Problems by Method

Use method routes when the problem’s action verb matters more than the model name. This is often the fastest way to repair a calculation habit.

Method Map

Method	Problem Route	Prerequisite Check
Normalize	Normalization Examples and Exercise Sets	measure and convention
Compute a probability	Exercises and Problems	Born rule and projectors
Compute an expectation value	Expectation-Value Examples	operator and state in same representation
Diagonalize	Matrix-Mechanics Examples	basis and Hermiticity
Apply boundary conditions	Exercise Sets	domain and matching rules
Calculate current or scattering data	Scattering Examples	flux ratios
Use perturbation theory	Perturbation-Theory Examples	degeneracy and small parameter
Trace out a subsystem	Partial Trace Exercises	tensor-factor ordering
Symmetrize or antisymmetrize	Identical-Particle Exercises	particle labels versus physical states
Build a benchmark	Benchmark Problems	reference result and convergence variable

Representative IDs

ID	Method	Canonical Page	Solution Status
QM-PROB-M101	Normalize and check dimensions	Self-Diagnostic Quiz	full solution
QM-PROB-M201	Apply Born rule in a rotated basis	Exercises and Problems	full solution
QM-PROB-M301	Match a derivative jump	Exercise Sets	partial full solution
QM-PROB-M401	Estimate a perturbative shift	Anharmonic Oscillator Perturbation	worked problem
QM-PROB-M501	Compute a reduced state	Partial Trace Exercises	full solution
QM-PROB-M601	Validate a numerical spectrum	Benchmark Problems	benchmark criteria

Method Pitfalls

• Diagonalization does not answer a probability question until the state is expanded in the measurement basis.
• A normalized wavefunction does not guarantee a self-adjoint Hamiltonian; boundary conditions still matter.
• Perturbation theory is not “small-looking algebra”; it is an expansion with a controlled parameter.
• A partial trace is not deleting indices. It is a trace over a specified tensor factor.
• A benchmark is not a plot. It is a comparison with a reference quantity under refinement.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.

Exercises

1. A problem asks for the probability that a spin prepared along $\hat z$ is found up along $\hat x$. Which method route should you use?

Solution

Use the probability and basis-change route, then the spin route if the spinor convention is unfamiliar. The main method is applying the Born rule in a different measurement basis.