For λ=0.1 and ⟨n∣V∣n⟩=3 eV, find the first-order energy shift.
Answer and explanation
0.3 eV
Multiply the matrix element by λ.
Subjects
Advanced quantum
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Sample exercises
Which quantity gives the nondegenerate first-order energy coefficient?
- Every off-diagonal entry added without denominators
- ⟨n∣V∣n⟩
- The state norm cubed
- Zero in all systems
Answer and explanation
⟨n∣V∣n⟩
The diagonal expectation in the unperturbed state is the first-order coefficient.
An unperturbed energy is 5 eV and first-order shift is −0.2 eV. Find the first-order corrected energy.
Answer and explanation
4.8 eV
Add the correction to the unperturbed value.
What should be done first within a degenerate subspace?
- Divide by zero energy gaps
- Diagonalize the perturbation restricted to that subspace
- Discard normalization
- Choose any one basis vector and ignore mixing
Answer and explanation
Diagonalize the perturbation restricted to that subspace
The appropriate combinations resolve first-order splitting.
Why can a tiny coupling produce a significant change near two nearly degenerate levels?
Answer and explanation
State-mixing corrections involve energy-gap denominators. A small gap can make the ratio large despite a small numerator.
Credit relative scales rather than coupling size alone.
For a normalized state with components (1/√2,1/√2) and V=diag(2,4) eV, find ⟨V⟩.
Answer and explanation
3 eV
The weighted diagonal mean is 2/2+4/2.
Study notes
Perturbation theory and approximation checks
Write H=H0+λV and expand observables or states about a solvable reference problem. Small λ is useful only relative to relevant spectral scales.
Advanced universityPerturbation theory and approximation checks · revision notes
Write H=H0+λV and expand observables or states about a solvable reference problem. Small λ is useful only relative to relevant spectral scales.
Advanced universityScattering, currents and probability conservation
Scattering models compare incoming and outgoing waves far from a localized interaction. Boundary conditions identify which waves are physically incident.
Advanced universityScattering, currents and probability conservation · revision notes
Scattering models compare incoming and outgoing waves far from a localized interaction. Boundary conditions identify which waves are physically incident.
Advanced universityVariational bounds and trial states
For a normalized trial state in the Hamiltonian domain, the expectation of a Hamiltonian bounded below is at least its ground energy.
Advanced universityVariational bounds and trial states · revision notes
For a normalized trial state in the Hamiltonian domain, the expectation of a Hamiltonian bounded below is at least its ground energy.
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