Approximation Power Tools: Perturbation Theory and Variational Methods

Big Picture: Why We Need Approximation Tools

Step 1: Setting Up Time-Independent Perturbation Theory

Step 2: First-Order Energy Shifts and State Corrections

Example 1: Stark Shift of the Hydrogen Ground State (Selection Rules)

Try It: Perturbing a 1D Infinite Square Well

Consider a particle in a 1D infinite square well of width `L`, with walls at `x = 0` and `x = L`. The unperturbed eigenstates are

`ψn(0)(x) = sqrt(2/L) sin(nπx/L)`, with energies `En(0) = (n² π² ħ²)/(2m L²)`.

Now add a small perturbation: a weak linear potential

`H' = α x`, where `α` is a small constant.

Use first-order perturbation theory to estimate the energy shift of the ground state `n = 1`.

1. Write the formula for the first-order energy correction `E_1(1)`.
2. Evaluate the integral symbolically (you can do it on paper or in a CAS):

`E1(1) = ∫₀ᴸ ψ1(0)(x) (α x) ψ_1(0)(x) dx`

1. Check if the answer is positive or negative. What does that tell you about the average position of the particle in the well?

Hint: you will need `∫₀ᴸ sin²(πx/L) x dx`.

Pause and work this out before checking a solution in a textbook or symbolic tool. Focus on setting up the integral correctly.

Step 3: The Variational Principle (Ground States Only)

Example 2: Variational Estimate for the Hydrogen Ground State

Try It: Variational Estimate for Helium (Effective Nuclear Charge)

Helium has two electrons. The exact solution of the full interacting Hamiltonian is not analytic, so approximation is essential.

A common variational ansatz is to treat each electron as if it sees an effective nuclear charge `Z_eff` (screened by the other electron):

`ψtrial(r₁, r₂; Zeff) = (Zeff³/π) e^{-Zeff r₁} * (Zeff³/π) e^{-Zeff r₂}`

(Then antisymmetrize with spin to satisfy the Pauli principle; for the energy expectation, this simple product captures the main radial effect.)

Your task (conceptual, not full integration):

1. Explain why `Z_eff` should be less than 2.
2. Predict qualitatively what happens to the variational energy as you increase `Z_eff` from 1 toward 2.
3. Sketch on paper a rough curve `E(Zeff)` vs `Zeff` showing a minimum.

Hint: Larger `Z_eff` pulls electrons closer (lower potential energy) but increases kinetic energy. The minimum balances these.

This is a simplified picture behind more sophisticated methods used in modern electronic-structure codes.

Optional: Numerically Minimizing a Variational Energy in Python

You can easily explore variational energies numerically. Here is a minimal Python example for the hydrogen-like trial wavefunction from Example 2.

We will:

• Define `E(α) = α2 / 2 - α` (in atomic units).
• Scan over a range of `α` values.
• Find the minimum numerically.

```python

import numpy as np

Define the variational energy function for hydrogen (atomic units)

def E(alpha):

return 0.5 alpha*2 - alpha

Scan a range of alpha values

alphas = np.linspace(0.1, 3.0, 200)

energies = E(alphas)

Find minimum

min_index = np.argmin(energies)

alphabest = alphas[minindex]

Ebest = energies[minindex]

print(f"Best alpha ≈ {alpha_best:.3f}")

print(f"Minimum energy ≈ {E_best:.3f} Hartree")

If you have matplotlib, you can also plot the curve:

try:

import matplotlib.pyplot as plt

plt.plot(alphas, energies)

plt.axvline(alpha_best, color='r', linestyle='--', label='Best alpha')

plt.xlabel('alpha')

plt.ylabel('E(alpha) [Hartree]')

plt.legend()

plt.show()

except ImportError:

print("matplotlib not installed; skipping plot.")

```

Quick Check: Perturbation vs Variational

Test your understanding of when to use each method.

Key Term Review

Flip through these cards to reinforce the core concepts.