Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.5

Particle in a Gravitational Field

Physics — Landau & Lifshitz · particle-in-gravitational-field

Problem

Consider a particle of mass

m

in the potential

V(z) = \begin{cases} mgz & z > 0 \\ \infty & z < 0 \end{cases}

Using the variational method with the trial wavefunction

\psi_\alpha(z) = A z e^{-\alpha z}

, find the best estimate for the ground state energy. Compare your result with the exact value

E_1 = 2.338 \left(\frac{\hbar^2 m g^2}{2}\right)^{1/3}