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 $(n + 1/2)\hbar\omega$.
a)
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 $|x_t |=\sqrt{(2E/k)}$
Hence
$Φ(x)=1/ℏ ∫_0^(√(2E/k))\sqrt{(2m(E-1/2 kx’^2 ) )} dx’=$
$=√m/ℏ [(x√(E-(kx^2)/2))/√2 +E/\sqrt{k} \arcsin(x√(k/2E))]$
b)
From the definition of probability
$P(x)~1/(dx/dt)=dt/dx$ we have $P(x)Δx~Δt$ or
$P(x)Δx=Δt/(T/2)=((Δx)v)/(T/2)=2Δx/vT=2mΔx/(p(x)*T)$
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.