Topic 5

Quantum Mechanics

The rules of the subatomic world, where particles are waves, uncertainty is fundamental, and observation changes reality. Quantum mechanics underlies chemistry, transistors, lasers, and most modern technology — even as it defies everyday intuition.

  1. The ultraviolet catastrophe & Planck
  2. Wave–particle duality
  3. The Schrödinger equation
  4. The uncertainty principle
  5. Quantum tunneling
  6. The measurement problem & interpretations

1. The Crisis That Started It All

By 1900, classical physics — Maxwell's equations and thermodynamics — was triumphant. But it faced a fatal flaw: blackbody radiation. When physicists tried to calculate the energy radiated by a hot object, classical theory predicted infinite energy at short wavelengths — the ultraviolet catastrophe. In 1900, Max Planck solved it by proposing a desperate fix: energy is emitted in discrete chunks.

$$\boxed{E = nh\nu = n\hbar\omega}$$

Energy of frequency $\nu$ comes in multiples of $h\nu$ — a quantum (plural: quanta). Planck called this a mathematical trick, unaware it was the birth of a revolution. A year later, Einstein applied the same idea to light itself, explaining the photoelectric effect:

$$hf = W_0 + K_{\text{max}}$$

Light of frequency $f$ ejects electrons from metal, but only if $hf$ exceeds the work function $W_0$ — no matter how intense the light. Brighter light gives more electrons, but each with the same maximum energy. Light is made of particles — photons.

Constants of quantum mechanics. Planck's constant $h = 6.626\times10^{-34}$ J·s is the quantum of action. The reduced constant $\hbar = h/2\pi = 1.055\times10^{-34}$ J·s appears in almost every quantum formula.

2. Wave–Particle Duality

If light is a particle, then surely electrons are particles too — right? The Davisson–Germer experiment (1927) fired electrons at a nickel crystal and found they diffracted, producing interference patterns — unmistakable wave behavior. de Broglie proposed that everything has a wavelength:

$$\boxed{\lambda_{\text{dB}} = \frac{h}{p}}$$
A profound symmetry of nature. Photons (light) are particle-like; electrons are wave-like. But both can behave as either. The question isn't "is it a wave or a particle?" — it's neither. Quantum entities are something deeper, revealed to us only through the approximations our classical minds and instruments impose.

For a baseball, $\lambda_{\text{dB}}$ is absurdly small ($10^{-34}$ m) — that's why we never see baseballs diffracting. But for an electron with 1 eV of energy, $\lambda \approx 1.2$ nm — comparable to atomic spacings, making electron microscopes possible.

3. The Schrödinger Equation

In 1926, Erwin Schrödinger wrote the equation that governs how quantum states evolve. It is the quantum version of Newton's $F=ma$ — a differential equation for the wavefunction $\Psi(x,t)$:

$$\boxed{i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right)\Psi}$$

The wave function $\Psi$ is a probability amplitude. The Born rule says $|\Psi|^2$ gives the probability density of finding the particle at a given location. The square modulus $|\Psi|^2$ is always real and positive — probabilities add up to 1.

$$\int |\Psi(x)|^2\,dx = 1$$
The particle in a box. Confine a particle to a region $[0,L]$ with infinite walls ($V=\infty$ outside). Inside, $V=0$ and the solutions are standing waves:
$\Psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$ with energies $E_n = n^2 h^2/(8mL^2)$, $n=1,2,3,\ldots$
The energy is quantized — only specific values are allowed. This explains why atoms have discrete energy levels and emit light only at specific frequencies.

4. The Uncertainty Principle

Werner Heisenberg showed that quantum mechanics has a fundamental limit on what we can know simultaneously. The more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa:

$$\boxed{\Delta x\,\Delta p \ge \frac{\hbar}{2}}$$
Uncertainty is not measurement error. This is not a limitation of our instruments. It's a statement about the quantum state itself — position and momentum are fundamentally linked quantities. A particle doesn't have a definite position and momentum simultaneously; the wavefunction encodes probabilities for each.

Another uncertainty relation links energy and time: $\Delta E\,\Delta t \ge \hbar/2$. This lets particles "borrow" energy from the vacuum for fleeting moments — the basis of virtual particles and the Casimir effect.

5. Quantum Tunneling

Classically, if you roll a ball at a hill and it doesn't have enough energy to crest the top, it rolls back. In quantum mechanics, the wavefunction doesn't drop to zero instantly — it evanesces into the barrier, giving a small probability the particle appears on the other side. This is tunneling — a purely quantum phenomenon with no classical analog.

$$T \approx e^{-2\kappa a},\qquad \kappa = \sqrt{\frac{2m(V - E)}{\hbar^2}}$$

The transmission probability decreases exponentially with barrier width $a$ and height $(V-E)$. Tunneling is essential to stellar fusion (the Sun's energy!), scanning tunneling microscopes, and radioactive decay.

Quantum tunneling. A wave packet (blue) approaches a barrier (orange). Classically the particle would bounce back, but quantum mechanics gives it a small chance to appear on the other side. Adjust the particle energy $E$ and barrier height $V$ to see how the transmission probability $T$ changes.

6. The Measurement Problem

If a quantum system can be in multiple states at once — the famous superposition — what happens when we measure it? Schrödinger's cat is simultaneously alive and dead (a poison vial triggered by a quantum random event), until the box is opened and the cat is found in one definite state. The collapse of the wavefunction is mysterious and has spawned decades of debate.

cat vial closed box: |alive⟩ + |dead⟩ open lid
Schrödinger's cat: the quintessential paradox showing how quantum superpositions clash with macroscopic reality.

Different interpretations offer different answers, none yet testable: the Copenhagen view (wavefunction collapses when observed), many-worlds (all branches are real, you split into each outcome), QBism, and others. What's certain is that quantum mechanics is the most precisely tested theory in science — every prediction checked so far matches experiment to extraordinary precision.

Key Takeaways

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