Approximation Methods

Recall prompts

Attempt these from memory before reading.

Write the first-order energy correction in non-degenerate perturbation theory.
What goes wrong when the unperturbed level is degenerate?
What does the variational principle give you, and what does it not?
What is the WKB validity condition near a turning point?
Which of these three methods gives an exact answer, and when?

Check your answers against the text. Prompt 2 is the one that separates a procedure from an understanding: degenerate perturbation theory is not non-degenerate theory with extra steps, it is a different problem.

Motivation: almost nothing in quantum mechanics is exactly soluble

Exactly soluble problems form a short list: the square well, the harmonic oscillator, the hydrogen atom, and a handful of others. Everything else — any atom with two electrons, any anharmonic oscillator, any real molecule — has no closed-form solution.

So the subject is organised around three ways of getting an answer anyway, and they are worth distinguishing because they give different kinds of answer:

Perturbation theory assumes the problem is close to one you can solve, and writes the answer as a series in a small parameter. It gives numbers, and it fails when the parameter is not small.
The variational principle makes a guess and proves the guess is an upper bound on the ground-state energy. It gives a bound, always valid, never exact.
WKB assumes the potential varies slowly compared with the wavelength. It gives the exponential behaviour that dominates tunnelling, and it breaks exactly where the physics is most interesting — at turning points.

None of these is a lesser version of the others. Perturbation theory is how the Lamb shift was predicted; the variational principle is how the ground-state energy of helium is bounded to high accuracy; WKB is why tunnelling has an exponential law at all.

Degeneracy is a separate case, not a special case. In non-degenerate theory the first-order correction is $\langle n^{(0)}|\hat{V}|n^{(0)}\rangle$ . That expression assumes $|n^{(0)}\rangle$ is uniquely determined, and a degenerate level has no unique choice. The fix is to diagonalise $\hat{V}$ inside the degenerate subspace first — which is not a patch on the method but the statement that the perturbation chooses which basis the problem wants.

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.

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.

Counterexamples worth knowing by name

Claim	True?	Counterexample
Perturbation theory always converges	no	the series is asymptotic; for most potentials it diverges for every nonzero coupling, and is still useful
The variational method gives the ground-state energy	no	it gives an upper bound, and without knowing the exact value you cannot say how far off it is
A good trial wave function needs to be close to the truth	no	the method only cares about the energy expectation; a poor trial function can give a good energy and bad everything else
Degenerate perturbation theory is optional	no	using the non-degenerate formula on a degenerate level divides by zero
WKB gives good wave functions everywhere	no	it fails at turning points, which is precisely where the connection formulae are needed
The variational principle works for excited states	no, not directly	any trial function orthogonal to the exact ground state bounds the first excited state; but you do not know the exact ground state
First-order perturbation theory is always the dominant correction	no	if the matrix element vanishes by symmetry, second order dominates
WKB counts bound states	yes	$\oint p\,dx = 2\pi\hbar(n + \tfrac12)$ reproduces the harmonic oscillator and square-well spectra exactly

The first row is the one that generalises furthest and surprises people most. Asymptotic series get better for the first few terms and then diverge, and the practical skill is knowing when to stop adding terms.

The last row is worth internalising as a check: the WKB quantisation condition reproduces the harmonic oscillator spectrum with no error at all, which is a good indication that the method is capturing something structural rather than approximating.

Where the research frontier is

Beyond Hartree-Fock. Coupled cluster and full configuration interaction are the systematic versions of perturbation theory, and their cost grows exponentially with system size.
Density functional theory. The exact functional exists but is unknown; every practical calculation uses an approximation to it, and improving the functional is an open problem.
Semiclassical methods. WKB and its modern descendants are used where the potential is smooth and the system is large; the mismatch with quantum behaviour at caustics is the active question.
Tensor networks and quantum computing. Both are attempts to get many-body answers without the exponential cost, and neither is a replacement for perturbation theory — they are a different kind of answer.
Renormalisation. Perturbation theory around an interacting fixed point requires reorganising the expansion, which is what the renormalisation group does; how that interacts with the asymptotic character of the series is the open question.

Interleaving problems

Mix these with one-dimensional problems, angular momentum, operators, Real Analysis and Optimisation.

(Operators.) Derive the first-order energy correction $E_n^{(1)} = \langle n^{(0)}|\hat{V}|n^{(0)}\rangle$ by inserting the perturbation expansion into the Schrodinger equation and matching powers of $\lambda$ . Identify the step that requires the level to be non-degenerate.
(Linear algebra.) Show that diagonalising $\hat{V}$ inside a degenerate subspace is the same as diagonalising the perturbation matrix $V_{ij} = \langle i^{(0)}|\hat{V}|j^{(0)}\rangle$ , and explain why the resulting basis is the one in which the perturbation is “already diagonal”.
(Real analysis.) Show that the perturbation series for the quartic oscillator diverges for every nonzero coupling, by estimating the growth of the coefficients. Then explain in what sense the series is still useful.
(One-dimensional problems.) Apply the variational method to the infinite square well with a trial function $\psi = x(L-x)$ and compare the resulting energy with the exact value. How much of the error comes from the trial function’s shape rather than its normalisation?
(WKB.) Derive the WKB quantisation condition $\oint p\,dx = 2\pi\hbar(n + \tfrac12)$ for the harmonic oscillator and confirm it reproduces the exact spectrum. Then explain where the extra $\tfrac12$ comes from.
(Research.) Density functional theory claims that the ground-state density determines everything. State the Hohenberg-Kohn theorem precisely, then explain why it is an existence result rather than a method — and what the practical gap is.

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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Approximation Methods

Recall prompts

Motivation: almost nothing in quantum mechanics is exactly soluble

8.1 Time-Independent Perturbation Theory

8.2 Degenerate Perturbation Theory

8.3 Worked Example: Perturbed Infinite Square Well

Counterexamples worth knowing by name

Where the research frontier is

Interleaving problems

Cross-References

Common Mistakes

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