10.9: Momentum-Space Variation Method for the Quartic Oscillator

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For unit mass the quartic oscillator has the following energy operator in atomic units in coordinate space.

\[ H = - \frac{1}{2} \frac{d^{2}}{dx^{2}} \nonumber \]

Suggested trial wavefunction:

\[ \psi (x, \beta ) = ( \frac{2 \beta}{ \pi})^{ \frac{1}{4}} exp( - \beta x^{2}) \nonumber \]

Demonstrate that the wavefunction is normalized.

\[ \int_{- \infty}^{ \infty} \psi (x, \beta )^{2} dx~assume, \beta >0 \rightarrow 1 \nonumber \]

Fourier transform the coordinate wavefunction into the momentum representation.

\[ \Phi (p, \beta ) = \frac{1}{ \sqrt{2 \pi}} \int_{- \infty}^{ \infty} exp (-i p x) \psi (x, \beta ) dx |_{simplify}^{assume,~ \beta > 1} \rightarrow \frac{1}{2} \frac{2^{ \frac{3}{4}}}{ \pi ^{ \frac{1}{4}}} \frac{ e^{ \frac{-1}{4} \frac{p^{2}}{ \beta}}}{ \beta ^{ \frac{1}{4}}} \nonumber \]

Demonstrate that the momentum wavefunction is normalized.

\[ \int_{- \infty}^{ \infty} \overline{ \Phi (p, \beta )} \Phi (p, \beta ) dp assume,~ \beta > 0 \rightarrow 1 \nonumber \]

The quartic oscillator energy operator in momentum space:

\[ H = \frac{p^{2}}{2} \blacksquare + \frac{d^{4}}{dp^{4}} \blacksquare \nonumber \]

Evaluate the variational energy integral.

\[ E ( \beta ) = \int_{- \infty}^{ \infty} \overline{ \Phi(p, \beta )} \frac{p^2}{2} \Phi (p, \beta ) dp + \int_{- \infty}^{ \infty} \overline{ \Phi (p, \beta )} \frac{d^{4}}{dp^4} \Phi (p, \beta ) dp |_{simplify}^{assume,~ \beta > 0} \rightarrow \frac{1}{16} \frac{8 \beta ^{3} + 3}{ \beta ^{2}} \nonumber \]

Minimize the energy with respect to the variational parameter \( \beta\) and report its optimum value and the ground-state energy.

\( \beta = 1\) \( \beta = Minimize (E, \beta )\) \( \beta = 0.90856\) \( E ( \beta ) = 0.688142\)

Plot the coordinate and momentum wavefunctions and the potential energy on the same graph.

Screen Shot 2019-02-11 at 12.25.15 PM.png

These results demonstrate the uncertainty principle. For the harmonic potential, x²/2, the coordinate and momentum wavefunctions are identical. Compared to the harmonic potential the quartic potential, x⁴, constrains the spatial wavefunction leading to less uncertainty in position. The uncertainty principle, therefore, requires an increase in the momentum uncertainty. This is clearly revealed in graph above.