First-Order Perturbation Theory
Split a Hamiltonian into a solved reference part and an added term. For an isolated nondegenerate level, the first-order energy correction is the added term's expectation in the reference state. Coupling matrix elements compared with energy gaps provide a first check of weak mixing.
What a learner can do afterwards
- Splits a Hamiltonian into a solved part and a small perturbation
- Computes a first-order energy shift as an expectation value
- Checks coupling-to-gap ratios, explains their limits, and states how degenerate reference levels require a separate treatment
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Shell models describe levels; here we use state vectors. Write H=H0+V. H0 is solved: H0|n>=E_n^0|n>. V is the added self-adjoint term. Choose orthonormal reference eigenstates |n>, so <n|n>=1. A matrix element <m|V|n> is the overlap of |m> with V|n>; <n|V|n> is V's expectation in |n>. A nondegenerate level differs in energy from every other reference level. First order keeps the energy correction linear in V.
For an isolated nondegenerate level, ΔE_n=<n|V|n> and E_n≈E_n^0+ΔE_n. For a position-dependent potential, ΔE_n=∫|psi_n(x)|² V(x) dx over the allowed region. The reference density supplies the weights. Thus V=F x gives ΔE_n=F<x>. For V=C I, with I the identity, normalization gives ΔE_n=C. This constant offset shifts every energy by C exactly and leaves the eigenstates unchanged.
An infinite well runs from x=0 to x=L. Each reference energy state's density is symmetric about L/2, so <x>=L/2. Adding V(x)=F x gives ΔE_n=F L/2. For L=12 nm and F=0.02 eV/nm, the shift is 0.12 eV. If the reference energy is 1.8 eV, the corrected energy is 1.92 eV. In a well centered at zero, the specified potential F x instead has zero first-order expectation.
For another level m, form r_m=|<m|V|n>|/|E_n^0-E_m^0|. Ratios much less than one are a first check of weak mixing; a ratio near one warns against that assumption. Zero diagonal expectation does not imply zero off-diagonal coupling. A constant offset has no such coupling, even if it exceeds a gap. With many states, combined mixing also matters. These ratios alone do not bound every error or guarantee that higher orders stay small.
Reference states with equal energy are degenerate. Form V's matrix within their shared-energy subspace and diagonalize it. This finds combinations that this matrix maps to multiples of themselves. Its eigenvalues give first-order shifts; its eigenvectors select starting combinations. Check coupling to outside levels separately. Nearly equal levels need a joint treatment when their coupling is not small compared with their gap. A narrow gap alone does not force mixing.
For a nondegenerate level, the first-order state correction adds reference states |m> with coefficients <m|V|n>/(E_n^0-E_m^0), for m different from n. Their magnitudes are the mixing ratios. This state correction is separate from the first-order energy calculation. That energy shift uses <n|V|n> in the known reference state. An average in a guessed corrected state does not implement this formula.
For an isolated nondegenerate level, use ΔE_n=<n|V|n>. Check mixing and higher-order effects. For degenerate levels, diagonalize V within their subspace.
Lesson, practice questions, worked solutions and flashcards written for this dataset.
Readings and flashcards written for this dataset. The readings discuss oscillator reference states, weighted expectations, and checks for mixing and degeneracy.
More to read
Reference states of an oscillator
The one-dimensional harmonic oscillator is a useful solved reference system. Its potential is m omega² x²/2 and its energies are E_n^0=(n+1/2) hbar omega, with n=0,1,2,... . Here omega is angular frequency and hbar is Planck's constant divided by 2 pi. Adjacent energy levels are separated by hbar omega.
The ground-state wavefunction is Gaussian. Excited eigenfunctions contain a polynomial factor as well as a Gaussian and can have nodes. They are not all bell curves. Although eigenfunctions alternate between even and odd parity, each eigenstate's probability density is even about the center.
Place that center at x=0 and add V=F x. The reference mean position is zero, so the first-order energy shift F<x> vanishes. This says nothing by itself about off-diagonal matrix elements. The added slope can change the state even when the first-order energy correction is zero.
To test mixing, compare each relevant coupling with its reference energy separation. The oscillator's level spacing supplies those denominators; its wavefunctions supply the matrix elements. A probability density describes measurement statistics, not a particle following a classical oscillating path.
The weights in an energy expectation
A normalized reference state assigns probability weights to positions. For a position-dependent perturbation, its first-order energy correction is the integral of |psi_n(x)|² V(x). The density supplies the weights and integrates to one. A uniform average of V over space generally uses different weights.
For an infinite well from zero to L, each energy eigenstate has a density symmetric about L/2. Its mean position is therefore L/2. An added potential F x has expectation F L/2. Specifying the well's interval matters: a centered well with potential F x instead has a zero first-order expectation.
Changing coordinates for one physical problem also requires rewriting the potential. Moving the coordinate origin while keeping the same formula F x can instead change the specified potential by a constant. Such constants shift the energy reference.
For V=C I, every eigenstate of H0 remains an eigenstate of H0+C I. Its energy increases by C exactly. All energy gaps remain unchanged. This case shows why the magnitude of an energy offset alone is not a test of state mixing.
Couplings, gaps and degenerate levels
The first-order energy shift uses the diagonal matrix element <n|V|n>. The first-order state correction uses off-diagonal elements. For a different reference state |m>, its coefficient is <m|V|n>/(E_n^0-E_m^0). A zero diagonal element does not determine these other elements.
Small coefficient magnitudes support weak mixing. If there are many coupled states, their combined contribution also needs attention. Neither a small diagonal shift nor small individual mixing ratios provide a universal bound on the error of a truncated expansion. Higher-order effects can still matter.
Equal reference energies make the nondegenerate denominator zero. In that case, form V's matrix within the shared-energy subspace and diagonalize it first. Its eigenvalues give the first-order energy shifts. Its eigenvectors identify the combinations of reference states that receive those shifts. Coupling to outside states remains a separate check.
For nearly equal levels, compare their coupling with their separation. Treat them together when the coupling is not small relative to that gap. A narrow gap alone does not prove mixing, because the coupling may vanish.
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