An electron is in the state ψ = (1/√2)ψ₁ + (1/√2)ψ₂, where ψ₁ and ψ₂ are momentum eigenstates with eigenvalues p₁ and p₂. What is the electron's momentum before measurement?
AThe average (p₁ + p₂)/2 — the superposition represents a definite momentum equal to the mean of the eigenvalues
BEither p₁ or p₂, with probability 1/2 each — there is no definite momentum before measurement, only a probability distribution
CZero — the two momentum components cancel because the coefficients are equal
DBoth p₁ and p₂ simultaneously — quantum mechanics allows particles to have multiple definite values at once
In quantum mechanics, a superposition state does not have a definite value for an observable unless it is an eigenstate of the corresponding operator. Before measurement, the electron's momentum is genuinely indefinite — not merely unknown. The Born rule tells us that measurement returns p₁ with probability 1/2 and p₂ with probability 1/2. Option A gives the expectation value (average over many measurements) but this is not the pre-measurement momentum. Option D gestures at superposition correctly but 'simultaneously' is misleading — each individual measurement returns exactly one definite eigenvalue.
Question 2 Multiple Choice
Why is it physically essential that every observable quantity in quantum mechanics be represented by a Hermitian operator?
ABecause only Hermitian operators can be applied to complex-valued wavefunctions
BBecause Hermitian operators always commute with each other, ensuring observables can be measured simultaneously
CBecause Hermitian operators guarantee real eigenvalues, and every measurement outcome must be a real number
DBecause Hermitian operators preserve the norm of the wavefunction, ensuring probability is conserved
When you measure momentum, energy, or position, the result is always a real number — you cannot measure a complex energy. A Hermitian operator is self-adjoint (∫ψ*(Âφ)dx = ∫(Âψ)*φ dx), and this property guarantees all eigenvalues are real. If observables were represented by non-Hermitian operators, eigenvalues could be complex, giving unphysical measurement outcomes. Note: Option B is false (Hermitian operators do not generally commute — non-commuting Hermitian operators represent complementary observables like position and momentum). Option D describes unitary operators.
Question 3 True / False
The momentum operator p̂ = −iℏ∂/∂x acts on a wavefunction by multiplying it by the particle's current momentum value, analogous to how the position operator multiplies by x.
TTrue
FFalse
Answer: False
The position operator x̂ acts by multiplication: x̂ψ(x) = xψ(x). The momentum operator is fundamentally different — it acts by differentiation: p̂ψ(x) = −iℏ ∂ψ/∂x. This derivative structure captures the quantum connection between momentum and spatial variation: a wavefunction oscillating rapidly in space has high momentum, just as a short-wavelength wave carries high frequency. The eigenfunctions of p̂ are plane waves e^{ipx/ℏ}, not delta functions. The multiplicative structure of x̂ versus the differential structure of p̂ is exactly why they satisfy the canonical commutation relation [x̂, p̂] = iℏ rather than commuting.
Question 4 True / False
If a quantum system is in an eigenstate of the Hamiltonian Ĥ with eigenvalue E, every measurement of the system's energy will return E with certainty.
TTrue
FFalse
Answer: True
This is the defining property of an eigenstate: Ĥψ = Eψ means ψ contains only one energy component. Expanding in the energy eigenbasis, all coefficients cₙ are zero except one (c_k = 1), so the Born rule assigns probability |c_k|² = 1 to measuring E_k and probability 0 to all other energies. These are stationary states — their probability distributions for all observables are time-independent. A general superposition state Σcₙψₙ does NOT have a definite energy and will yield different eigenvalues on different measurements with probabilities |cₙ|².
Question 5 Short Answer
Explain what happens to a quantum state when a measurement is made, using the operator/eigenfunction framework. How do the expansion coefficients determine the probability of each outcome?
Think about your answer, then reveal below.
Model answer: Before measurement, a general state ψ is expanded in the eigenbasis of the observable's operator: ψ = Σcₙψₙ, where Âψₙ = aₙψₙ. The Born rule states that measuring observable A returns eigenvalue aₙ with probability |cₙ|². The measurement collapses the superposition — after returning eigenvalue aₙ, the state is now the corresponding eigenstate ψₙ. The expansion coefficients cₙ encode how much of each eigenstate the original wavefunction contained, and their squared magnitudes give the probability distribution over possible outcomes. Measuring does not reveal a pre-existing value; it collapses a genuine indefiniteness into one definite eigenvalue.
The key insight is that measurement is not passive revelation of a pre-existing fact — it is an interaction that projects the state onto an eigenstate. The operator defines what is measured, eigenfunctions define the possible definite-value states, and the decomposition of ψ in the eigenbasis determines the probability distribution. Every quantum calculation — energy levels, selection rules, expectation values — flows from this framework.