- Why do we need Quantum Theory?
- Evidence for Quantum Theory
- Planck's Quantum Hypothesis
- Bohr atom;Electron Wave
- Basic Postulates of Quantum Theory
- Wave Function
- Quantum Particle in a Box
- Two State System
- Quantum Superposition Principle
- Heisenberg's Uncertainty Principle
- *Path Integral Quantum Mechanics
- *Energy in Quantum Mechanics
- *Quantum Mechanics and other Disciplines
- *Quantum Theory and Philosophy

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.

Bohr atom;Electron Wave

where is given in eq.(3.97). Bohr made the

By merely 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 m.

Bohr further made the incorrect conjecture that the electron inside the hydrogen atom moves in an

(11.11) |

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

(11.12) |

The energy of the hydrogen atom comes out to be negative, as expected since the electrons are in a

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.

(11.16) | |||

(11.17) |

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

(11.18) | |||

(11.19) |

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.

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

(11.20) |

(11.21) | |||

(11.22) |

Its complex conjugate, denoted by , is formed by everywhere changing the sign of to , and which yields

(11.23) |

(11.24) |

(11.25) | |||

(11.26) |

So we need to understand what does stand for? Note in general we have

(11.27) | |||

(11.28) | |||

(11.29) |

Motivated by the well known trigonometric identity

(11.30) |

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.

(11.32) | |||

(11.33) |

[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 , we need 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 the following.

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

(11.36) | |||

(11.37) |

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

(11.38) | |||

(11.39) |

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

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'.

(11.40) |

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.

(11.41) | |||

(11.42) |

(11.43) |

(11.44) | |||

(11.45) |

The fact that a quantum particle can be in two states

(11.46) |

(11.47) | |||

(11.48) |

Recall the reason that the intensity of the interference pattern, namely , is

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.

(11.50) | |||

(11.51) |

When

(11.52) |

The superposition principle is the unique feature of quantum mechanics, and shows graphically that, under some circumstances, particles behave as probability waves. Note from Figure11.17 that the points of minima, say , of the interference pattern indicate that

(11.55) |

Recall the energy of a free particle

Combining eqs.(11.70)and (11.70) yields

(11.60) | |||

(11.61) |

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
the momentum
can be known
only to a precision of which satisfy the celebrated
Heisenberg Uncertainty relation.

*Path Integral Quantum Mechanics

(11.64) |

(11.65) |

(11.66) |

The above equation states that for

(11.67) |

(11.68) |

(11.69) |

(11.70) | |||

(11.71) |

Energy in quantum mechanics is the physical quantity that determines how the system will evolve in time, and becomes an

(11.72) | |||

(11.73) | |||

(11.74) | |||

(11.75) |