Quantum Mechanics Advanced

Paul Dirac introduced a concise notation for quantum mechanics that makes the algebra transparent and basis-independent.

Ket, bra, and bracket

• A ket |ψ⟩ represents a quantum state (a vector in Hilbert space).
• A bra ⟨φ| is the dual (conjugate transpose of a ket).
• The inner product ⟨φ|ψ⟩ is a complex number.

Hilbert space

The space of physically allowed quantum states is an (often infinite-dimensional) complex vector space with an inner product. Orthonormal bases {|n⟩} satisfy ⟨n|m⟩ = δ_nm and Σ|n⟩⟨n| = 1.

Operators in Dirac notation

Matrix element of  in a basis: ⟨m|Â|n⟩. Expectation value: ⟨Â⟩ = ⟨ψ|Â|ψ⟩.

The infinite and finite square wells are the canonical 1D potentials.

Infinite square well

ψ_n(x) = √(2/L) sin(nπx/L),   E_n = n²π²ℏ²/(2mL²)

Finite square well

V = −V₀ inside, V = 0 outside. Inside: oscillatory wave function. Outside: exponentially decaying tails (tunnelling). Boundary conditions give a transcendental equation for the allowed energies — only finitely many bound states exist.

Penetration depth

The decay constant κ = √(2m|E|)/ℏ characterises how far the wave function penetrates the classically forbidden region.

The harmonic oscillator (V = ½mω²x²) appears throughout physics — molecular vibrations, phonons, the electromagnetic field. It is exactly solvable via algebraic methods.

Ladder operators

Define â = √(mω/2ℏ)(x̂ + ip̂/mω) and ↠(raising). Then Ĥ = ℏω(â†â + ½).

â†|n⟩ = √(n+1)|n+1⟩,   â|n⟩ = √n|n−1⟩

Energy levels

E_n = (n + ½)ℏω,   n = 0, 1, 2, …

The ground state has zero-point energy ½ℏω — a consequence of the uncertainty principle.

Coherent states

Eigenstates of â are minimum-uncertainty wave packets that most closely resemble classical oscillation. Laser light is a coherent state of the electromagnetic field.

Angular momentum in quantum mechanics is deeply connected to rotational symmetry and conservation laws.

Orbital angular momentum

L̂ = r̂ × p̂. Components satisfy [L̂_x, L̂_y] = iℏL̂_z (cyclic). Total magnitude: L̂² with eigenvalues ℏ²ℓ(ℓ+1). Projection L̂_z with eigenvalues mℏ, −ℓ ≤ m ≤ ℓ.

General angular momentum

Any operators satisfying [Ĵ_x, Ĵ_y] = iℏĴ_z (and cyclic) form an angular momentum algebra. Eigenvalues of Ĵ² = ℏ²j(j+1) with j = 0, ½, 1, …; eigenvalues of Ĵ_z = mℏ, −j ≤ m ≤ j.

Addition of angular momenta

Two systems with j₁ and j₂ combine to give total j from |j₁−j₂| to j₁+j₂. The Clebsch–Gordan coefficients express combined eigenstates in the product basis.

Most real quantum systems cannot be solved exactly. Perturbation theory provides systematic corrections when Ĥ = Ĥ⁰ + λH'.

First-order corrections

Energy: E_n^(1) = ⟨n⁰|H'|n⁰⟩

State: |n^(1)⟩ = Σ_{k≠n} [⟨k⁰|H'|n⁰⟩ / (E_n⁰ − E_k⁰)] |k⁰⟩

Second-order energy

E_n^(2) = Σ_{k≠n} |⟨k⁰|H'|n⁰⟩|² / (E_n⁰ − E_k⁰)

Applications

• Fine structure of hydrogen (relativistic + spin-orbit coupling)
• Stark effect (hydrogen in an electric field)
• Zeeman effect (hydrogen in a magnetic field)

Degenerate perturbation theory

When unperturbed levels are degenerate, diagonalise H' within the degenerate subspace first.

The variational principle provides upper bounds on ground-state energies without solving the Schrödinger equation directly.

The theorem

⟨ψ_trial|Ĥ|ψ_trial⟩ ≥ E_ground for any normalised |ψ_trial⟩

Method

1. Choose a trial wave function |ψ(α)⟩ with free parameters α.
2. Compute ⟨Ĥ⟩(α).
3. Minimise over α to get the best estimate of E_ground.

Example: helium atom

A simple trial function with effective nuclear charge Z_eff gives E_ground ≈ −77.5 eV vs. exact −79.0 eV — under 2% error from a one-parameter calculation.

Modern use

Density-functional theory (DFT) and variational quantum eigensolvers (VQE) on quantum hardware are rooted in this principle.

When two particles are truly identical, quantum mechanics demands a strict symmetry condition on their combined wave function.

Bosons vs fermions

• Bosons (integer spin): wave function is symmetric under particle exchange. Ψ(1,2) = +Ψ(2,1).
• Fermions (half-integer spin): wave function is antisymmetric. Ψ(1,2) = −Ψ(2,1).

Pauli exclusion from antisymmetry

If two fermions are in the same state, antisymmetry forces Ψ = 0. Two identical fermions cannot occupy the same quantum state.

Consequences

• The Periodic Table structure
• Degeneracy pressure in white dwarfs and neutron stars
• Fermi–Dirac statistics for electrons in metals
• Bose–Einstein condensation for bosons

The density matrix ρ̂ generalises the wave function to describe both pure and mixed quantum states.

Pure state

ρ̂ = |ψ⟩⟨ψ|,   tr(ρ̂²) = 1

Mixed state

ρ̂ = Σ p_i |ψ_i⟩⟨ψ_i|,   tr(ρ̂²) ≤ 1

Von Neumann equation

iℏ dρ̂/dt = [Ĥ, ρ̂]

Why density matrices matter

• Open quantum systems — tracing over the environment gives a reduced density matrix.
• Entanglement entropy S = −tr(ρ̂ log ρ̂) quantifies entanglement.
• Decoherence — off-diagonal elements (coherences) decay as the system interacts with its environment.

In 1964 John Bell derived inequalities that any local hidden variable theory must satisfy. Quantum mechanics violates them.

The CHSH inequality

For two observers making binary measurements along two settings each:

|⟨AB⟩ + ⟨AB'⟩ + ⟨A'B⟩ − ⟨A'B'⟩| ≤ 2

Quantum mechanics predicts a maximum of 2√2 ≈ 2.83.

Experimental tests

From Clauser (1972) to Aspect (1982) to the loophole-free experiments of 2015 (Hensen et al., Giustina et al., Shalm et al.) — all confirm quantum violation. No local realistic theory can explain the correlations.

Implications

Nature is either non-local (influences propagate instantaneously) or non-real (particles do not have definite properties before measurement) — or both.

Decoherence explains why quantum superpositions of macroscopic objects are never observed in practice.

The mechanism

A quantum system S in superposition α|0⟩ + β|1⟩ entangles with its environment E:

|Ψ_SE⟩ = α|0⟩|E₀⟩ + β|1⟩|E₁⟩

Tracing over the inaccessible environment gives ρ̂_S with off-diagonal terms suppressed — the system appears classical to any local observer.

Decoherence timescales

For a dust particle in air: ~10⁻³¹ s. For a single atom: milliseconds. Quantum computers must maintain coherence long enough to complete computation.

Does decoherence solve the measurement problem?

Decoherence explains why we don't see superpositions — but not why one branch becomes "the" outcome. The measurement problem remains philosophically open.