Finding Points of Inflection in Harmonic Oscillator Wave Function

[ynuo]

Homework Statement

Apply direct differentiation to the ground state wave function for the harmonic oscillator,

Psi=A*e^(-sqrt(mk)x^2/(2*h_bar))*e^(-i*w*t/2) (unnormalized)

and show that Psi has points of inflection at the extreme positions of the particle's classical motion.

The Attempt at a Solution

My understanding of this question is first I have to normalize the wave function to get the value of the constant A. Then I must differentiate twice and set the second derivative of the wave function to zero and solve the resulting equation. When I do that I get an equation whose solution is complex and is different than that required by the question.

Normalization:

after normalization I get A=(m*k)^1/8 / (2*(pi*h_bar)^1/4)

Differentiation:

d(Psi)/dx=A*(-sqrt(m*k) / h_bar) e^(-i*w*t/2) * x *e^(-sqrt(m*k)x^2/(2*h_bar))

d^2(Psi)/dx=A * e^(-i*w*t/2) * [(-sqrt(m*k) / h_bar) * e^(-sqrt(m*k)x^2/(2*h_bar)) + 4 * x^2 * (-sqrt(m*k) / 2*h_bar)^2 * e^(-sqrt(m*k)x^2/(2*h_bar))]

Setting d^2(Psi)/dx=0 I get:

(1+0.5*x^2) * e^(-sqrt(mk)x^2/(2*h_bar)) = 0

Whose solution is: 1.414213562 i, -1.414213562 i

 

[Dick]

In the first place, neither the normalization nor the time-dependent part will have anything to do with location of the inflection points. Just drop them. You are not including them consistantly anyway. Second, in your first version of the second derivative I see a sign difference between the x^2 term and the other one. In your last form it has disappeared. Do it again. Much more carefully.