Quantum Symmetries & Applications

When a Hamiltonian $\hat{H} = \hat{H}_0 + \hat{H}'$ can be split into a solvable part $\hat{H}_0$ and a small perturbation $\hat{H}'$, we use perturbation theory to find corrections to the energy levels and wave functions.

For a nondegenerate unperturbed state $|n^{(0)}\rangle$ with energy $E_n^{(0)}$, the first-order energy correction is:

The first-order correction to the wave function is:

The second-order energy correction is:

Perturbation theory is an expansion in powers of $\hat{H}'/\Delta E$, where $\Delta E$ is the level spacing. It converges when the perturbation is small compared to the gaps between unperturbed energy levels.

When the unperturbed energy level $E_n^{(0)}$ is degenerate, the standard perturbation theory breaks down because the denominators $E_n^{(0)} - E_m^{(0)}$ vanish for states in the degenerate subspace.

The resolution is to diagonalize the perturbation within the degenerate subspace. Let $\{|k\rangle\}$ be the degenerate eigenstates of $\hat{H}_0$ with energy $E$. We form the matrix $W_{ij} = \langle i|\hat{H}'|j\rangle$ and diagonalize it.

The eigenvalues of $W$ are the first-order energy corrections, and the eigenvectors are the 'good' states — the correct zero-order states that diagonalize the perturbation. After finding the good states, we can apply nondegenerate perturbation theory to find higher-order corrections.

The key insight is that the perturbation typically lifts the degeneracy, splitting the energy level into distinct levels. The size of the splitting depends on the matrix elements of $\hat{H}'$ in the degenerate subspace.

The variational principle provides an upper bound on the ground state energy of any quantum system. For any normalized trial state $|\psi\rangle$, the expectation value of the Hamiltonian satisfies:

The proof is simple: expand $|\psi\rangle$ in the energy eigenbasis: $|\psi\rangle = \sum_n c_n |n\rangle$. Then $\langle\psi|\hat{H}|\psi\rangle = \sum_n |c_n|^2 E_n \geq E_0 \sum_n |c_n|^2 = E_0$, since $E_n \geq E_0$ for all $n$.

To use the variational principle, we choose a trial wave function $\psi_{\text{trial}}(x; \alpha)$ with one or more variational parameters $\alpha$. We compute $E(\alpha) = \langle\psi_{\text{trial}}|\hat{H}|\psi_{\text{trial}}\rangle$ and minimize with respect to $\alpha$. The minimum value is our best upper bound on $E_0$.

The quality of the bound depends on how well the trial function captures the physics of the true ground state. A cleverly chosen trial function with few parameters can give excellent results.

The helium atom Hamiltonian cannot be solved exactly because of the electron-electron repulsion. Perturbation theory and the variational principle provide accurate approximations.

First-order perturbation theory treats the electron-electron repulsion as a perturbation. The unperturbed ground state energy is $E_0^{(0)} = -108.8\text{ eV}$ (two non-interacting electrons in a $Z=2$ Coulomb field). The first-order correction is:

Evaluating the integral gives $K \approx 34\text{ eV}$, so $E_{\text{gs}} \approx -108.8 + 34 = -74.8\text{ eV}$. The experimental value is $-79.0\text{ eV}$, so first-order perturbation theory gives about 5% error.

The variational principle does better. Using a trial function with effective nuclear charge $Z_{\text{eff}}$ as a variational parameter accounts for screening. The optimal value is $Z_{\text{eff}} \approx 27/16 \approx 1.69$, giving $E \approx -77.5\text{ eV}$, much closer to the experimental value.

The principles developed in this curriculum form the foundation of all quantum physics. The infinite square well and harmonic oscillator provide model systems whose solutions appear repeatedly. The hydrogen atom explains atomic structure and spectroscopy.

Identical particles and exchange symmetry explain the periodic table, chemical bonding, and the properties of solids. The Pauli exclusion principle forces electrons to occupy successively higher energy levels, creating the shell structure of atoms.

Perturbation theory and the variational principle are essential tools for treating realistic systems that cannot be solved exactly. They are used throughout atomic, molecular, nuclear, and condensed matter physics.

Beyond the material covered here, quantum mechanics extends to: time-dependent perturbation theory and quantum transitions; scattering theory; relativistic quantum mechanics and quantum field theory; and many-body quantum systems. Each of these builds on the foundations laid in this curriculum.

Use a Gaussian trial function $\psi(x) = (2\beta/\pi)^{1/4} e^{-\beta x^2}$ to estimate the ground state energy of the harmonic oscillator $H = p^2/(2m) + (1/2)m\omega^2 x^2$.

Step 1 — $\langle T \rangle$. For a Gaussian, $\langle p^2 \rangle = \hbar^2 \beta$. Thus $\langle T \rangle = \hbar^2 \beta / (2m)$.

Step 2 — $\langle V \rangle$. For a Gaussian, $\langle x^2 \rangle = 1/(4\beta)$. Thus $\langle V \rangle = (1/2)m\omega^2/(4\beta) = m\omega^2/(8\beta)$.

Step 3 — Total energy. $E(\beta) = \hbar^2 \beta/(2m) + m\omega^2/(8\beta)$.

Step 4 — Minimize. $dE/d\beta = \hbar^2/(2m) - m\omega^2/(8\beta^2) = 0$. Solving: $\beta^2 = m^2\omega^2/(4\hbar^2)$, so $\beta = m\omega/(2\hbar)$.

Step 5 — Minimum energy. $E_{\min} = \hbar^2(m\omega/2\hbar)/(2m) + m\omega^2/(8 \cdot m\omega/2\hbar) = \hbar\omega/4 + \hbar\omega/4 = \hbar\omega/2$. This is exactly the ground state energy! The Gaussian trial function happened to be the exact ground state for the harmonic oscillator.