1. The Harmonic Oscilloi The harmonic oscillator is probably the simplest and in
ID: 1794778 • Letter: 1
Question
1. The Harmonic Oscilloi The harmonic oscillator is probably the simplest and in some ways the most important ex ample in quantum mechanics. It is the system in which a particle acts as if it were oscillating on a spring: the force that pulls it back to the middle is proportional to the displacement, F =-k(Az). The corresponding potential is simply V(r) = r2 (a) What is the frequency of oscillations for a classical particle with mass m in the (b) Write down the time independent Schrodinger equation for a particle with mass m (c) Show that harmonic oscillator potential above? acted on by the harmonic oscillator potential above is a solution to the time independent Schrodinger equation you wrote down, with enery This solution is not normalized, for simplicity sake.] (d) What is the relationship between the energy above the the classical frequency from part (a)? (e) Write the corresponding solution to the time dependendent Schrodinger equation.Explanation / Answer
for the given case
F = -k(dx)
V(x) = 0.5kx^2
a. hence frequency of osscilation of classical particle
with mass m in the harmonic potential above is
f = w/2*pi
w = sqroot(k/m)
hence
f = sqroot(k/m)/2*pi
b. Hamiltonian for the case = H
H = p^2/2m + V(x)
p = -ih'd/dx
where h' = h/2*pi
the time independent schrodinger's equation is
-h'^2/2m * d^2(phi(x))/dx^2 + V(x)phi(x) = E*phi(x)
c. consider phi = e^(-sqroot(mk)x^2/2h')
d(phi)/dx = -2x*sqroot(mk)e^(-sqroot(mk)x^2/2h')/2h'
d^(phi)/dx^2 = -2sqroot(mk)e^(-sqroot(mk)x^2/2h')/2h' +4x^2*mk*e^(-sqroot(mk)x^2/2h')/4h'^2
hence
substituting
-h'^2/2m * (-2sqroot(mk)e^(-sqroot(mk)x^2/2h')/2h' +4x^2*mk*e^(-sqroot(mk)x^2/2h')/4h'^2) + 0.5kx^2*e^(-sqroot(mk)x^2/2h') = E*e^(-sqroot(mk)x^2/2h')
-h'/m * (-sqroot(mk)/2 +2x^2*mk/4h') + 0.5kx^2 = E
0.5kx^2 + h'*sqroot(k/m)/2 - 0.5*x^2k = h'*sqroot(k/m)/2 = E
d. E = h'*2*pi*f/2 = hf/2
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