For $H=H_0+\lambda V$, nondegenerate perturbation theory expands $E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2 E_n^{(2)}+\cdots$. First-order shifts are $E_n^{(1)}=\langle n^{(0)}|V|n^{(0)}\rangle$.

When a subspace of $H_0$ is degenerate, one must diagonalize $V$ within that subspace. This determines the correct zeroth-order basis and explains level splitting in atoms, molecules, and solids.

Given a normalized trial state $|\psi(\alpha)\rangle$, one minimizes $E(\alpha)=\langle\psi(\alpha)|H|\psi(\alpha)\rangle$. Sophisticated ansatz design is crucial in many-body theory, quantum chemistry, and tensor networks.

In the semiclassical regime, the wavefunction has the form $\psi(x)\sim A(x)e^{\frac{i}{\hbar}S(x)}$. The leading order yields the Hamilton–Jacobi equation, while quantization conditions emerge from matching across turning points.

For a weak drive $V(t)$, transition amplitudes are computed with Dyson series or interaction-picture evolution. Fermi's golden rule gives $\Gamma_{i\to f}=\dfrac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)$ in the appropriate continuum limit.