Quantum Mechanics: Formalism
CollegeThe Schrödinger equation and operators
Chapter 17 introduced quantization qualitatively. The full formalism replaces a particle's trajectory with a complex-valued wavefunction ψ(x,t), whose evolution is governed by the time-dependent Schrödinger equation:
Physical quantities become operators acting on ψ: position x̂ = x, momentum p̂ = −iℏ∂/∂x. They do not commute:
This single algebraic fact is the rigorous origin of the uncertainty principle Chapter 17 stated: for any two operators, ΔAΔB ≥ ½|⟨[Â,B̂]⟩|, and substituting the canonical commutator recovers ΔxΔp ≥ ℏ/2. The Born rule interprets |ψ(x,t)|² as a probability density: the probability of finding the particle in [x, x+dx] is |ψ|²dx, requiring normalization ∫|ψ|²dx = 1.
For a time-independent potential, separable solutions ψ(x,t) = φ(x)e−iEt/ℏ satisfy the time-independent Schrödinger equation Ĥφ = Eφ — an eigenvalue equation whose allowed energies E are often discrete.
Why boundaries force quantization
Quantization is not an extra postulate bolted onto waves — it falls straight out of requiring a wave to fit consistently inside boundaries, exactly as a guitar string (Chapter 12) can only vibrate at discrete harmonics fn = nv/2L because it must vanish at both fixed ends. A confined quantum particle's wavefunction must likewise satisfy boundary conditions, and only discrete standing-wave-like solutions survive — hence discrete energy levels. An unconfined (free) particle has no such boundary and its energy spectrum is continuous, matching the classical intuition that quantization is a signature of confinement, not a universal graininess.
The Born rule forces a genuine break from classical thinking: before a measurement, a particle described by a superposition of states does not merely have unknown position — it has no definite position at all. Measurement is described as "collapsing" the wavefunction onto a definite outcome with the corresponding probability. This is not a limitation of our instruments; interference experiments (Chapter 13's double slit, and its single-particle version) show a particle really does explore multiple paths simultaneously until measured.
ΔxΔp ≥ ℏ/2 is not an isolated rule to memorize — it is a direct, provable consequence of [x̂,p̂]=iℏ, the same way every result in this chapter traces back to treating x and p as non-commuting operators rather than ordinary numbers.Worked example — the particle in a box
An electron is confined to an infinite square well of width L = 1.0 nm (V=0 inside, V=∞ at the walls). Find the allowed energy levels.
- Inside the well the time-independent Schrödinger equation reduces to
−ℏ²/2m φ″ = Eφ, with general solutionφ(x) = A sin(kx) + B cos(kx),k = √(2mE)/ℏ. - Boundary condition
φ(0) = 0forcesB = 0. Boundary conditionφ(L) = 0forcessin(kL) = 0, sokL = nπfor integern = 1, 2, 3, …— quantization from the boundary, exactly as in the explanation above. - Solve for energy:
k = nπ/L, soEn = ℏ²k²/2m = n²π²ℏ²/(2mL²). - Plug in numbers (
m = 9.11×10⁻³¹ kg,ℏ = 1.055×10⁻³⁴ J·s,L = 1.0×10⁻⁹ m):E₁ = π²(1.055×10⁻³⁴)²/(2×9.11×10⁻³¹×10⁻¹⁸) = 6.02×10⁻²⁰ J ≈ 0.376 eV. - Higher levels scale as
n²:E₂ = 4E₁ ≈ 1.50 eV,E₃ = 9E₁ ≈ 3.38 eV— sharply discrete, macroscopically-measurable steps, unlike anything in classical mechanics where a confined particle could have any energy at all.
Normalize the ground-state wavefunction φ₁(x) = A sin(πx/L) for 0 ≤ x ≤ L.
Require ∫₀L A²sin²(πx/L) dx = 1. Since ∫₀L sin²(πx/L)dx = L/2, we get A²(L/2) = 1, so A = √(2/L).
Derive Δx·Δp ≥ ℏ/2 from the commutator [x̂,p̂] = iℏ using the general uncertainty relation.
The general relation is ΔAΔB ≥ ½|⟨[Â,B̂]⟩|. With Â=x̂, B̂=p̂, the expectation of the commutator is simply iℏ (a constant), so |⟨[x̂,p̂]⟩| = ℏ. Substituting gives ΔxΔp ≥ ℏ/2 exactly.
Why does the particle-in-a-box ground state have E₁ > 0, unlike a classical particle at rest?
A classical particle can sit motionless with E=0. Quantum mechanically, confining a particle to width L forces some momentum uncertainty Δp ≳ ℏ/L by the uncertainty principle, which forces nonzero kinetic energy — this "zero-point energy" is a direct, testable consequence of quantization and shows up in real confined systems (quantum dots, nuclei).
What physically happens to a superposition state when a measurement is made?
Before measurement, ψ can be a superposition of eigenstates, each with its own probability amplitude. Upon measuring the corresponding observable, the outcome is one of the eigenvalues, selected randomly with probability given by the Born rule (|amplitude|²), and the wavefunction "collapses" to the corresponding eigenstate. Repeated identical measurements on identically-prepared systems reproduce the predicted probability distribution, not a single deterministic value.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1Find the ground-state energy of an electron in a 1D box of width L (n=1).
- E_n = n^2*h^2/(8*m*L^2)
- For n=1: E_1 = h^2/(8*m*L^2)
Example 2What is the ratio of the second to first energy level in a box?
- E_n scales as n^2
- E_2/E_1 = 2^2 / 1^2
Now you try
Work each one out first, then tap to reveal the worked answer.