# 447: Momentum-Space Variation Method for Particle in a Gravitational Field

The following problem deals with a particle of unit mass in a gravitational field with acceleration due to gravity equal to 1.

Energy operator for particles near Earth's surface:

$\frac{-1}{2 \mu} \frac{d^2}{dz^2} \blacksquare + z \blacksquare$

Trial wave function:

$\Psi ( \alpha , z) := 2 \alpha ^{ \frac{3}{2}} z exp(- \alpha z)$

Fourier the position wave function into momentum space:

$\Phi ( \alpha , p) := \frac{1}{ \sqrt{2 \pi}} \int_{0}^{ \infty} exp(-1 p z) \Psi ( \alpha , z) dz |_{simplify}^{assume,~ \alpha > 0} \rightarrow \frac{2 ^{ \frac{1}{2}}}{ \pi ^{ \frac{1}{2}}} \frac{ \alpha ^{ \frac{3}{2}}}{(ip + a)^2}$

Demonstrate that the momentum wave function is normalized.

$\int_{- \infty}^{ \infty} \overline{ \Phi ( \alpha , p)} \Phi ( \alpha , p) dp~~assume,~ \alpha > 0 \rightarrow 1$

Energy operator in momentum space:

$\frac{p^2}{2} \blacksquare + i \frac{d}{dp} \blacksquare$

Evaluate the variational expression for the energy:

$E( \alpha ) := \int_{- \infty}^{ \infty} \overline{ \Phi ( \alpha , p)} \frac{p^2}{2} \Phi ( \alpha , p) dp ... + \int_{- \infty}^{ \infty} \overline{ \Phi ( \alpha , p)} i ( \frac{d}{dp} \Phi ( \alpha , p)) dp$

Minimize energy with respect to variational parameter $$\alpha$$:

$$\alpha$$ := 1 $$\alpha$$ := Minimize (E, $$\alpha$$ $$\alpha$$ = 1.145 E( $$\alpha$$) = 1.966

This momentum space result is in exact agreement with the coordinate-space result. The exact value for the energy is 1.856.

$\frac{E ( \alpha) - 1.856}{1.856} = 5.9$