← Part VI/Variational Method

6. Variational Method

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Upper bound on ground state energy through trial wave functions.

The Variational Principle

For any normalized trial wave function $|\psi\rangle$:

$$\langle\psi|\hat{H}|\psi\rangle \geq E_0$$

where $E_0$ is the true ground state energy

Equality holds only if $|\psi\rangle$ is exact ground state

Proof of Variational Principle

Expand trial function in eigenstates:

$$|\psi\rangle = \sum_n c_n|n\rangle, \quad \hat{H}|n\rangle = E_n|n\rangle, \quad E_0 \leq E_1 \leq E_2 \leq \cdots$$

Expectation value:

$$\langle\psi|\hat{H}|\psi\rangle = \sum_n |c_n|^2 E_n \geq E_0\sum_n|c_n|^2 = E_0$$

since all $E_n \geq E_0$ and $\sum_n|c_n|^2 = 1$

The Variational Method

Step 1: Choose trial wave function with parameters:

$$|\psi(\alpha_1, \alpha_2, \ldots)\rangle$$

Step 2: Compute expectation value:

$$E(\alpha_1, \alpha_2, \ldots) = \frac{\langle\psi|\hat{H}|\psi\rangle}{\langle\psi|\psi\rangle}$$

Step 3: Minimize with respect to parameters:

$$\frac{\partial E}{\partial\alpha_i} = 0$$

Step 4: Best estimate: $E_{min} \geq E_0$

Example: Ground State of Helium

Hamiltonian (neglecting electron-electron repulsion initially):

$$\hat{H} = -\frac{\hbar^2}{2m}(\nabla_1^2 + \nabla_2^2) - \frac{2e^2}{4\pi\epsilon_0}\left(\frac{1}{r_1} + \frac{1}{r_2}\right) + \frac{e^2}{4\pi\epsilon_0 r_{12}}$$

Trial wave function with effective charge $Z_{eff}$:

$$\psi(r_1, r_2) = \frac{Z_{eff}^3}{\pi a_0^3}e^{-Z_{eff}(r_1+r_2)/a_0}$$

Variational energy:

$$E(Z_{eff}) = \left(Z_{eff}^2 - 4Z_{eff} + \frac{5Z_{eff}}{4}\right) \times 13.6\text{ eV}$$

Minimize: $Z_{eff} = 2 - \frac{5}{16} = 1.6875$

Result: $E = -77.5$ eV (experimental: -79.0 eV, 2% error!)

Linear Variational Method

Expand in basis of fixed functions:

$$|\psi\rangle = \sum_{i=1}^N c_i|\phi_i\rangle$$

Variational equation becomes matrix eigenvalue problem:

$$\sum_j (H_{ij} - ES_{ij})c_j = 0$$

where $H_{ij} = \langle\phi_i|\hat{H}|\phi_j\rangle$, $S_{ij} = \langle\phi_i|\phi_j\rangle$

Lowest eigenvalue gives best variational estimate

Choice of Trial Functions

Good trial functions should:

Respect symmetries of problem (parity, angular momentum, etc.)
Satisfy boundary conditions
Have correct asymptotic behavior
Be simple enough to evaluate integrals
Have flexible parameters

Common choices:

Gaussian functions: $e^{-\alpha r^2}$
Exponential: $e^{-\alpha r}$
Polynomial × exponential

Example: Harmonic Oscillator with Gaussian

Trial function:

$$\psi(x) = Ae^{-\alpha x^2}$$

Energy functional:

$$E(\alpha) = \frac{\hbar^2\alpha}{2m} + \frac{m\omega^2}{8\alpha}$$

Minimize: $\frac{dE}{d\alpha} = 0 \Rightarrow \alpha = \frac{m\omega}{2\hbar}$

Result: $E_{min} = \frac{\hbar\omega}{2}$ - exact!

Lucky: Gaussian is exact ground state form

Excited States

To get upper bound on first excited state $E_1$:

Method 1: Enforce orthogonality to ground state:

$$\langle\psi_0|\psi_{trial}\rangle = 0$$

Method 2: Use symmetry (e.g., odd function for first excited state)

Result: $\langle\psi_{trial}|\hat{H}|\psi_{trial}\rangle \geq E_1$

Rayleigh-Ritz Method

Linear variational method with large basis:

Diagonalize Hamiltonian in finite basis
Lowest $n$ eigenvalues approximate lowest $n$ energy levels
Larger basis → better approximation
Widely used in quantum chemistry (Hartree-Fock, CI, DFT)

Advantages and Limitations

Advantages:

Always gives upper bound - systematic improvement possible
Works for any Hamiltonian (no small parameter needed)
Can handle complex many-body systems
Provides wave function, not just energy

Limitations:

Quality depends on trial function choice
No systematic way to estimate error
Best for ground state (excited states harder)
Integrals can be difficult to evaluate

← Fermi's Golden Rule Adiabatic Approximation →

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