WKB Versus Exact Spectrum Notebook

This notebook compares WKB quantization with numerical eigenvalues for a smooth one-dimensional potential well. It is designed to show where Bohr-Sommerfeld quantization is exact, where it is asymptotically accurate, and where low-lying states expose its limitations. WKB Bound States in a Smooth Potential gives the complementary analytic calculation for the pure quartic limit; this notebook owns the quadratic-plus-quartic parameter sweep and numerical convergence workflow.

Physical Problem

Use dimensionless units with

$$\hbar=m=\omega=1$$

and study the smooth confining potential

$$V(x) = \frac12x^2+gx^4, \qquad g\ge0.$$

The harmonic case $g=0$ is a validation problem because the leading WKB quantization rule gives the exact spectrum. The anharmonic case tests WKB away from that special solvable limit.

Analytic Target

For a bound state between two turning points $x_-(E)$ and $x_+(E)$, WKB gives

$$\int_{x_-(E)}^{x_+(E)} dx\, \sqrt{2(E-V(x))} = \pi \left( n+\frac12 \right).$$

For each $n$, solve this equation for $E_n^{\mathrm{WKB}}$.

For $g=0$, the integral gives

$$E_n^{\mathrm{WKB}} = n+\frac12,$$

which should match the exact harmonic oscillator spectrum.

Numerical Spectrum

Compute reference eigenvalues either by finite differences on a large box or by diagonalization in a harmonic-oscillator basis. The basis method is natural here because the potential is polynomial.

For a finite-difference check, choose a box

$$-L\le x\le L$$

with Dirichlet boundaries far enough into the forbidden region. Refine the grid spacing $\Delta x$ and increase $L$ until the low-lying eigenvalues are stable.

For a basis calculation, construct the Hamiltonian matrix in oscillator states and increase the basis size $N$ until the same stability is reached.

WKB Integral

For a trial energy $E$, find the turning points by solving

$$V(x)=E.$$

For the even potential used here,

$$x_+(E)>0, \qquad x_-(E)=-x_+(E).$$

The WKB action is

$$I(E) = 2 \int_0^{x_+(E)} dx\, \sqrt{2(E-V(x))}.$$

Then solve

$$I(E) = \pi \left( n+\frac12 \right).$$

The integrand has square-root behavior at the turning point. Use an integration method stable for endpoint square-root singularities, or change variables to smooth the endpoint.

Parameter Sweep

Use representative couplings such as

$$g=0,\quad 0.01,\quad 0.1,\quad 1.$$

For each $g$, compare the first several levels:

$$n=0,1,\ldots,20$$

when the numerical method is converged for those levels.

WKB should generally improve with increasing $n$ because the classical action grows compared with $\hbar$.

Expected Figures

The notebook should generate:

• numerical and WKB energies versus $n$ for each $g$;
• relative error versus $n$;
• WKB action $I(E)$ as a monotone function of $E$;
• convergence of selected numerical eigenvalues with grid size or basis size.

The relative-error plot should not hide sign. Plot either signed fractional error or absolute fractional error with a clear label.

Validation Checks

For $g=0$, WKB and exact numerical energies should agree with

$$E_n=n+\frac12$$

within numerical tolerance.

The WKB action must be monotone:

$$\frac{dI}{dE}>0.$$

Numerical eigenvectors should have the correct node count: the $n$th one-dimensional bound state has $n$ nodes.

The finite box should not affect the low-lying states. Increase $L$ and verify stability.

Known Failure Modes

• Using the smooth-turning-point WKB rule for hard walls or discontinuous potentials.
• Under-resolving the turning-point integral.
• Mistaking finite-box error for WKB error.
• Comparing WKB to high numerical eigenvalues near the grid cutoff.
• Expecting leading WKB to be highly accurate for the ground state in a generic potential.

Reproducibility Metadata

Record:

• potential and dimensionless units;
• numerical method, grid spacing or basis size;
• box size if using finite differences;
• quadrature method and tolerance;
• root-finding tolerance for WKB energies;
• number of levels retained;
• eigenvalue residual checks;
• software versions.

Cross-Links

• Computational Notebooks
• WKB Bound States in a Smooth Potential
• Bohr-Sommerfeld Quantization
• Turning Points and Connection Formulas
• Anharmonic Oscillator
• Quantum Harmonic Oscillator
• Matrix Diagonalization

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.

Exercises

1. Why is the harmonic oscillator a useful validation case for this notebook?

Solution

For the harmonic oscillator, the leading Bohr-Sommerfeld rule with the turning-point correction gives the exact energies

$$E_n=n+\frac12$$

in the chosen units. Therefore disagreement at $g=0$ indicates a numerical, integration, or convention error rather than a WKB limitation.

2. Why should WKB generally improve for larger $n$?

Solution

Larger $n$ corresponds to larger classical action. WKB is an expansion in the smallness of $\hbar$ compared with the action scale. As the action grows, the phase varies rapidly and the semiclassical approximation typically becomes more accurate, away from singularities and special boundary effects.