Examples by Method

Use method routes when you know what kind of calculation is needed but can choose among several model systems. This page points to the most transferable example families.

Method Map

Method	Example Route	Canonical Check
Normalize a state	Normalization Examples	Normalization Formula
Compute an expectation value	Expectation-Value Examples	Expectation Value Formula
Diagonalize a matrix Hamiltonian	Matrix-Mechanics Examples	Diagonalization
Use spinor algebra	Spin Examples	Spin-Half Matrices
Add angular momenta	Spin Examples	Addition of Angular Momentum
Apply perturbation theory	Perturbation-Theory Examples	First-Order Perturbation Theory
Compute scattering data	Scattering Examples	Scattering Cross Section
Trace out a subsystem	Density-Matrix Examples	Partial Trace

Method Before Formula

The formula card is usually too compressed to teach the method. For example, the first-order energy shift

$$E_n^{(1)}=\langle n^{(0)}\rvert V\lvert n^{(0)}\rangle$$

is only usable after checking that the unperturbed state is normalized, nondegenerate or treated in the correct degenerate subspace, and paired with a small parameter. The example route should expose those decisions.

Common Method Confusions

• Normalization of a probability density is not the same as normalization of a scattering amplitude.
• Diagonalizing a Hamiltonian is not enough; measurement probabilities require expanding the state in the eigenbasis.
• A spin measurement along $\hat z$ and one along $\hat x$ use different projectors.
• Perturbation theory requires a controlled expansion parameter, not just a hard exact problem.
• A partial trace requires a tensor-factor convention before any indices are summed.

Practice Sequence

1. Start with Normalization Examples and Expectation-Value Examples.
2. Move to Matrix-Mechanics Examples and Spin Examples for finite-dimensional bookkeeping.
3. Use Perturbation-Theory Examples after exact eigenvalue examples are comfortable.
4. Use Scattering Examples and Density-Matrix Examples once currents and traces are no longer mysterious.

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 of measuring spin up along $\hat x$ after preparing spin up along $\hat z$. Should you begin with a matrix-diagonalization example or a spin example?

Solution

Begin with Spin Examples. The calculation uses finite-dimensional vectors, but the central issue is the relation between measurement axes and spinor bases.