Perturbation Theory

Perturbation theory lets us find approximate solutions to quantum systems that are "close" to a solvable one. We write the Hamiltonian as:

$H = H_0 + \lambda H'$

where $H_0$ has known solutions $H_0 |n\rangle = E_n^{(0)} |n\rangle$ and $\lambda H'$ is a small perturbation.

First-Order Energy Correction

The first-order correction to the energy of state $|n\rangle$ is simply the expectation value of the perturbation:

$E_n^{(1)} = \langle n | H' | n \rangle = \int \psi_n^*(x)\, H'(x)\, \psi_n(x)\, dx$

Particle in a Box

The 1D particle in a box (infinite square well) on $[0, L]$ is a classic exactly solvable system:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{n\pi x}{L}\right), \quad E_n^{(0)} = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}$

with $n = 1, 2, 3, \ldots$ (quantum number starts at 1).

Common Perturbations

• Constant perturbation $H' = V_0$: correction is $E_n^{(1)} = V_0$ for all $n$ (the wavefunctions are already orthonormal and $\int \psi_n^2 = 1$).
• Linear ramp $H' = V_0 x/L$: correction is $E_n^{(1)} = V_0/2$ for all $n$ (by symmetry of the sin² integral).

Physical Constants

Using SI units with $\hbar = 1.055 \times 10^{-34}$ J·s, $m_e = 9.109 \times 10^{-31}$ kg, and $1\,\text{eV} = 1.602 \times 10^{-19}$ J.

Implementation

• pib_wavefunction(x, n, L) — $\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$
• pib_energy_eV(n, L_nm) — unperturbed energy in eV for a box of width $L_{\text{nm}}$ nanometres
• first_order_correction(V_func, n, L, N=1000) — numerically integrates $\langle n|V|n\rangle$ over $[0,L]$

import math

def pib_wavefunction(x, n, L):
    return math.sqrt(2.0 / L) * math.sin(n * math.pi * x / L)

def pib_energy_eV(n, L_nm):
    hbar = 1.055e-34
    m = 9.109e-31
    eV = 1.602e-19
    L = L_nm * 1e-9
    return n**2 * math.pi**2 * hbar**2 / (2 * m * L**2) / eV