Approximate wavefunction (Homework 7, Physics 325)

1. Consider the WKB solution for a ball of mass m with energy E in a gravitational potential $V (z) = mgz$,  $z > 0$, and $V = +\infty$,  $z < 0$, with a hard surface at $z = 0$.

(a) Estimate the normalization constant of the wave function in terms of the period of oscillation T of the ball $T = 2\int_0^{z0} dz/v(z)$, where $v(z) = p(z)/m$. Assuming  $\psi(z)$ is rapidly oscillating, you may assume $sin^2(\theta ) \approx  1/2 in your estimate.

Momentum is

$p(z)=\sqrt{2m(E-V(z)}=\sqrt{(2m(E-mgz)}$   so that $v(z)=(p(z))/m=\sqrt{2(E/m-gz)}$

The turning point (where $E=V$) is $z_0$. The shape of the wavefunction in a triangular well can be approximated with (eq. 8.46)


$=2D/\sqrt{p(z)}*sin⁡(1/ℏ ∫_z^{z_0} p(z’)dz’+π/4)$ ,in the well

$=D/\sqrt{|p(z)|}*exp⁡(-1/ℏ ∫_z^{z_0} |p(z’ ) |dz’)$  ,ouside the well

This can be further written as


$=2D/√(p(z) )*sin⁡(2π/T*z+π/4)$   ,in the well
$=D/√(|p(z) | )*exp⁡(-2π/T*z)$  ,outside the well

The normalization constant comes from ($z_0=E/mg$)

$1=∫_{-∞}^∞ ||ψ|^2*dx=D^2 (∫_{-∞}^0 1/√∞*exp⁡(…)*dz+$

$+∫_0^{z_0} 4/\sqrt{(2m(E-mgz))}*sin^2⁡ (2π/T*z+π/4)*dz+$

$+∫_{z_0}^∞ 1/\sqrt{(2m(mgz-E)}*exp⁡(-2*2π/T*z)dz$)

$1=D^2 [∫_0^{z_0} 1/2*4*1/\sqrt{2m(E-mgz)}*dz+\text{something small}$

$1= D^2/√2m*[-(4*2√(E-mgz))/mg*1/2  (\text{from 0 to z_0}) ]=[(4D^2)/(mg√2m)] √E$


The average period of oscillation is ($z_0=E/mg$)

$T=2∫_0^{z_0} dz/v(z) =2∫_0^{z_0} dz/\sqrt {2(E/m-gz)}=$

$=-2 \sqrt{2(E/m-gz)} /g  (\text {from 0 to z_0})=2/g*(\sqrt{2E/m}-\sqrt{2(E/m-gz_0 )} =$

$=2/g*\sqrt{2E/m}=1/g \sqrt{8E/m}$