Quantum Mechanics and Atomic Physics

Introduction

Classical physics explains many macroscopic phenomena, but it fails for atomic-scale systems. It cannot correctly account for blackbody radiation, the photoelectric effect, atomic stability, discrete spectra, or the behavior of electrons in solids. Quantum mechanics provides the framework for understanding matter and radiation when action is comparable with Planck’s constant, h = 6.626×10^-34\,J·s.

For electrical engineers, quantum mechanics is not merely an abstract theory. Semiconductor diodes, transistors, lasers, LEDs, photodetectors, solar cells, electron microscopes, magnetic sensors, and quantum processors all depend on quantum principles.

Learning Objectives

After completing this chapter, students should be able to:

1. Explain wave–particle duality and the experimental evidence for quantization.
2. Use the Schrödinger equation for simple one-dimensional systems.
3. Interpret wavefunctions, probability density, operators, eigenvalues, and eigenstates.
4. Apply the uncertainty principle and explain quantum tunneling.
5. Relate atomic energy levels to line spectra and laser operation.
6. Describe orbital and intrinsic angular momentum, spin, and quantum statistics.
7. Connect quantum concepts with electronic and photonic technologies.

1. The Origins of Quantum Physics

1.1 Blackbody radiation and energy quantization

A blackbody is an ideal absorber and emitter of electromagnetic radiation. Classical theory predicted that the emitted energy would become infinite at short wavelengths, a result called the ultraviolet catastrophe. Max Planck resolved this difficulty by proposing that electromagnetic oscillators exchange energy in discrete packets:

Here, E_n is the allowed energy, n is an integer, h is Planck’s constant, and f is frequency. The energy of one quantum, or photon, is therefore:

where c is the speed of light and λ is wavelength. This equation applies to a free photon in vacuum or, with the appropriate phase velocity, in a medium.

1.2 The photoelectric effect

When light strikes a metal, electrons may be emitted. The maximum kinetic energy depends on frequency rather than intensity:

Here, φ is the metal’s work function. Emission occurs only when hf ≥ φ. Increasing intensity increases the number of emitted electrons, whereas increasing frequency increases their maximum kinetic energy. This is evidence that light transfers energy in localized quanta.

1.3 Matter waves

De Broglie proposed that particles also possess wave properties. The wavelength associated with a particle of momentum p is

For a nonrelativistic particle, p = mv, so λ = h/(mv). Electron diffraction confirms this prediction. Matter-wave effects are important when the wavelength is comparable with the dimensions of the system; they are negligible for ordinary macroscopic objects.

2. The Quantum State and Wavefunction

The state of a particle is represented by a wavefunction, commonly written ψ(x,t). The wavefunction itself is generally complex and is not directly observable. According to the Born interpretation, the probability density is

where ψ* is the complex conjugate. For a particle known to be somewhere in a region, the wavefunction must be normalized:

A physically acceptable wavefunction is finite, single-valued, continuous in ordinary finite potentials, and square-integrable. The probability of finding the particle between x₁ and x₂ is

Quantum mechanics predicts probabilities for measurement outcomes, not definite classical trajectories.

3. Schrödinger’s Equation

3.1 Time-dependent equation

For a nonrelativistic particle of mass m moving in a potential energy V(x,t), the one-dimensional time-dependent Schrödinger equation is

Here, i = √(−1) and ℏ = h/(2π). The equation is valid in nonrelativistic quantum mechanics and preserves normalization when the Hamiltonian is Hermitian.

The two terms on the right represent kinetic and potential energy. Their sum is represented by the Hamiltonian operator Ĥ.

3.2 Time-independent equation

If the potential is independent of time, stationary states can be written as ψ(x,t)=φ(x)e^−iEt/ℏ. Substitution gives

or explicitly,

This is an eigenvalue equation. The allowed energies E are eigenvalues, and the corresponding functions φ are energy eigenstates.

3.3 Infinite square well

Consider a particle confined between x=0 and x=L, with infinitely high potential outside. The boundary conditions are φ(0)=φ(L)=0. The normalized stationary states are

and the energy levels are

The ground-state energy is not zero. This zero-point energy follows from confinement and the uncertainty principle. The energy spacing increases with decreasing well width and decreasing particle mass, a fact central to quantum wells and nanostructures.

4. Operators, Observables, and Measurement

A measurable physical quantity is represented by a linear Hermitian operator. Important one-dimensional operators include

If an operator acts on an eigenstate according to

then a is the measured eigenvalue. A measurement of A in that state gives a with certainty.

For a general state, the expectation value is

assuming a normalized state and suitable boundary conditions. The expectation value is an ensemble average, not necessarily the result of every individual measurement.

Two operators may be compatible when their commutator vanishes:

Compatible observables can possess simultaneous eigenstates. Position and momentum do not commute:

This noncommutation leads directly to the uncertainty principle.

5. Uncertainty and Complementarity

The standard deviations of position and momentum obey

More generally, for two observables A and B,

The uncertainty principle is not a limitation caused by poor instruments. It is a property of quantum states and noncommuting observables. A narrow wave packet in position requires a broad range of momentum components.

Wave–particle duality is expressed through complementarity: an experiment arranged to determine a particle’s path may reduce the visibility of interference, while an experiment arranged to observe interference does not provide classical which-path information. Quantum objects are neither simply classical particles nor classical waves.

6. Quantum Tunneling

A particle can penetrate a classically forbidden region where its energy is less than the potential energy. For a rectangular barrier of height V₀ and width a, with E<V₀, the approximate transmission probability for a sufficiently thick barrier is

This approximation neglects prefactors and is most useful when κa ≫ 1. Exact transmission requires matching wavefunctions at the boundaries. Tunneling probability decreases exponentially with barrier width, particle mass, and barrier height.

Applications include tunnel diodes, scanning tunneling microscopes, flash memory, Josephson junctions, alpha decay, and field emission. In a scanning tunneling microscope, a sharp tip is brought near a conducting surface; the tunneling current changes strongly with separation, enabling atomic-scale imaging.

7. Atomic Structure and Spectra

7.1 Hydrogen atom

The hydrogen atom consists of a proton and an electron interacting through the Coulomb potential

where e is the elementary charge and ε₀ is the vacuum permittivity. Solving the three-dimensional Schrödinger equation gives the energy levels

The negative sign indicates a bound state. A transition between levels produces or absorbs a photon with

for emission from initial state i to lower final state f. Equivalently, the spectral wavelength obeys the Rydberg relation

for n_i>n_f, where R_H is the hydrogen Rydberg constant.

7.2 Quantum numbers

Atomic states are described by quantum numbers:

• n, principal quantum number, determines the major energy scale.
• l, orbital angular momentum quantum number, with 0≤l≤n−1.
• m_l, magnetic quantum number, with −l≤m_l≤l.
• m_s, spin projection, equal to +1/2 or −1/2 for an electron.

Orbital angular momentum has magnitude

and its z-component is L_z=m_lℏ. The quantum numbers explain atomic shells, subshells, and spectral selection rules.

7.3 Selection rules and spectra

Electric-dipole transitions commonly obey ∆l=±1 and ∆m_l=0,±1. These rules arise from the symmetry of the transition matrix element. Spectral lines therefore contain information about atomic structure, composition, temperature, and motion. Spectroscopy is used in lighting, plasma diagnostics, chemical analysis, astronomy, and optical communication components.

8. Spin and Quantum Statistics

Spin is intrinsic angular momentum; it is not literally a small sphere rotating in space. An electron has spin quantum number s=1/2 and magnetic moment associated with its spin. In a magnetic field, spin states can split in energy, producing phenomena such as the Stern–Gerlach effect, electron spin resonance, and magnetic resonance imaging.

The Pauli exclusion principle states that no two identical fermions can occupy the same complete quantum state. Electrons are fermions and therefore fill atomic orbitals according to this principle. Bosons, such as photons, can occupy the same state in unlimited numbers.

The two distributions are:

The Pauli principle explains the periodic table, chemical bonding, electron degeneracy, and the existence of semiconductor valence and conduction bands.

9. Quantum Physics in Electronic and Photonic Technologies

9.1 Semiconductors and band structure

In a crystal, closely spaced atomic levels form bands. The valence band is usually occupied, while the conduction band can support mobile electrons. The energy gap E_g determines optical and electrical behavior. A photon can promote an electron across a direct band gap when

In an LED, electron–hole recombination emits light with approximate wavelength

when most of the energy is released optically. In a photodiode or solar cell, absorbed photons generate electron–hole pairs if their energy is at least the band gap.

9.2 Lasers

A laser requires stimulated emission, population inversion, and optical feedback. The photon energy equals the difference between two allowed states. Coherent output results because stimulated photons match the frequency, phase, polarization, and direction of the stimulating field. Semiconductor lasers use engineered quantum transitions in a gain region and are fundamental to fiber-optic communication.

9.3 Quantum wells and confinement

When carriers are confined to nanometer-scale layers, allowed energies become quantized. Changing well width changes the energy spacing and therefore the emission wavelength. Quantum-well LEDs and lasers exploit this design freedom to improve efficiency and wavelength control.

9.4 Quantum information

A classical bit is either 0 or 1, whereas a qubit can be in a superposition of basis states. Measurement produces probabilistic outcomes, and entanglement creates correlations that have no classical equivalent. Practical quantum computers remain limited by decoherence, gate errors, fabrication variability, and the need for error correction. These technologies complement rather than universally replace conventional computers.

10. Worked Examples

Example 1: Photon energy

Find the energy of a photon of wavelength 600 nm.

Using E=hc/λ, with h=6.626×10^-34\,J·s, c=3.00×10⁸\,m/s, and λ=600×10^-9\,m,

E=(6.626×10^-34)(3.00×10⁸)/(600×10^-9)=3.31×10^-19\,J.

In electronvolts, using 1\,eV=1.602×10^-19\,J, the energy is approximately 2.07\,eV.

Example 2: Ground-state energy in a nanowire model

An electron is confined in a one-dimensional infinite well of width 1.0 nm. Find its ground-state energy.

Taking m=9.11×10^-31\,kg and L=1.0×10^-9\,m,

E₁≈6.02×10^-20\,J≈0.376\,eV.

The result demonstrates why quantum confinement becomes significant at nanometer dimensions.

11. Applied Case: Quantum-Well Laser for Fiber Communication

Background

An illustrative engineering case concerns a semiconductor laser intended for a 1.55 μm optical-fiber link, a wavelength region associated with low fiber attenuation.

Problem

A conventional bulk semiconductor active region provides inadequate control over carrier confinement and optical gain. The design team must obtain efficient stimulated emission near the target wavelength while limiting threshold current.

Analysis

The photon energy required is E=hc/λ≈0.80\,eV. The semiconductor composition is selected so that its effective transition energy is near this value. Thin quantum wells are placed between higher-band-gap barrier layers. The barriers confine electrons and holes, increasing their overlap and improving radiative recombination.

Approach and solution

The device is designed with separate optical and carrier confinement, ohmic contacts, a cleaved or grating-defined resonant cavity, and temperature control. The quantum-well thickness and alloy composition are adjusted using band-structure models. Testing measures threshold current, slope efficiency, spectral linewidth, modulation bandwidth, and temperature sensitivity.

Outcome

Compared with an equivalent bulk active region, the quantum-well structure can provide lower threshold current and improved wavelength engineering, although fabrication tolerances and thermal effects remain important.

Learning points

1. Energy quantization is directly used to design emission wavelength.
2. Carrier confinement improves the probability of radiative recombination.
3. Quantum theory must be combined with electromagnetic cavity design, heat transfer, materials science, and manufacturing control.
4. Device specifications must be verified experimentally because ideal models omit defects, scattering, leakage, and temperature dependence.

12. Limitations, Safety, and Professional Practice

Quantum models are idealizations. The infinite well, isolated hydrogen atom, and simple barrier neglect lattice vibrations, many-body interactions, defects, electromagnetic loss, and fabrication variation. Engineers should state assumptions, check dimensional consistency, and use numerical or experimental validation for real devices.

Quantum technologies also raise governance issues. Secure handling of sensitive quantum-encryption data, responsible export control, accurate claims about quantum advantage, and protection of intellectual property are professional responsibilities. Quantum devices may require cryogenic systems, vacuum equipment, lasers, high voltages, or strong magnetic fields; appropriate laboratory safety procedures are essential.

Contemporary Developments

Current engineering research includes two-dimensional materials, quantum-dot displays, topological materials, single-photon sources, superconducting and semiconductor qubits, quantum sensors, and integrated photonics. Artificial intelligence is increasingly used to optimize materials, calibrate quantum devices, identify spectral features, and control experiments. AI predictions must nevertheless be checked against physical laws, uncertainty estimates, reproducibility requirements, and experimental evidence.

Chapter Summary

Quantum mechanics describes matter and radiation through quantized states, wavefunctions, operators, and probabilities. Planck’s hypothesis and the photoelectric effect establish photon energy E=hf, while de Broglie’s relation λ=h/p assigns wave behavior to matter. The Schrödinger equation determines allowed states and their time evolution. Operators represent observables; eigenvalues are possible measurement results, and noncommuting operators produce uncertainty.

Confinement produces discrete energies, while tunneling permits transmission through classically forbidden barriers. Hydrogen spectra arise from transitions between quantized levels. Quantum numbers, spin, and the Pauli principle explain atomic structure and the distinction between fermions and bosons. These principles underpin semiconductor bands, LEDs, lasers, photodiodes, quantum wells, and emerging quantum information systems.

Key Terms

Wave–particle duality: The complementary wave and particle behavior of quantum objects.

Photon: A quantum of electromagnetic radiation.

Work function: Minimum energy required to remove an electron from a material.

De Broglie wavelength: Matter wavelength given by λ=h/p.

Wavefunction: Mathematical representation of a quantum state.

Probability density: Quantity |ψ|² giving probability per unit position or volume.

Normalization: Condition that total probability equals one.

Hamiltonian: Operator representing total energy.

Eigenstate: State that returns a scalar multiple when acted on by an operator.

Eigenvalue: Allowed measurement value associated with an eigenstate.

Expectation value: Statistical average predicted for repeated measurements.

Commutator: Operator difference [A,B]=AB−BA.

Uncertainty principle: Constraint on simultaneous spreads of noncommuting observables.

Quantum tunneling: Transmission through a classically forbidden potential region.

Quantum number: Integer or half-integer specifying a quantum state.

Spin: Intrinsic angular momentum of a particle.

Fermion: Particle obeying Fermi–Dirac statistics.

Boson: Particle obeying Bose–Einstein statistics.

Band gap: Energy separation between relevant valence and conduction bands.

Stimulated emission: Radiation emission induced by an incident photon.

Review Questions

Short Answer Questions

1. What experimental observation supports the particle nature of light?
2. What does |ψ|² represent?
3. State the one-dimensional time-independent Schrödinger equation.
4. Why can electrons not all occupy the same atomic state?
5. What is quantum tunneling?

Long Answer Questions

1. Explain wave–particle duality using blackbody radiation, the photoelectric effect, and electron diffraction.
2. Describe operators, eigenstates, eigenvalues, and expectation values with suitable equations.
3. Derive the energy levels of an infinite one-dimensional square well and interpret the ground-state energy.
4. Explain the origin of hydrogen spectral lines and the role of quantum numbers.
5. Discuss how quantum mechanics enables LEDs, photodiodes, and semiconductor lasers.

Analytical / Application Questions

1. Calculate the wavelength of a 3.0 eV photon and identify a likely region of the electromagnetic spectrum.
2. Compare tunneling probabilities for two barriers of equal height but widths 1 nm and 2 nm using the exponential approximation. Explain why the ratio is highly sensitive to width.
3. A quantum-well laser is emitting at 1.3 μm. Estimate the photon energy in electronvolts and discuss two practical reasons why the actual semiconductor band structure may not equal this simple estimate.

Multiple Choice Questions with Answers

Question 1

Which equation gives the energy of a photon?

A. E=hf

B. E=mc²

C. E=pV

D. E=½mv²

Correct Answer: A. E=hf

Question 2

What physical quantity is represented by |ψ(x,t)|²?

A. Potential energy

B. Probability density

C. Momentum density only

D. Electric field intensity

Correct Answer: B. Probability density

Question 3

For an infinite square well, how does the ground-state energy change when the well width is doubled?

A. It doubles

B. It becomes four times larger

C. It becomes one-quarter as large

D. It is unchanged

Correct Answer: C. It becomes one-quarter as large

Question 4

Which commutator is associated with the position–momentum uncertainty relation?

A. [x̂,p̂]=0

B. [x̂,p̂]=iℏ

C. [x̂,p̂]=ℏ²

D. [x̂,p̂]=−1

Correct Answer: B. [x̂,p̂]=iℏ

Question 5

An electron can tunnel through a barrier even when its energy is:

A. Greater than the barrier height only

B. Equal to zero only

C. Less than the barrier height

D. Relativistic only

Correct Answer: C. Less than the barrier height

Question 6

Which principle prevents two electrons from occupying the same complete quantum state?

A. Correspondence principle

B. Pauli exclusion principle

C. Equivalence principle

D. Superposition principle

Correct Answer: B. Pauli exclusion principle

Question 7

What process produces light in an LED?

A. Electron–hole recombination

B. Nuclear fission

C. Elastic collision with a wall

D. Gravitational acceleration

Correct Answer: A. Electron–hole recombination

Question 8

For hydrogen, the energy of the n=2 level is approximately:

A. −13.6 eV

B. −6.8 eV

C. −3.4 eV

D. +3.4 eV

Correct Answer: C. −3.4 eV

Question 9

Which particles obey Bose–Einstein statistics?

A. Electrons

B. Neutrons

C. Photons

D. Protons only

Correct Answer: C. Photons

Question 10

The approximate wavelength of a photon from a 1.55 μm optical-fiber laser is determined primarily by:

A. The photon’s rest mass

B. The transition energy of the active material

C. The electron charge alone

D. The gravitational field

Correct Answer: B. The transition energy of the active material