Marche quantique

Brillouin-Wigner Theory

1. Projection

Consider H = H_0 + V, where the unperturbed states satisfy H_0 |\Phi_n\rangle = E_n^0 |\Phi_n\rangle. We partition the Hilbert space using projection operators P = |\Phi_n\rangle\langle\Phi_n| and Q = I - P = \sum_{k \neq n} |\Phi_k\rangle\langle\Phi_k|.

Adopting intermediate normalization \langle \Phi_n | \Psi \rangle = 1, the exact state |\Psi\rangle is decomposed as:

|\Psi\rangle = (P+Q) |\Psi\rangle = |\Phi_n\rangle + Q|\Psi\rangle

2. Wavefunction

From the Schrödinger equation (H_0 + V) |\Psi\rangle = E |\Psi\rangle, projecting onto Q yields the recursive relation Q|\Psi\rangle = \frac{Q}{E - H_0} V |\Psi\rangle. By substituting this relation into the exact state decomposition:

|\Psi\rangle = |\Phi_n\rangle + \frac{Q}{E - H_0} V |\Psi\rangle

We can iteratively substitute this expression for |\Psi\rangle back into the right-hand side. The first iteration yields:

\begin{aligned} |\Psi\rangle &= |\Phi_n\rangle + \frac{Q}{E - H_0} V \left[ |\Phi_n\rangle + \frac{Q}{E - H_0} V |\Psi\rangle \right] \\ &= |\Phi_n\rangle + \frac{Q}{E - H_0} V |\Phi_n\rangle + \left( \frac{Q}{E - H_0} V \right)^2 |\Psi\rangle \end{aligned}

By continuing this process, we obtain the infinite series expansion in terms of the unperturbed state |\Phi_n\rangle:

|\Psi\rangle = \sum_{k=0}^{\infty} \left( \frac{Q}{E - H_0} V \right)^k |\Phi_n\rangle

3. Energy Correction

The energy shift is given by E - E_n^0 = \langle\Phi_n| V |\Psi\rangle. Substituting the series for |\Psi\rangle into this expression, we can derive the corrections at various orders:

  • First-order correction:
    E_n^{(1)} = \langle\Phi_n| V |\Phi_n\rangle = V_{nn}
  • Second-order correction:
    E_n^{(2)} = \langle\Phi_n| V \frac{Q}{E - H_0} V |\Phi_n\rangle = \sum_{k \neq n} \frac{|\langle\Phi_k| V |\Phi_n\rangle|^2}{E - E_k^0}

Note: In Brillouin-Wigner theory, the energy E appears on both sides of the equation, making it an implicit equation that typically requires numerical methods to solve.

Xu Dong Liu

Leave a comment

Instagram / TikTok / X

Designed with WordPress