Approximation Methods | Physics

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.

Solution

The unperturbed states are $\phi_n^{(0)}(x) = \sqrt{2/L}\sin(n\pi x/L)$ .

$E_n^{(1)} = \langle n^{(0)} | H' | n^{(0)} \rangle = \int_0^{L/2} V_0 \frac{2}{L}\sin^2\!\left(\frac{n\pi x}{L}\right) dx$

$= \frac{2V_0}{L}\int_0^{L/2} \frac{1 - \cos(2n\pi x/L)}{2}\, dx = \frac{V_0}{L}\!\left[\frac{L}{2} - \frac{L}{4n\pi}\sin(n\pi)\right] = \frac{V_0}{2}$

The first-order correction is $E_n^{(1)} = V_0/2$ for all $n$ . $\blacksquare$

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 $V(x) = \frac{1}{2}m\omega^2 x^2 + \lambda x^4$ .

Solution

The normalised Gaussian is $\phi(x) = (\pi\alpha^2)^{-1/4}\exp(-x^2/(2\alpha^2))$ with $\langle x^2 \rangle = \alpha^2/2$ and $\langle x^4 \rangle = 3\alpha^4/4$ .

$\langle T \rangle = \frac{\langle p^2 \rangle}{2m} = \frac{\hbar^2}{4m\alpha^2}$

(by using $p = -i\hbar\,d/dx$ and integrating by parts).

$\langle V \rangle = \frac{1}{2}m\omega^2\langle x^2 \rangle + \lambda\langle x^4 \rangle = \frac{m\omega^2\alpha^2}{4} + \frac{3\lambda\alpha^4}{4}$

$E(\alpha) = \frac{\hbar^2}{4m\alpha^2} + \frac{m\omega^2\alpha^2}{4} + \frac{3\lambda\alpha^4}{4}$

Minimising: $dE/d\alpha = 0$ gives $-\hbar^2/(2m\alpha^3) + m\omega^2\alpha/2 + 3\lambda\alpha^3 = 0$ .

For $\lambda = 0$ (harmonic oscillator), this gives $\alpha^2 = \hbar/(m\omega)$ and $E = \hbar\omega/2$ Which is exact. For small $\lambda$ Expand $\alpha^2 = \hbar/(m\omega)(1 - \delta)$ :

$\alpha^2 \approx \frac{\hbar}{m\omega}\!\left(1 - \frac{3\lambda\hbar}{2m^2\omega^3}\right)$

$E_{\mathrm{var} \approx \frac{\hbar\omega}{2}\!\left(1 + \frac{3\lambda\hbar}{4m^2\omega^3}\right)}$

This agrees with second-order perturbation theory to first order in $\lambda$ .

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.

Solution

For $V(x) = \frac{1}{2}m\omega^2 x^2$ The turning points are at $x_{1,2} = \pm\sqrt{2E/(m\omega^2)}$ .

$\int_{x_1}^{x_2} \sqrt{2m\!\left(E - \frac{1}{2}m\omega^2 x^2\right)}\,dx = \sqrt{2mE}\int_{-a}^{a}\sqrt{1 - (x/a)^2}\,dx$

Where $a = \sqrt{2E/(m\omega^2)}$ . The integral equals $\pi a/2$ times $\sqrt{2mE}$ :

$= \sqrt{2mE}\cdot\frac{\pi a}{2} = \frac{\pi E}{\omega}$

Setting this equal to $(n + 1/2)\pi\hbar$ :

$\frac{\pi E}{\omega} = \left(n + \frac{1}{2}\right)\pi\hbar \implies E_n = \left(n + \frac{1}{2}\right)\hbar\omega$

This is the exact result! The WKB approximation gives the exact energy levels for the harmonic Oscillator because the potential is quadratic, so the connection formulas are exact.

Validity and limitations. The WKB method fails near turning points (where $E = V(x)$ ) because $p(x) \to 0$ and the local wavelength diverges. The linear approximation of the potential near Turning points (used to derive the connection formulas) breaks down when the potential is not smooth. The method also fails for potentials with discontinuities or cusps.

Despite these limitations, WKB is remarkably useful for estimating energy levels in potentials Where exact solutions are not available, and it forms the basis of the JWKB approximation in Scattering theory.

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    A[8_Approximation Methods] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Intuition

Approximation methods tackle quantum problems that cannot be solved exactly. Perturbation theory treats a complicated Hamiltonian as a simple known part plus a small correction, then systematically calculates how the correction shifts energy levels and wavefunctions. The key assumption is that the correction is small compared to the energy spacing of the unperturbed system. The variational method provides an upper bound on the ground state energy by testing trial wavefunctions and minimizing the expectation value. WKB approximates the wavefunction as a locally oscillating function whose wavelength varies with position, working best when the potential changes slowly compared to the wavelength. These methods turn intractable problems into calculable ones.

Cross-References

Wave Functions and the Schrodinger Equation: The Schrodinger equation provides the foundation for approximation methods.
One-Dimensional Problems: Exactly solvable 1D problems provide test cases for approximation methods.
Variational Methods: The variational principle provides a systematic approach to estimating ground state energies.
Calculus
Linear Algebra
Quantum Computing

Common Mistakes

Using perturbation theory when the perturbation is not small: Perturbation theory assumes $V' \ll E_n - E_m$ . If the perturbation is comparable to energy spacings, the series diverges and the results are meaningless. Check the validity condition before applying.
Forgetting that the variational method gives an upper bound, not the exact energy: The variational principle states $E_0 \leq \langle \psi | H | \psi \rangle$ for any trial state. A lower trial energy is always better, but you never know how close you are to the true ground state energy.
Neglecting degenerate perturbation theory when eigenvalues are degenerate: Non-degenerate perturbation theory gives infinite corrections when two unperturbed states have the same energy. You must diagonalise the perturbation matrix in the degenerate subspace first.
Applying WKB near classical turning points: The WKB approximation breaks down where $p(x) = 0$ (turning points) because the semiclassical wavefunction diverges. Use connection formulas to match WKB solutions across turning points.

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