Glossary

For the energy, to first order, you can assume that the wavefunction is unchanged and evalulate the energy shift from the matrix element

\[\Delta E_0 \approx \langle n_0|\hat{V} |n_0 \rangle \equiv V_{00} \nonumber\]

For the wavefunctions, the amount of other unperturbed wavefunction mixed in depends on the matrix element between them, and their energy difference.

\[|\phi_0 \rangle = |n_0 \rangle + \sum_{k \neq 0} \frac{\langle n_k|\hat{V} |n_0 \rangle}{(E_0 − E_k)} |n_k \rangle \equiv |n_0 \rangle + \sum_{k \neq 0} \frac{V_{k0}}{(E_0 − E_k)} |n_k \rangle \nonumber\]

For the energy, to second order, we consider mixing of the wavefunctions:

\[\Delta E_{0} = 0+\sum_{i=1, \infty} \langle n_{i}|\hat{V}| n_{0} \rangle \frac{V_{0 i}}{\left(E_{0}-E_{i}\right)} = \sum_{i=1, \infty} \frac{\left|V_{i 0}\right|^{2}}{\left(E_{0}-E_{i}\right)} \nonumber\]

The perturbed wavefunction is a linear combination of the unperturbed, degenerate states. Calculate eigenstates and eigenvalues from the matrix:

\[\begin{vmatrix}
V_{11}-\Delta E_{j} & V_{12} & \ldots & V_{1 N} \\ V_{21} & V_{22}-\Delta E_{j} & \ldots & V_{2 N} \\ \ldots & \ldots & \ldots & \ldots \\ V_{N 1} & V_{N 2} & \ldots & V_{N N}-\Delta E_{j} \end{vmatrix} = 0 \nonumber\]

With every degeneracy there is an associated symmetry and conservation law.

If a system starts in state \(|k\rangle , (c_m(t) = 0)\) the probability that it will shift to state \(|k\rangle\) in time \(t\) depends on the matrix element

\[c_m(t) = (i\hbar )^{−1} \int^t_0 V_{mk} \text{ exp}(i\omega_{mk}t) dt \nonumber\]

For a harmonic perturbation we get a resonance when the perturbing frequency matches the energy difference between the states \(m\) and \(k\):

\[|c_{m}(t)|^{2}=\frac{V_{m k}^{2} \sin ^{2}\left[\left(\omega_{m k}-\omega\right) t / 2\right]}{\hbar^{2}\left(\omega_{m k}-\omega\right)^{2}}=\frac{4}{\hbar^{2}} V_{m k}^{2} f\left(t, \omega_{m k}-\omega\right) \nonumber\]

Energy is extracted from the perturbing potential.

If the transition is to a group of near-degenerate states, The transition rate depends on the square of the matrix element, and the density of states, as given by Fermi’s Golden rule:

\[R = \frac{2\pi}{\hbar} \left[ |V_{mk}|^2 g(E_m) \right]_{E_m=E_k} \nonumber\]

A surprisingly good estimate of the ground state energy comes from minimising the expectation value of the energy for an arbitrary, normalised trial wavefunction \(\phi (a_n)\).

\[E[a_n] \approx M in_{a_n} \left( \frac{\langle \phi (a_n)|\hat{H} |\phi (a_n)\rangle}{\langle \phi (a_n)|\phi (a_n)\rangle} \right) \nonumber\]

Excited states can also be found using trial wavefunctions orthogonal to the ground state.

At high energy, scattering can be treated by perturbation theory. The scattered wave depends on a matrix element which is the Fourier transform of the potential.

\[\langle {\bf k'} |\hat{V} |{\bf k} \rangle = \int \int \int V ({\bf r}) \text{ exp }(−i\chi.{\bf r}) d\tau ; \quad \chi \equiv {\bf k'} − {\bf k} \nonumber\]

For elastic scattering from a central potential this becomes:

\[\frac{d\sigma}{d\Omega} = \frac{m^2}{(k \sin \frac{\theta}{2} )^2 \hbar^4} \left| \int^{\infty}_{0} rV (r) \sin(2kr \sin \frac{\theta}{2} )dr \right|^2 \nonumber\]