Approximation Methods

8.1 Time-Independent Perturbation Theory

For a Hamiltonian $\hat{H} = \hat{H}_0 + \lambda \hat{H}"$ where $\hat{H}'$ is “small” and $\hat{H}_0$ Has known eigenstates $|n^{(0)}\rangle$ and eigenvalues $E_n^{(0)}$.

First-order energy correction:

$E_n^{(1)} = \langle n^{(0)} | \hat{H}' | n^{(0)} \rangle$

Second-order energy correction:

$E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m^{(0)} | \hat{H}' | n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}}$

First-order state correction:

$|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)} | \hat{H}' | n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle$

Physical interpretation. The first-order energy correction is the expectation value of the Perturbation in the unperturbed state. The second-order correction accounts for virtual transitions To other states: if the perturbation mixes in state $|m\rangle$ with amplitude proportional to $V_{mn}/(E_n - E_m)$The energy shift is the sum of $|V_{mn}|^2/(E_n - E_m)$ over all Intermediate states. Lower-energy intermediate states ($E_m \lt E_n$) always lower the energy, While higher-energy ones raise it.

Higher-order corrections. The perturbation series can be extended to arbitrary order:

$E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \lambda^3 E_n^{(3)} + \cdots$

$|n\rangle = |n^{(0)}\rangle + \lambda|n^{(1)}\rangle + \lambda^2|n^{(2)}\rangle + \cdots$

The series converges if $\lambda|\langle m|\hat{H}'|n\rangle| \ll |E_n^{(0)} - E_m^{(0)}|$ for all $m \neq n$. In practice, low-order corrections often give excellent results for weak perturbations.

8.2 Degenerate Perturbation Theory

When $E_n^{(0)}$ is degenerate, the corrections are found by diagonalising the perturbation matrix in The degenerate subspace.

Theorem 8.1. The correct zeroth-order states are the eigenvectors of the matrix $W_{ij} = \langle n_i^{(0)} | \hat{H}' | n_j^{(0)} \rangle$ within the degenerate subspace.

Proof. In a $d$-dimensional degenerate subspace spanned by $\\{|n_1^{(0)}\rangle, \ldots, |n_d^{(0)}\rangle\\}$ The first-order correction to the states is undetermined by the non-degenerate formula (denominators Vanish). The correct approach is to note that $\hat{H}$ restricted to this subspace is:

$\hat{H}_{\mathrm{sub} = E_n^{(0)}\hat{I} + \lambda \hat{W}}$

Where $W_{ij} = \langle n_i^{(0)}|\hat{H}'|n_j^{(0)}\rangle$. Diagonalising $\hat{W}$ gives the correct Zeroth-order states and first-order energy splittings. $\blacksquare$

8.3 Worked Example: Perturbed Infinite Square Well

Problem. A one-dimensional infinite square well of width $L$ has a small perturbation $H' = V_0$ for $0 \lt x \lt L/2$ and $H' = 0$ for $L/2 \lt x \lt L$. Find the first-order energy Corrections.

:::caution Common Pitfall Perturbation theory assumes the perturbation is “small” compared to the level spacing. If $|\langle m | H' | n \rangle| \sim |E_n^{(0)} - E_m^{(0)}|$The perturbation series may diverge. The Method also fails for systems where the unperturbed Hamiltonian has closely spaced or degenerate Levels that are not handled correctly.

8.4 Variational Principle

Theorem 8.2 (Variational Principle). For any normalised trial state $|\phi\rangle$:

$\langle\phi|\hat{H}|\phi\rangle \geq E_0$

Where $E_0$ is the true ground state energy. The equality holds if and only if $|\phi\rangle = |0\rangle$.

Proof. Expand the trial state in the energy eigenbasis:

$|\phi\rangle = \sum_{n=0}^{\infty} c_n |n\rangle, \quad \sum_n |c_n|^2 = 1$

Then:

$\langle\phi|\hat{H}|\phi\rangle = \sum_n |c_n|^2 E_n \geq E_0\sum_n |c_n|^2 = E_0$

Since $E_n \geq E_0$ for all $n$And the inequality is strict unless $c_n = 0$ for all $n \geq 1$. $\blacksquare$

Procedure. Choose a trial wave function $\phi(x; \alpha_1, \alpha_2, \ldots)$ depending on variational Parameters $\alpha_i$. Compute $E(\alpha_i) = \langle\phi|\hat{H}|\phi\rangle / \langle\phi|\phi\rangle$ And minimise with respect to $\alpha_i$. The minimum provides an upper bound on $E_0$.

Example 8.1. Use a Gaussian trial function $\phi(x) = A\exp(-x^2/(2\alpha^2))$ to estimate the Ground state energy of the anharmonic oscillator $V(x) = \frac{1}{2}m\omega^2 x^2 + \lambda x^4$.

8.5 WKB Approximation

The WKB (Wentzel-Kramers-Brillouin) method provides approximate solutions to the one-dimensional Schrodinger equation when the potential varies slowly compared to the de Broglie wavelength.

Ansatz. Write $\psi(x) = A(x)\exp(iS(x)/\hbar)$ and substitute into the Schrodinger equation:

$-i\hbar A'S - \hbar^2 A S'' + \hbar^2 A S'^2 - A S' \cdot \hbar^2 = \mathrm{(terms)}$

Actually, substituting directly into $-\hbar^2\psi''/(2m) + V\psi = E\psi$ and separating orders of $\hbar$:

Leading order ($\hbar^0$): $S'(x) = \pm p(x) = \pm\sqrt{2m(E - V(x))}$

Next order ($\hbar^1$): $A'(x)/A(x) = -S''(x)/(2S'(x))$Giving $A(x) \propto 1/\sqrt{p(x)}$.

Therefore, in the classically allowed region ($E \gt V$):

$\psi(x) \approx \frac{C}{\sqrt{p(x)}}\cos\!\left(\frac{1}{\hbar}\int_{x_1}^x p(x')\,dx' + \frac{\pi}{4}\right)$

Where $x_1$ is a turning point ($E = V(x_1)$).

Connection formulas. At a turning point, the WKB solutions must be matched. The standard Connection formula (for a linear turning point, $V(x) \approx E + V'(x_1)(x-x_1)$) gives:

• Approaching from the classically allowed side: $\frac{2C}{\sqrt{p(x)}}\cos\!\left(\frac{1}{\hbar}\int_{x_1}^x p(x')\,dx' - \frac{\pi}{4}\right) \longleftrightarrow \frac{C}{\sqrt{|p(x)|}}\exp\!\left(-\frac{1}{\hbar}\int_x^{x_1} |p(x')|\,dx'\right)$

Quantization condition. For a potential well with turning points $x_1$ and $x_2$Applying the Connection formulas at both ends yields:

$\int_{x_1}^{x_2} p(x)\,dx = \left(n + \frac{1}{2}\right)\pi\hbar, \quad n = 0, 1, 2, \ldots$

Equivalently, using the closed phase-space integral:

$\oint p\,dx = \left(n + \frac{1}{2}\right)h$

This is the Bohr-Sommerfeld quantization condition, corrected by the $1/2$ term from the connection Formulas.

Validity. The WKB approximation requires $|d\lambda/dx| \ll 1$Where $\lambda = h/p(x)$ is the Local de Broglie wavelength. Equivalently, the change in potential over one wavelength must be small Compared to the kinetic energy: $|\hbar\,|V'(x)|/(2mp(x))| \ll 1$.

Example 8.2. Apply the WKB quantization condition to the harmonic oscillator.

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