PHY3251 Quantum Mechanics - Homework Set 1 (Vocabulary Flashcards)

Vocabulary flashcards covering key concepts, definitions, and formulas from PHY3251 lecture notes.

Wavefunction

A function Ψ(x,t) that encodes the quantum state of a particle; |Ψ|^2 gives the probability density for position (and similarly for other observables).

Normalizable

A wavefunction for which ∫|Ψ|^2 dx is finite and can be set to 1 (normalized) to interpret probabilities properly.

Physical acceptability of a wavefunction

A wavefunction that is square-integrable, single-valued, and satisfies the system’s boundary conditions and physical constraints.

Normalization constant A

A real constant chosen so that ∫|Ψ|^2 dx = 1; for Ψ(x,0) in the problem, |A|^2 = 3/(2 a^3).

Stationary state

A state whose probability density is time-independent; Ψ(x,t) = φ(x) e^{-iEt/ħ} up to a global phase.

Infinite potential well

A 1D box with V(x)=0 for 0<x<a and V(x)=∞ outside; eigenfunctions are proportional to sin(nπx/a).

Ground state ϕ1(x)

Lowest-energy eigenfunction of the well: ϕ1(x) = sqrt(2/a) sin(πx/a) on 0<x<a.

First excited state ϕ2(x)

Next higher-energy eigenfunction: ϕ2(x) = sqrt(2/a) sin(2πx/a) on 0<x<a.

Orthogonality of eigenfunctions

∫0^a φn^*(x) φm(x) dx = δnm; eigenfunctions with different n are orthogonal.

Expectation value ⟨X⟩

⟨X⟩ = ∫ Ψ^*(x,t) x Ψ(x,t) dx; the average position.

Position uncertainty Δx

Δx = sqrt(⟨X^2⟩ − ⟨X⟩^2); the spread in position measurements.

Momentum operator P̂

P̂ = −iħ ∂/∂x; the operator corresponding to momentum in one dimension.

Momentum-squared operator P̂^2

P̂^2 = −ħ^2 ∂^2/∂x^2; used in ⟨P̂^2⟩ calculations.

Expectation ⟨P^2⟩

⟨P̂^2⟩ = ∫ Ψ^*(−ħ^2 ∂^2Ψ/∂x^2) dx; alternatively ħ^2 ∫ |∂Ψ/∂x|^2 dx with appropriate boundary terms.

Momentum uncertainty Δp

Δp = sqrt(⟨P̂^2⟩ − ⟨P̂⟩^2); the spread in momentum measurements.

Uncertainty principle

Δx Δp ≥ ħ/2; fundamental limit on simultaneously precise knowledge of position and momentum.

Two-state superposition in a well

Ψ(x,t) = (1/√2) φ1(x) e^{-iE1 t/ħ} + (1/√2) φ2(x) e^{-iE2 t/ħ}; a state with two energy components.

Cross term integral ∫_0^a x φ1 φ2 dx

Integral needed for ⟨X⟩ in the two-state system; for φ1=√(2/a) sin(πx/a) and φ2=√(2/a) sin(2πx/a), it equals −16 a/(9π^2).

⟨X⟩(t) for the two-state superposition

⟨X⟩(t) = −8a/(9π^2) cos[(E2−E1)t/ħ]; a time-varying expectation value due to beating between eigenstates.

E1

Ground-state energy: E1 = π^2 ħ^2/(2 m a^2).

E2

First excited energy: E2 = 4π^2 ħ^2/(2 m a^2) = 2π^2 ħ^2/(m a^2).

ΔE

Energy difference: ΔE = E2 − E1 = 3π^2 ħ^2/(2 m a^2).

ω (beat frequency)

ω = ΔE/ħ = 3π^2 ħ/(2 m a^2); the angular frequency of the ⟨X⟩ oscillation.

⟨P^2⟩ at t=0 for the two-state state

⟨P̂^2⟩ = (5/2) π^2 ħ^2 / a^2 for Ψ ∝ (φ1+φ2)/√2 at t=0.

⟨P⟩ at t=0

Zero for the symmetric real superposition at t=0.

Separation of variables

A method to solve the time-dependent Schrödinger equation by writing Ψ(x,t) = ψ(x) ϕ(t) and solving a spatial eigenproblem and a separate temporal equation.

Time-dependent Schrödinger equation

iħ ∂Ψ/∂t = [−(ħ^2/2m) ∂^2/∂x^2 + V(x)] Ψ.

ψi(x) and ϕi(t)

Spatial eigenfunction ψi(x) and temporal factor ϕi(t) in a separable solution; ψi satisfy the stationary equation, ϕi(t) evolves with Ei.

Ei (separation constant)

Energies from the spatial equation: [−(ħ^2/2m) d^2/dx^2 + V(x)] ψi = Ei ψ_i.

General solution in eigenbasis

Ψ(x,t) = ∑i ci ψi(x) e^{-iEi t/ħ}; a superposition of stationary states.

Coefficients c_i

Expansion amplitudes in the eigenbasis; |ci|^2 gives the probability of finding energy Ei; ∑|c_i|^2 = 1.

Probability of an outcome q_n

If ψ = ∑ cn φn, the probability of measuring qn is |cn|^2 when φn is the eigenfunction with eigenvalue qn.

Post-measurement state (collapse)

After measuring qk, the state collapses to the eigenstate φk (normalized) corresponding to q_k.

Ehrenfest theorem

d⟨p⟩/dt = −⟨∂V/∂x⟩; connects quantum expectation values to classical equations of motion.

Lx, Ly, Lz

Angular momentum components: Lx = y pz − z py, Ly = z px − x pz, Lz = x py − y px.

Commutation relations of angular momentum

[Lx, Ly] = iħ Lz, [Ly, Lz] = iħ Lx, [Lz, Lx] = iħ Ly; angular momentum components do not commute.

Incompatible observables

Two observables that do not commute cannot be simultaneously measured with arbitrary precision; no common eigenbasis.

Pauli matrices

2×2 matrices used for spin-1/2; examples: σx = [[0,1],[1,0]], σz = [[1,0],[0,−1]].

Operator ordering

For noncommuting operators, the order matters; (A+B)(A−B) generally ≠ A^2 − B^2 unless [A,B]=0.

Jacobi identity

[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 for any operators A,B,C.

Product rule for commutators

[AB, C] = A[B, C] + [A, C] B; similarly for other permutations.

[f(X), P] = iħ f′(X)

Functional relationship between X and P: the commutator with P yields iħ times the derivative of the function f evaluated at X.

[X, P] = iħ

Canonical commutation relation between position and momentum.

[X, P^2] = 2iħ P

Higher-order canonical relation obtained from [X,P^2] = [X,P]P + P[X,P].

Linear operator

An operator L that satisfies L(αf + βg) = α Lf + β Lg for all complex α,β and functions f,g.

Linearity of maps

Property that preserves addition and scalar multiplication in the operator acting on functions.