
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 which acts on the wavefunction, causing it to
change. In other words, for a small interval , the
Hamiltonian is defined by
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
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 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.
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
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
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 ; in other words, the extent to which position and momentum
interpenetrate and make
transitions into each other is quantitatively expressed by .
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.
Next: Atoms and the Periodic
Up: Laws of Physics :
Previous: *Statistical Mechanics