A normalized two-outcome state has amplitude sqrt(3)/2 for outcome 0. Find its probability.
Answer and explanation
0.75
Square the amplitude magnitude: 3/4.
Subjects
Quantum mechanics
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Sample exercises
Probabilities of energies 2 and 6 eV are 0.25 and 0.75. Find mean energy.
Answer and explanation
5 eV
The weighted mean is 2×0.25+6×0.75=5.
Does multiplying an entire state by exp(iφ) change its outcome probabilities?
- Yes, all probabilities double
- Only for φ=0
- No; this is a global phase
- Yes, normalization fails always
Answer and explanation
No; this is a global phase
A common unit-magnitude phase leaves all squared magnitudes unchanged.
Which values must probabilities over a complete orthonormal basis sum to?
- The energy
- The number of outcomes
- Zero
- One
Answer and explanation
One
Normalization requires a total probability of one.
Outcome probabilities are 0.2, 0.3 and p. Find p.
Answer and explanation
0.5
Subtract the first two probabilities from one.
Explain why an expectation value can differ from every allowed outcome.
Answer and explanation
It is an ensemble mean, such as 5 eV from outcomes 2 and 6 eV. Individual measurements still return allowed outcomes.
Credit ensemble averaging and allowed eigenvalues.
Study notes
Quantum states, amplitudes and measurement
A normalized state vector encodes probability amplitudes. Overall global phase leaves measurement probabilities unchanged.
UniversityQuantum states, amplitudes and measurement · revision notes
A normalized state vector encodes probability amplitudes. Overall global phase leaves measurement probabilities unchanged.
UniversitySchrodinger equations and a particle in a box
The time-dependent equation is iℏ ∂ψ/∂t=Hψ. A time-independent Hamiltonian permits stationary-state analysis.
UniversitySchrodinger equations and a particle in a box · revision notes
The time-dependent equation is iℏ ∂ψ/∂t=Hψ. A time-independent Hamiltonian permits stationary-state analysis.
UniversityOperators, commutators and uncertainty
Observables are represented by suitable self-adjoint operators. In finite dimensions they are Hermitian matrices with real eigenvalues.
UniversityOperators, commutators and uncertainty · revision notes
Observables are represented by suitable self-adjoint operators. In finite dimensions they are Hermitian matrices with real eigenvalues.
UniversityInfinite-well energy ratio
A one-dimensional infinite well has n=1,2,...; this question changes quantum number and well width.
UniversityInfinite-well energy ratio · formula reference
A one-dimensional infinite well has n=1,2,...; this question changes quantum number and well width.
UniversityPhoton energy from wavelength
Use hc = 1239.84 eV·nm for the stated numerical convention.
UniversityPhoton energy from wavelength · formula reference
Use hc = 1239.84 eV·nm for the stated numerical convention.
UniversityExpectation value in an orthogonal eigenbasis
The squared amplitudes are probabilities. Orthogonality and normalization are specified.
Advanced universityExpectation value in an orthogonal eigenbasis · formula reference
The squared amplitudes are probabilities. Orthogonality and normalization are specified.
Advanced universityVariance of a two-outcome observable
Compute the second moment before subtracting the square of the mean.
Advanced universityVariance of a two-outcome observable · formula reference
Compute the second moment before subtracting the square of the mean.
Advanced universityFirst-order stationary perturbation
Nondegenerate first-order perturbation theory; the perturbation is diagonal in the stated orthonormal basis.
Advanced universityFirst-order stationary perturbation · formula reference
Nondegenerate first-order perturbation theory; the perturbation is diagonal in the stated orthonormal basis.
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