Quantum Mechanics Intermediate

The central object in quantum mechanics is the wave function ψ (psi). Its square gives the probability density of finding a particle at a location:

P(x) = |ψ(x)|²

Because the particle must be somewhere, the wave function must be normalised: ∫|ψ|² dx = 1.

Properties of ψ

• ψ is generally complex-valued.
• It obeys the Schrödinger equation.
• |ψ|² gives probabilities, not definite outcomes.
• Upon measurement, ψ collapses to a state consistent with the result obtained.

For a particle in a time-independent potential V(x), energy eigenstates satisfy the TISE:

−(ℏ²/2m) d²ψ/dx² + V(x)ψ = Eψ

Infinite square well

V = 0 inside (0 ≤ x ≤ L), V = ∞ elsewhere. Allowed energies:

E_n = n²π²ℏ²/(2mL²),   n = 1, 2, 3, …

Only discrete energies are allowed — quantisation emerges directly from boundary conditions.

In quantum mechanics, every measurable quantity is represented by a linear operator acting on the wave function.

Key operators

• Position: x̂ψ = xψ
• Momentum: p̂ψ = −iℏ dψ/dx
• Hamiltonian: Ĥψ = −(ℏ²/2m) d²ψ/dx² + Vψ

Expectation values

⟨Q⟩ = ∫ψ* Q̂ψ dx

Eigenvalues

If ψ is an eigenstate of Q̂ with eigenvalue q, measurement always returns q with certainty.

Electrons carry an intrinsic angular momentum called spin — a purely quantum property with no classical analogue.

Spin-½

When you measure the spin along any axis, you get exactly one of two values:

m_s = +½ ("spin up", ↑) or m_s = −½ ("spin down", ↓)

Pauli matrices

Spin is represented by 2×2 matrices (σ_x, σ_y, σ_z). The spin states are 2-component column vectors called spinors.

Why spin matters

• Explains the Pauli exclusion principle.
• Underlying physics of ferromagnetism.
• The qubit in quantum computing is often a spin-½ system.

In 1922 Stern and Gerlach sent a beam of silver atoms through a non-uniform magnetic field.

What they expected (classical)

A continuous smear on the detector screen.

What they got (quantum)

Two discrete spots — spin up and spin down. Direct experimental proof that angular momentum is quantised.

Sequential measurements

If you filter spin-up atoms and then measure spin along a perpendicular axis, you again get two spots (±½ along the new axis), each with 50% probability. Measuring the perpendicular component randomises the original — a vivid demonstration of non-commuting observables.

The uncertainty principle has a precise algebraic origin. Two operators do not commute if:

[Â, B̂] = ÂB̂ − B̂Â ≠ 0

For position and momentum: [x̂, p̂] = iℏ. The Robertson uncertainty relation gives:

σ_A σ_B ≥ ½ |[Â, B̂]|

For x and p: Δx Δp ≥ ℏ/2.

Compatible observables

If [Â, B̂] = 0, then A and B can be simultaneously measured with arbitrary precision and share common eigenstates.

The hydrogen atom — one proton plus one electron — is the simplest real atom and the one quantum mechanics solves exactly.

Energy levels

E_n = −13.6 eV / n²,   n = 1, 2, 3, …

The orbitals

Each state is labelled (n, ℓ, m_ℓ). Wave functions are products of radial functions R_nℓ(r) and spherical harmonics Y_ℓ^m(θ, φ). The s, p, d, f labels correspond to ℓ = 0, 1, 2, 3.

Spectral series

Lyman series (to n=1): ultraviolet. Balmer (to n=2): visible. Paschen (to n=3): infrared.

Selection rules state which transitions between energy levels are strongly allowed.

Electric dipole rules for hydrogen

• Δn: any value
• Δℓ = ±1 (must change by exactly one)
• Δm_ℓ = 0, ±1

Why?

Photons carry angular momentum ℏ. Conservation of angular momentum requires Δℓ = ±1.

Forbidden transitions

Transitions that violate electric-dipole rules can still occur via higher-order processes but are much slower. The 2s → 1s transition takes ~0.12 seconds instead of ~1 ns.

Entanglement is a correlation between two or more quantum systems that has no classical equivalent.

A simple entangled state

|ψ⟩ = (1/√2)(|↑↓⟩ − |↓↑⟩)

Neither electron has a definite spin. If you measure electron A and find spin-up, electron B is instantly spin-down — no matter how far apart they are.

What entanglement is NOT

• It does not transmit information faster than light.
• It is not a hidden-variable effect (Bell's theorem rules this out).

Applications

• Quantum key distribution
• Quantum teleportation of states
• Bell tests of quantum mechanics

Quantum mechanics makes extraordinarily precise predictions, but physicists disagree about what it means.

Copenhagen interpretation

The wave function is a calculation tool. When measured, it collapses. Questions about what happens before measurement are meaningless.

Many-worlds (Everett, 1957)

Every measurement causes the universe to branch. All outcomes occur in different branches — no collapse.

Pilot-wave (de Broglie–Bohm)

Particles have definite positions, guided by a real pilot wave. Reproduces all quantum predictions deterministically.

Relational QM

A system's state is always relative to another system (the observer). There is no absolute, observer-independent state.

All interpretations give identical experimental predictions.