# 433: Variation Method for the Harmonic Oscillator

This exercise deals with a variational treatment for the ground state of the simple harmonic oscillator which is, of course, an exactly soluble quantum mechanical problem.

The energy operator for a harmonic oscillator with unit effective mass and force constant is:

$$H = \frac{-1}{2} \frac{d^2}{dx^2} \blacksquare + \frac{x^2}{2} \blacksquare$$

The following trial wavefunction is selected:

$\psi (x, \beta ) = \frac{1}{1 + \beta x^2}$

The variational energy integral is evaluated (because of the symmetry of the problem it is only necessary to integrate from 0 to ∞, rather than from ‐∞ to ∞:

$$E( \beta ) = \dfrac{ \int_{0}^{ \infty} \psi (x, \beta ) \frac{-1}{2} \frac{d^2}{dx^2} \psi (x, \beta ) dx + \int_{0}^{ \infty} \psi (x, \beta ) \frac{x^2}{2} \psi (x, \beta ) dx}{ \int_{0}^{ \infty} \psi (x, \beta )^2 dx} |^{assume,~ \beta > 0}_{simplify} \rightarrow \frac{1}{4} \frac{ \beta ^2 + 2}{ \beta}$$

The energy integral is minimized with respect to the variational parameter:

$$\beta$$ := 1 $$\beta$$ := Minimize (E, $$\beta$$) $$\beta$$ = 1.414 E( $$\beta$$) = 0.707

The % error is calculated given that the exact result is $$0.50 E_h$$.

$$\frac{E( \beta ) - 0.5}{0.5} = 41.421$$%

The optimized trial wavefunction is compared with the SHO ground-state eigenfunction.

Now a second trial function is chosen:

$$\psi (x, \beta ) := \frac{1}{(1 + \beta x^2)^2}$$

Evaluate the variational energy integral:

$$E( \beta ) := \frac{ \int_{0}^{ \infty} \psi (x, \beta ) \frac{-1}{2} \frac{d^2}{dx^2} \psi (x, \beta ) dx + \int_{0}^{ \infty} \psi (x, \beta ) \frac{x^2}{2} \psi (x, \beta ) dx}{ \int_{0}^{ \infty} \psi (x, \beta )^2 dx} |^{assume,~ \beta > 0}_{simplify} \rightarrow \frac{1}{10} \frac{7 \beta ^2 + 1}{ \beta}$$

Minimize the energy integral with respect to the variational parameter:

$$\beta$$ := 1 $$\beta$$ := Minimize (E, $$\beta$$) $$\beta$$ = 0.378 E( $$\beta$$) = 0.529

Calculate the % error given that the exact result is 0.50 Eh.

$$\frac{E( \beta ) - 0.5}{0.5} = 5.83$$%

The optimized trial wavefunction is compared with the SHO ground-state eigenfunction.

Suggestion: Continue this exercise with the following trial wavefunction and interpret the improved agreement with the exact solution.

$$\psi (x, \beta ) - \frac{1}{(1 + \beta x^2)^n}$$

where n is an integer greater than 2.