Unit 4: Quantum Mechanics - Subjective Questions

Need for quantum mechanics: Classical mechanics and classical electromagnetic theory successfully describe macroscopic objects but fail at atomic and subatomic scales.

Major failures of classical physics:

• Black-body radiation: Classical theory predicted infinite energy emission at short wavelengths, known as the ultraviolet catastrophe.
• Photoelectric effect: Classical wave theory could not explain threshold frequency, instantaneous electron emission, or the dependence of maximum kinetic energy on frequency.
• Atomic stability: According to classical electrodynamics, an orbiting electron should continuously radiate energy and collapse into the nucleus.
• Discrete atomic spectra: Classical physics could not explain why atoms emit or absorb only specific frequencies.
• Wave-particle duality: Experiments show that photons and material particles possess both wave-like and particle-like properties.

Quantum mechanics resolves these problems by introducing energy quantization, matter waves, wave functions, and a probabilistic description of microscopic systems.

A black body is an ideal body that absorbs all incident radiation and emits the maximum possible radiation at a given temperature. Its spectral distribution describes how emitted energy varies with wavelength.

Characteristics of the distribution:

• At a fixed temperature, the spectral intensity initially increases with wavelength, reaches a maximum at , and then decreases.
• As temperature increases, the total emitted energy increases.
• The wavelength of maximum intensity shifts toward shorter wavelengths as temperature rises.
• According to Wien's displacement law,
  
  where .
• According to the Stefan-Boltzmann law, the total radiant power per unit area is
  

The spectral curve is continuous, but its correct explanation requires Planck's quantum hypothesis.

Planck assumed that the oscillators in a black body can exchange energy only in discrete packets called quanta:

The number of electromagnetic modes per unit volume in the frequency interval to is

Using Boltzmann statistics, the average energy of an oscillator is

Multiplying the mode density by the average energy gives Planck's energy-density law:

In wavelength form,

Resolution of the ultraviolet catastrophe: At high frequencies, the exponential term becomes very large, so the energy density approaches zero rather than increasing without limit. At low frequencies, Planck's law reduces to the Rayleigh-Jeans result. Thus, it agrees with experiments over the complete spectrum.

The photoelectric effect is the emission of electrons from a material surface when electromagnetic radiation of sufficiently high frequency falls on it.

Einstein proposed that radiation consists of photons, each having energy

A part of this energy is used to overcome the work function of the surface, and the remainder appears as the maximum kinetic energy of the emitted electron. Conservation of energy gives

Therefore,

If is the stopping potential,

and hence

At threshold frequency , the emitted electron has zero kinetic energy, so

Thus,

This equation explains the threshold frequency and the linear dependence of stopping potential on incident frequency.

Experimental laws:

• Photoelectric emission occurs only when the incident frequency satisfies , where is the threshold frequency.
• The maximum kinetic energy of photoelectrons depends on frequency, not on intensity.
• The photoelectric current is proportional to the intensity of incident radiation when .
• Emission begins almost instantaneously, without measurable time delay.
• A stopping potential can reduce the photocurrent to zero.

Failure of classical theory:

• Classical wave theory predicts that electron energy should increase with intensity, contrary to observation.
• It predicts no threshold frequency because energy could supposedly accumulate from radiation of any frequency.
• At low intensity, it predicts a time delay while electrons absorb enough energy, but emission is observed to be instantaneous.

The photon model explains these facts because one electron absorbs energy from one photon of energy .

According to de Broglie, every moving material particle is associated with a wave called a matter wave or de Broglie wave. Its wavelength is

where is Planck's constant and is the momentum of the particle.

For a non-relativistic particle of mass moving with speed ,

The associated frequency may be written as

Significance:

• The wavelength is inversely proportional to momentum.
• Wave effects are appreciable for microscopic particles but negligible for macroscopic bodies.
• Matter waves are not ordinary mechanical or electromagnetic waves; they represent the quantum behavior of particles.

The Davisson-Germer electron-diffraction experiment verified de Broglie's hypothesis by showing that electrons produce diffraction patterns consistent with .

The general de Broglie relation is

For a non-relativistic particle,

Therefore,

and

If a particle of charge magnitude is accelerated from rest through a potential difference , then

Thus,

For an electron, , giving

In convenient units for a non-relativistic electron,

where is measured in volts.

For relativistic kinetic energy , momentum satisfies

so the relativistic wavelength is

The kinetic energy gained by an electron accelerated through potential is

For non-relativistic motion,

Therefore,

Using ,

Numerically,

Thus, wavelength varies as

If the potential is increased from to , then

Hence, increasing the accelerating potential fourfold reduces the electron wavelength to one-half of its original value.

Phase velocity: It is the velocity with which a point of constant phase of a single wave travels:

Group velocity: It is the velocity with which the envelope of a wave packet travels:

For a non-relativistic free particle:

so

while

Physical significance:

• Phase velocity describes the motion of individual wave phases and does not represent particle or information speed.
• Group velocity describes the motion of the wave packet and equals the particle velocity.
• In a relativistic treatment including total energy, and , so . A phase velocity greater than does not violate relativity because it does not transmit information.

Heisenberg's uncertainty principle states that certain pairs of physical quantities cannot be measured simultaneously with unlimited precision.

For position and momentum,

where and are standard deviations and

For energy and time, the commonly used uncertainty relation is

Interpretation:

• Greater localization of a particle produces greater uncertainty in its momentum.
• The principle is not caused merely by defective instruments; it is intrinsic to quantum systems.
• A perfectly definite position requires a superposition of many wavelengths and therefore a wide range of momenta.
• It rules out a classical trajectory having simultaneously exact position and momentum.

The energy-time relation indicates that a state existing for a short characteristic time may possess a comparatively large energy spread.

If a particle is confined to a region of width , its position uncertainty is approximately

From Heisenberg's uncertainty principle,

so the minimum momentum uncertainty is approximately

If the average momentum is zero, the momentum spread still gives the particle a nonzero kinetic energy. Its minimum order-of-magnitude kinetic energy is

Substitution gives

Conclusion:

• A confined particle cannot remain at rest.
• Its minimum energy increases as the confinement length decreases.
• This estimate explains qualitatively the existence of zero-point energy.

The exact ground-state energy for a one-dimensional infinite box is ; the uncertainty estimate correctly predicts the dependence on and .

A wave function is a generally complex function that contains the available information about a quantum state. The wave function itself is not directly observable.

According to Born's interpretation,

represents the probability density. Thus,

is the probability of finding the particle in the volume element .

For a normalized wave function,

Conditions for an acceptable wave function:

• It must be single-valued.
• It must be finite.
• It must be continuous wherever the potential is finite.
• Its first spatial derivative is normally continuous wherever the potential is finite.
• It must be square-integrable and normalizable.
• It must satisfy the relevant boundary conditions.

Multiplication of by a constant global phase factor does not change and therefore does not alter measurable predictions.

Probability density: For a state , the probability density is

The probability of finding the particle between and is

Normalization: The certainty of finding the particle somewhere requires

If an unnormalized function is , then is chosen so that

Expectation value: The average result of many measurements of an observable represented by operator is

For example,

while momentum is represented by

These concepts connect the mathematical wave function with measurable probabilities and statistical averages.

The time-dependent Schrödinger equation in three dimensions is

In one dimension, it becomes

The quantity in brackets is the Hamiltonian operator:

Meaning of the terms:

• is the energy operator.
• is the kinetic-energy operator.
• represents potential energy.
• The equation determines how the quantum state evolves with time.

It is linear, so any linear combination of solutions is also a solution. This is the mathematical basis of quantum superposition.

For a time-independent potential , the time-dependent Schrödinger equation is

Assume a separable solution

Substitution gives

Dividing by ,

The left side depends only on time and the right side only on position, so both must equal a constant . The spatial equation is

which is the time-independent Schrödinger equation.

The temporal equation is

with solution

Therefore, a stationary state is

Time-dependent Schrödinger equation:

• Describes the complete time evolution of a quantum system.
• Applies generally, including when the potential depends on time.
• Its solution depends on both position and time.
• It is first order in time.

Time-independent Schrödinger equation:

• Applies when the potential and Hamiltonian have no explicit time dependence.
• It is an energy eigenvalue equation.
• Its solutions give allowed energies and corresponding spatial eigenfunctions.
• In coordinate space, it is generally second order in position.

For a stationary state,

Although the wave function has a time-dependent phase, its probability density is constant:

Consider the potential

Inside the box, the time-independent Schrödinger equation is

Rearranging,

where

The general solution is

The boundary condition gives . The condition gives

so

Hence,

and the allowed energies are

Normalization gives the eigenfunctions

Outside the box, .

For an infinite box of length , the allowed energies are

Properties of energy levels:

• Energy is quantized and only discrete values are allowed.
• The ground-state energy is nonzero:
  
• There is no state with , because it would give everywhere.
• Energy varies as and inversely as and .
• Adjacent level spacing is
  

The normalized eigenfunctions are

Properties of eigenfunctions:

• They vanish at the walls.
• The state has internal nodes.
• Different eigenfunctions are orthogonal:
  
• Each stationary state has a time-independent probability density.

The normalized ground-state wave function is

The required probability is

Substituting the wave function,

Using

we obtain

Since and ,

Thus, the probability of finding the particle in the left half of the box is . This also follows from the symmetry of the probability density about .

Similarities:

• Both exhibit wave-particle duality.
• Both obey the relations
  
  and
  
• Both can produce interference and diffraction.
• Their behavior requires a quantum description.

Differences:

• A photon is a quantum of electromagnetic radiation, whereas a matter wave is associated with a material particle.
• A photon has zero rest mass and travels at speed in vacuum.
• A massive particle travels at a speed less than .
• For a photon, phase and group velocities in vacuum are both .
• For a massive free particle, group velocity corresponds to particle velocity, while phase velocity generally differs from it.
• Photon polarization is an electromagnetic property, whereas a scalar spatial matter-wave function does not represent an oscillating electromagnetic field.

Matter waves therefore extend wave-particle duality from radiation to material particles.