FAD1022 L45 — Introduction to Quantum Mechanics

This lecture introduces the formal mathematical framework of quantum mechanics, building on the wave-particle duality foundations from previous lectures. Topics span from photon momentum and the Compton effect through de Broglie's matter waves and the Heisenberg uncertainty principle, to the time-independent Schrödinger equation and the particle in a 1D infinite square well.

Lecture File

  • Lecture 45 - Photon Momentum, Compton Effect, de-Broglie waves and Heisenberg Uncertainty Principle.pdf (28 slides)
  • Lecturer: Nurul Izzati (NIA)

1. Wave-Particle Duality Recap

Wave-particle duality is the cornerstone of quantum mechanics: all matter and radiation exhibit both wave-like and particle-like behaviour depending on the experimental context.

De Broglie Wavelength

Louis de Broglie (1924) proposed that any particle with momentum $p$ has an associated wavelength:

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$

Where:

  • $h = 6.626 \times 10^{-34}$ J·s (Planck's constant)
  • $p$ = momentum of the particle
  • $m$ = mass, $v$ = velocity

Key Insight: For macroscopic objects (e.g., a 0.1 kg baseball at 40 m/s), $\lambda \approx 1.7 \times 10^{-34}$ m — far too small to detect. For electrons ($m_e = 9.11 \times 10^{-31}$ kg) moving at $10^6$ m/s, $\lambda \approx 7.3 \times 10^{-10}$ m — comparable to atomic dimensions.

Double-Slit Experiment

The double-slit experiment provides direct evidence of wave-particle duality:

  • Classical expectation: Particles produce two bands (one per slit)
  • Actual result: Even when electrons pass through one at a time, an interference pattern builds up over many detections
  • Interpretation: Each particle interferes with itself, behaving as a wave that passes through both slits simultaneously, then collapsing to a point-like detection

Complementarity Principle (Niels Bohr)

The wave and particle aspects of a quantum entity are complementary — both are needed for a complete description, but they cannot be observed simultaneously in a single experiment.

A quantum object exhibits wave behaviour when it is not being "watched" (interference) and particle behaviour when a which-path measurement is made.

2. Wave Functions and Probability Density

The Wave Function $\Psi(x, t)$

In quantum mechanics, the state of a particle is described by a wave function $\Psi(x, t)$, a complex-valued function of position and time. The wave function itself has no direct physical interpretation — it is a mathematical tool.

Probability Density (Born Interpretation)

Max Born (1926) proposed that $|\Psi(x, t)|^2$ gives the probability density of finding the particle at position $x$ at time $t$:

$$P(x, t),dx = |\Psi(x, t)|^2,dx = \Psi^*(x, t)\Psi(x, t),dx$$

Where $\Psi^*$ is the complex conjugate of $\Psi$.

Normalization Condition

Since the particle must be found somewhere in space, the total probability over all space must equal 1:

$$\int_{-\infty}^{\infty} |\Psi(x, t)|^2,dx = 1$$

A wave function satisfying this condition is said to be normalized. This requirement constrains the mathematical form of $\Psi$ and leads directly to quantized energy levels in bound systems.

Properties of Valid Wave Functions

A physically admissible wave function must be:

  1. Single-valued — one value at each point
  2. Continuous — no jumps or breaks
  3. Finite — cannot blow up to infinity
  4. Square-integrable — $\int_{-\infty}^{\infty} |\Psi|^2,dx$ must be finite

3. Heisenberg Uncertainty Principle

Werner Heisenberg (1927) showed that certain pairs of physical quantities cannot be simultaneously known with arbitrary precision.

Position-Momentum Uncertainty

$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$

Where:

  • $\Delta x$ = uncertainty in position
  • $\Delta p$ = uncertainty in momentum
  • $\hbar = h/2\pi = 1.0546 \times 10^{-34}$ J·s (reduced Planck constant)

Energy-Time Uncertainty

$$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$

Where:

  • $\Delta E$ = uncertainty in energy
  • $\Delta t$ = uncertainty in time / lifetime of a state

Physical Meaning

The uncertainty principle is not a measurement limitation — it is a fundamental property of nature:

  • If you confine a particle to a very small region (small $\Delta x$), its momentum becomes highly uncertain (large $\Delta p$)
  • Short-lived quantum states have inherently uncertain energies
  • This is why electrons in atoms do not spiral into the nucleus: confining an electron to the nuclear volume would give it enormous momentum uncertainty and thus enormous kinetic energy

Example — Electron Confinement

For an electron confined to $\Delta x = 10^{-10}$ m (atomic size): $$\Delta p \geq \frac{\hbar}{2\Delta x} = \frac{1.0546 \times 10^{-34}}{2 \times 10^{-10}} \approx 5.27 \times 10^{-25}\text{ kg·m/s}$$

$$\Delta v \geq \frac{\Delta p}{m_e} \approx 5.8 \times 10^5\text{ m/s}$$

This minimum velocity is comparable to the orbital velocity in hydrogen, confirming that quantum uncertainty governs atomic structure.

4. Time-Independent Schrödinger Equation (TISE)

The Wave Equation for Matter

Erwin Schrödinger (1926) proposed a wave equation governing the evolution of the wave function. The full time-dependent Schrödinger equation is:

$$i\hbar\frac{\partial\Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t)$$

Separation of Variables

For a time-independent potential $V(x)$, we can separate the wave function:

$$\Psi(x, t) = \psi(x) \cdot e^{-iEt/\hbar}$$

Substituting into the time-dependent equation yields the Time-Independent Schrödinger Equation (TISE):

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$$

Hamiltonian Form

The TISE can be written compactly using the Hamiltonian operator $\hat{H}$:

$$\hat{H}\psi(x) = E\psi(x)$$

Where $\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$

This is an eigenvalue equation: $\psi(x)$ is the eigenfunction and $E$ is the eigenvalue (the allowed energy). Only certain values of $E$ yield physically acceptable solutions — this is the origin of energy quantization.

Terms in the TISE

Term Symbol Meaning
Kinetic energy operator $-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$ From de Broglie relation $p = \hbar k$
Potential energy $V(x)$ Depends on the physical situation
Energy eigenvalue $E$ Allowed energy of the state
Wave function $\psi(x)$ Spatial part of the full wave function

Additional Examples from Lecture Slides

Example — De Broglie Wavelength of an Electron

Calculate the de Broglie wavelength of an electron moving at $2.0 \times 10^6$ m/s.

$$\lambda = \frac{h}{mv} = \frac{6.63 \times 10^{-34}}{(9.11 \times 10^{-31})(2.0 \times 10^6)} = \boxed{3.63 \times 10^{-10} \text{ m}}$$

Example — De Broglie Wavelength from Kinetic Energy

Calculate the de Broglie wavelength of an electron with kinetic energy of 150 eV.

Step 1: Convert eV to joules $$KE = 150 \text{ eV} = 150 \times 1.602 \times 10^{-19} = 2.403 \times 10^{-17} \text{ J}$$

Step 2: Use formula $\lambda = \frac{h}{\sqrt{2mKE}}$

$$\lambda = \frac{6.626 \times 10^{-34}}{\sqrt{2(9.11 \times 10^{-31})(2.403 \times 10^{-17})}} = \frac{6.626 \times 10^{-34}}{6.62 \times 10^{-24}} \approx \boxed{1.00 \times 10^{-10} \text{ m} = 0.1 \text{ nm}}$$

This is within the atomic scale — explaining electron diffraction.

Example — Heisenberg Uncertainty Principle

Case 1 (Macroscopic): A pitcher throws a 0.1-kg baseball at 40 m/s. Momentum is $0.1 \times 40 = 4$ kg·m/s. Suppose the momentum is measured to an accuracy of 1%, i.e., $\Delta p = 0.01p = 4 \times 10^{-2}$ kg·m/s.

$$\Delta x \geq \frac{h}{4\pi \Delta p} = 1.3 \times 10^{-33} \text{ m}$$

No wonder one does not observe the effects of the uncertainty principle in everyday life!

Case 2 (Microscopic): Same situation, but baseball replaced by an electron ($m = 9.11 \times 10^{-31}$ kg) moving at $4 \times 10^6$ m/s.

$$p = 3.6 \times 10^{-24} \text{ kg·m/s}, \quad \Delta p = 3.6 \times 10^{-26} \text{ kg·m/s}$$

$$\Delta x \geq \frac{h}{4\pi \Delta p} = \boxed{1.4 \times 10^{-4} \text{ m}}$$

This is a macroscopic uncertainty — 0.14 mm — showing that quantum effects are significant for electrons.

5. Particle in a 1D Box (Infinite Square Well)

The simplest non-trivial quantum system: a particle of mass $m$ confined to a one-dimensional region $0 < x < L$ by infinitely high potential walls.

Potential Definition

$$V(x) = \begin{cases} 0 & \text{for } 0 < x < L \ \infty & \text{for } x \leq 0 \text{ or } x \geq L \end{cases}$$

Boundary Conditions

Since the potential is infinite outside the box, the wave function must be zero at the walls:

$$\psi(0) = 0, \quad \psi(L) = 0$$

Solving the TISE Inside the Box

For $0 < x < L$, $V = 0$, so the TISE becomes:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$$

The general solution is $\psi(x) = A\sin(kx) + B\cos(kx)$ where $k = \sqrt{2mE}/\hbar$.

Applying $\psi(0) = 0$ forces $B = 0$, so $\psi(x) = A\sin(kx)$.

Applying $\psi(L) = 0$ gives $\sin(kL) = 0$, which requires $kL = n\pi$ for $n = 1, 2, 3, \ldots$

Energy Quantization

From $k_n = n\pi/L$ and $k = \sqrt{2mE}/\hbar$:

$$E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}$$

Where $n = 1, 2, 3, \ldots$ is the quantum number.

Key properties:

  • Energy is quantized — only discrete values are allowed
  • $E_n \propto n^2$ — energy spacing increases with $n$
  • Zero-point energy: $E_1 = h^2/(8mL^2) > 0$ — the particle can never be at rest ($n=0$ is forbidden by the uncertainty principle)
  • As $L$ increases, energy levels become closer together, approaching a classical continuum

Normalized Wave Functions

Applying the normalization condition $\int_0^L |\psi_n(x)|^2,dx = 1$:

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right), \quad n = 1, 2, 3, \ldots$$

Nodes

Each wave function $\psi_n(x)$ has $(n-1)$ nodes — points inside the box where $\psi_n = 0$ (excluding the walls). The probability of finding the particle at a node is zero.

$n$ Nodes inside box Shape description
1 0 Half sine wave (ground state)
2 1 Full sine wave (first excited state)
3 2 1.5 sine waves
4 3 2 full sine waves

Probability Distributions

The probability density for state $n$ is:

$$|\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)$$

For the ground state ($n=1$), the particle is most likely found near the centre of the box. For higher $n$, the probability distribution shows multiple peaks, approaching the classical uniform distribution as $n \to \infty$ (Bohr's correspondence principle).

Worked Example 1 — Electron in a Box

An electron is confined in a 1D box of length $L = 1.0 \times 10^{-10}$ m (approximately atomic size). Find: (a) The ground state energy $E_1$ (b) The wavelength of a photon emitted when the electron transitions from $n=3$ to $n=1$

Solution (a): $$E_1 = \frac{h^2}{8mL^2} = \frac{(6.626 \times 10^{-34})^2}{8(9.11 \times 10^{-31})(1.0 \times 10^{-10})^2}$$ $$E_1 = \frac{4.39 \times 10^{-67}}{7.29 \times 10^{-50}} = 6.02 \times 10^{-18}\text{ J} = 37.6\text{ eV}$$

Solution (b): $$E_3 = 9E_1 = 9(6.02 \times 10^{-18}) = 5.42 \times 10^{-17}\text{ J}$$ $$\Delta E = E_3 - E_1 = 5.42 \times 10^{-17} - 6.02 \times 10^{-18} = 4.82 \times 10^{-17}\text{ J}$$ $$\lambda = \frac{hc}{\Delta E} = \frac{(6.626 \times 10^{-34})(3.0 \times 10^8)}{4.82 \times 10^{-17}} = 4.12 \times 10^{-9}\text{ m} = 4.12\text{ nm}$$

This wavelength is in the X-ray region of the electromagnetic spectrum.

Worked Example 2 — Proton in a Box

A proton ($m_p = 1.67 \times 10^{-27}$ kg) is confined in a 1D box of length $L = 1.0 \times 10^{-15}$ m (nuclear dimensions). Find the ground state energy.

$$E_1 = \frac{h^2}{8mL^2} = \frac{(6.626 \times 10^{-34})^2}{8(1.67 \times 10^{-27})(1.0 \times 10^{-15})^2}$$ $$E_1 = \frac{4.39 \times 10^{-67}}{1.34 \times 10^{-56}} = 3.28 \times 10^{-11}\text{ J} \approx 205\text{ MeV}$$

This enormous energy scale (MeV range) is why nuclear processes involve such high energies.

6. Energy Quantization — Classical vs Quantum

Why is Energy Discrete at Quantum Scales?

The TISE is a differential equation with boundary conditions, similar to a standing wave on a string. Only specific wavelengths — and therefore specific energies — can satisfy the boundary conditions. This is the mathematical origin of quantization.

Contrast Between Classical and Quantum

Property Classical Physics Quantum Mechanics
Energy Continuous (any value) Discrete (quantized) at small scales
Position Definite trajectory $x(t)$ Probability distribution $
Momentum Definite $p = mv$ Uncertain; $\Delta x\Delta p \geq \hbar/2$
State Determined by $x$ and $p$ Determined by wave function $\psi$
Measurement Passive observation Affects the system (wave function collapse)
Causality Deterministic Probabilistic

When Does Classical Physics Recover?

As the quantum number $n$ becomes large, the energy spacing relative to the energy becomes small:

$$\frac{E_{n+1} - E_n}{E_n} = \frac{(n+1)^2 - n^2}{n^2} = \frac{2n+1}{n^2} \approx \frac{2}{n} \to 0 \text{ as } n \to \infty$$

This is Bohr's Correspondence Principle: quantum mechanics approaches classical mechanics in the limit of large quantum numbers.

Zero-Point Energy

Even in the lowest energy state ($n=1$), the particle has non-zero energy ($E_1 > 0$). This is a direct consequence of the uncertainty principle:

  • Confining the particle to the box gives it a minimum $\Delta x \approx L$
  • The uncertainty principle then requires a minimum $\Delta p \approx \hbar/L$
  • This minimum momentum implies a minimum kinetic energy

Classically, a particle could be at rest ($E=0$) inside the box. Quantum mechanically, this is impossible.

Key Equations Summary

Equation Description
$\lambda = h/p$ De Broglie wavelength
$P(x) = \Psi(x,t)
$\int_{-\infty}^{\infty} \Psi
$\Delta x \cdot \Delta p \geq \hbar/2$ Heisenberg uncertainty (position-momentum)
$\Delta E \cdot \Delta t \geq \hbar/2$ Heisenberg uncertainty (energy-time)
$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$ Time-Independent Schrödinger Equation
$\hat{H}\psi = E\psi$ TISE in operator form
$E_n = \frac{n^2 h^2}{8mL^2}$ Particle in 1D box energy
$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$ Particle in 1D box wave function

Related Concepts

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Lecturer

Nurul Izzati (NIA) — PASUM Physics Lecturer

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