Worked Examples, Exercises, and Notebooks

This chapter helps you practice the methods used throughout wave mechanics. Choose a worked example when you need to see a calculation unfold, an exercise when you want to reconstruct it yourself, or a notebook when you want to explore how the answer changes with parameters.

Turning a model into a solved problem

Start with a question you can answer quantitatively: an allowed energy, a probability in a region, a transmitted flux, or the width of a moving packet. Write down the potential, coordinate range, and boundary conditions before choosing a technique.

For example, the infinite well on $0<x<L$ satisfies

$$-\frac{\hbar^2}{2m}\psi''=E\psi, \qquad \psi(0)=\psi(L)=0.$$

For positive energy the local solution is $A\sin(kx)+B\cos(kx)$. The left wall removes $B$; the right wall requires $kL=n\pi$. Normalization then gives

$$\psi_n(x)=\sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}, \qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2},\qquad n\geq1.$$

Each step answers a different part of the problem: the differential equation gives local shapes, boundaries select wavelengths, and normalization fixes amplitude. The full well calculation explains the remaining spectral and physical details.

When you get stuck, identify the missing move. Piecewise potentials suggest matching; an even potential suggests parity; a central potential suggests angular and radial separation; reflection and transmission require current. The problem-solving patterns below explain when each method applies.

Reading sequence

Page	What you will learn
Worked Examples Index	Task-oriented route map to examples distributed across the volume
Exercise Sets	Practice from introductory calculations to more demanding extensions
Problem-Solving Patterns	Matching, parity, scaling, current, asymptotics, separation, gauge, and limits
Numerical Notebooks Index	Calculations to rerun, required software, expected outputs, and checks
Benchmark Problems	Known answers and refinement studies for testing numerical calculations
Visualization Gallery	Figures illustrating wavefunctions, probabilities, motion, and model comparisons

Practicing effectively

Try an exercise before opening its solution. Compare the reasoning as well as the final formula, then change one assumption: use a finite wall, alter a mass, or take a limiting case. Being able to explain why a familiar method stops working is part of understanding it.

For numerical work, begin with a model whose answer you can independently calculate. Vary the grid spacing, basis size, box size, or time step according to the suspected error. Norm conservation is useful, but a simulation can preserve norm while accumulating the wrong phase or spreading at the wrong rate. Compare the observable you actually want to predict.

Exercises

A finite-difference energy calculation has relative errors $8\times10^{-4}$ at spacing $h$ and $2\times10^{-4}$ at spacing $h/2$. What convergence order do these two results suggest?

Solution

If the leading error is $Ch^p$, halving the spacing reduces it by $2^p$. The observed ratio is four, suggesting $p=2$. More refinements are needed to establish a consistent trend; a single pair does not rule out cancellation or another source of error.

A normalized wave packet scatters from a barrier. At a finite observation time the integrated probabilities well to the left and right sum to $0.97$. Must the simulation have lost probability?

Solution

No. The remaining $0.03$ may still lie near the barrier or outside the chosen left and right windows. Integrate over the whole domain and inspect the packet separation. For a real potential with norm-preserving boundaries, total probability should remain one. An absorber or probability flux through an open boundary requires separate accounting. Late-time reflected and transmitted probabilities should be assigned only after the relevant packets have separated.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.
• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.