5.1.2 Non-Degenerate Perturbation Theory

5.1.2 Non-Degenerate Perturbation Theory#

Prompts

  • What is the Hellmann-Feynman theorem, and how does it give the first-order energy correction as an expectation value without expanding eigenstates?

  • Derive the first-order state correction. Why does the admixture of state \(m\) scale as coupling/gap?

  • Why does the second-order energy correction always lower the ground state? Connect this to the variational principle.

  • When does non-degenerate perturbation theory break down? What signals the need for degenerate perturbation theory?

Lecture Notes#

Overview#

Non-degenerate perturbation theory provides a systematic expansion of eigenvalues and eigenstates in powers of a small parameter \(\lambda\), for the Hamiltonian \(H(\lambda) = H_0 + \lambda V\). The Hellmann-Feynman theorem is the organizing principle: the exact rate at which an eigenvalue changes with \(\lambda\) equals the expectation value of \(V\) in the exact eigenstate.

Hellmann-Feynman Theorem#

Hellmann-Feynman Theorem

For a parameter-dependent Hamiltonian \(H(\lambda)\) with eigenvalue \(E_n(\lambda)\) and normalized eigenstate \(|\psi_n(\lambda)\rangle\):

(81)#\[\frac{\mathrm{d}E_n}{\mathrm{d}\lambda} = \langle \psi_n(\lambda) | \frac{\partial H}{\partial \lambda} | \psi_n(\lambda) \rangle\]

At \(\lambda = 0\) with \(\partial H/\partial\lambda = V\), this immediately gives the first-order energy:

\[E_n^{(1)} = \frac{\mathrm{d}E_n}{\mathrm{d}\lambda}\bigg|_{\lambda=0} = \langle \psi_n^{(0)} | V | \psi_n^{(0)} \rangle\]

Requires: adiabatic continuity — eigenstates can be tracked continuously as \(\lambda\) varies (no level crossings).

First-Order Energy Correction#

First-Order Energy

(82)#\[\boxed{E_n^{(1)} = \langle \psi_n^{(0)} | V | \psi_n^{(0)} \rangle}\]

The first-order shift is the diagonal matrix element of \(V\) in the unperturbed basis — the expectation value of the perturbation in the unperturbed state.

If \(V\) has no diagonal elements (e.g., \(\hat{\sigma}^x\) in the computational basis), the first-order correction vanishes and the leading effect is second-order.

Derivation

Expand \(H|\psi_n\rangle = E_n|\psi_n\rangle\) to order \(\lambda^1\):

\[V|\psi_n^{(0)}\rangle + H_0|\psi_n^{(1)}\rangle = E_n^{(0)} |\psi_n^{(1)}\rangle + E_n^{(1)} |\psi_n^{(0)}\rangle\]

Project onto \(\langle \psi_n^{(0)}|\) and use \(\langle \psi_n^{(0)}|H_0 = E_n^{(0)}\langle \psi_n^{(0)}|\) plus the normalization choice \(\langle \psi_n^{(0)}|\psi_n^{(1)}\rangle = 0\):

\[\langle \psi_n^{(0)} | V | \psi_n^{(0)} \rangle = E_n^{(1)}\]

First-Order State Correction#

First-Order State

(83)#\[\boxed{|\psi_n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} |\psi_m^{(0)}\rangle}\]

The admixture of state \(m\) is proportional to the coupling \(\langle m|V|n\rangle\) and inversely proportional to the energy gap \(E_n^{(0)} - E_m^{(0)}\). States mix strongest when they are close in energy and strongly coupled.

Derivation

Expand \(|\psi_n^{(1)}\rangle = \sum_{m\neq n} c_m^{(1)}|\psi_m^{(0)}\rangle\) (excluding \(m=n\) by orthogonality). Insert into the order-\(\lambda^1\) equation and project onto \(\langle\psi_m^{(0)}|\):

\[\langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle = c_m^{(1)}(E_m^{(0)} - E_n^{(0)})\]

Solving: \(c_m^{(1)} = \langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle / (E_n^{(0)} - E_m^{(0)})\).

Second-Order Energy Correction#

Second-Order Energy

(84)#\[\boxed{E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}}}\]

  • Ground state: all denominators are negative, so \(E_0^{(2)} \leq 0\) — perturbations always lower the ground state energy (consistent with the variational principle).

  • Level repulsion: nearby levels push each other apart through virtual transitions.

  • Convergence: requires \(\lambda|\langle m|V|n\rangle| \ll |E_n^{(0)} - E_m^{(0)}|\) for all \(m\).

Derivation

From the order-\(\lambda^2\) equation, \(E_n^{(2)} = \langle\psi_n^{(0)}|V|\psi_n^{(1)}\rangle\). Substituting the first-order state and using Hermiticity of \(V\):

\[E_n^{(2)} = \sum_{m\neq n} \frac{\langle\psi_n^{(0)}|V|\psi_m^{(0)}\rangle\langle\psi_m^{(0)}|V|\psi_n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}} = \sum_{m\neq n}\frac{|\langle m^{(0)}|V|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}\]

Physical Intuitions#

The perturbation series has a clean physical interpretation at each order:

  • First-order energy = expectation value of the perturbation (average energy shift in the unperturbed state)

  • First-order state = hybridization with other levels, weighted by coupling/gap

  • Second-order energy = virtual transitions to intermediate states and back; always lowers the ground state

  • Breakdown: when \(E_n^{(0)} - E_m^{(0)} \to 0\) (degeneracy), denominators diverge → signal to switch to degenerate perturbation theory (\S5.1.3)

Example: Verification Against the Qubit Model

Problem. Verify perturbation theory on the qubit \(H(\lambda) = \hat{\sigma}^z + \lambda\hat{\sigma}^x\) from \S5.1.1.

Solution. Unperturbed: \(E_+^{(0)} = +1\) (\(|0\rangle\)), \(E_-^{(0)} = -1\) (\(|1\rangle\)). Since \(\langle 0|\hat{\sigma}^x|0\rangle = 0\), the first-order energy vanishes: \(E_+^{(1)} = 0\). The second-order correction:

\[E_+^{(2)} = \frac{|\langle 1|\hat{\sigma}^x|0\rangle|^2}{E_+^{(0)} - E_-^{(0)}} = \frac{1}{2}\]

So \(E_+(\lambda) = 1 + \tfrac{1}{2}\lambda^2 + O(\lambda^3)\), matching the exact result \(\sqrt{1+\lambda^2} = 1 + \tfrac{1}{2}\lambda^2 - \tfrac{1}{8}\lambda^4 + \cdots\). \(\checkmark\)

The first-order state: \(|\psi_+^{(1)}\rangle = \tfrac{1}{2}|1\rangle\), so \(|\psi_+(\lambda)\rangle \approx |0\rangle + \tfrac{\lambda}{2}|1\rangle\), again matching the exact eigenvector expansion. \(\checkmark\)

Discussion

The second-order formula \(E_n^{(2)} = \sum_m |V_{mn}|^2/(E_n - E_m)\) diverges when two levels become degenerate (\(E_n = E_m\)). Is this a failure of quantum mechanics, or just a signal that you need a different computational method? What is the correct procedure when the energy gap \(\Delta E\) approaches zero while the coupling \(V_{mn}\) stays finite?

Summary#

  • The Hellmann-Feynman theorem \(\mathrm{d}E_n/\mathrm{d}\lambda = \langle\psi_n|V|\psi_n\rangle\) is the organizing principle: energy derivatives are expectation values.

  • First-order energy: \(E_n^{(1)} = \langle n^{(0)}|V|n^{(0)}\rangle\) — expectation value of perturbation.

  • First-order state: admixture of other levels, weighted by coupling over energy gap.

  • Second-order energy: virtual transitions; always lowers the ground state; series diverges at degeneracy.

See Also

Homework#

1. Show that the normalization choice \(\langle\psi_n^{(0)}|\psi_n(\lambda)\rangle = 1\) implies \(\langle\psi_n^{(0)}|\psi_n^{(1)}\rangle = 0\). Why does this simplify the expansion?

2. A hydrogen atom in a uniform electric field \(\mathcal{E}\) along \(z\) has perturbation \(V = e\mathcal{E}z\). (a) Show \(E_{1s}^{(1)} = 0\) using parity. (b) The second-order shift is \(E_{1s}^{(2)} = -\frac{9}{4}a_0^3\epsilon_0\mathcal{E}^2\). Identify the atomic polarizability \(\alpha\).

3. For a 1D symmetric potential with perturbation \(V = \lambda x\) (odd parity), show that \(E_n^{(1)} = 0\) for all \(n\). What is the leading correction?

4. For the harmonic oscillator with perturbation \(V = \lambda x^4\), compute \(E_n^{(1)}\) using \(x = \sqrt{\hbar/(2m\omega)}(a + a^\dagger)\). Keep only diagonal terms in \(x^4\) expressed via ladder operators.

5. For the qubit \(H(\lambda) = \hat{\sigma}^z + \lambda\hat{\sigma}^x\): (a) compute \(E_+^{(1)}\) and \(E_+^{(2)}\), (b) compute \(|\psi_+^{(1)}\rangle\), (c) compare both to the exact eigenvalues \(E_\pm = \pm\sqrt{1+\lambda^2}\) expanded to \(O(\lambda^2)\).

6. Show that \(E_0^{(2)} \leq 0\) for the ground state by analyzing the sign of each term in the sum. Connect this to the variational principle: why must the ground state energy decrease under any perturbation?

7. A 3-level system has \(E_1 = 0\), \(E_2 = \Delta\), \(E_3 = 10\Delta\) with couplings \(V_{12}\) and \(V_{23}\) (all other matrix elements zero). (a) Write \(E_1^{(2)}\). (b) As \(\Delta \to 0\), which term diverges? (c) What is the correct procedure when the divergence signals near-degeneracy?