Harmonic Oscillator (Homework 6-325)

3. WKB Approximation for a Harmonic Oscillator

a) Using the WKB approximation for a Harmonic oscillator with spring constant k and mass m, write down the form of theWKB wave-function in the classical region for distances less than the classical turning point $|x| < x_{tp}$.

b) Determine the classical probability $P(x)$ of finding a particle at point x, where $P(x) =1/(dx/dt)$, so that $P(x) x$ is is proportional to the time t the particle spends in the interval x. Compare with part a) in the limit of large energies $(n + 1/2)\hbar\omega$.


For a Hamiltonian of the type

$H=T+U=-ℏ^2/2m ∇^2+1/2*mω^2 x^2=-ℏ^2/2m ∇^2+1/2 kx^2$

The eigenfunctions in the position space (x) can be written as

$ψ(x)=1/√p*{A*\sin⁡ [Φ(x) ]+B*\cos⁡[Φ(x)]}$

Where Φ(x) are the momentum space eigenfunctions

$Φ(x)=1/ℏ ∫_0^xp(x’ )dx’$ and $p(x)=\sqrt{(2m(E-1/2 kx^2 )} )$

The turning point is where

$E-1/2*kx^2=0$ so that $|x_t |=\sqrt{(2E/k)}$


$Φ(x)=1/ℏ ∫_0^(√(2E/k))\sqrt{(2m(E-1/2 kx’^2 ) )} dx’=$

$=√m/ℏ [(x√(E-(kx^2)/2))/√2 +E/\sqrt{k} \arcsin⁡(x√(k/2E))]$


From the definition of probability

$P(x)~1/(dx/dt)=dt/dx$ we have $P(x)Δx~Δt$ or


where $p(x)=\sqrt{(2m(E-1/2 kx^2 ))}$ and $T=2π/ω$

So that

$P(x)=2m/(p(x)*T)=mω/π*1/\sqrt{(2m(E-1/2 kx^2 ))}$

Normally from above (point a) the expected probability is

$P(x)=|ψ|^2=1/(p(x))=1/\sqrt{(2m(E-1/2 kx^2 ))}$

so that WKB approximation is good for large energies.