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Verifying an equation

  • Thread starter ynuo
  • Start date
18
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1. Homework Statement

Consider the wave function

Psi(x, t)=1/sqrt(a) * [sin(2*pi*x/a)*e^(-i*E2*t/h_bar) + cos(3*pi*x/a)*e^(-i*E3*t/h_bar)]

for the particle in the one-dimensional box.

a) Calculate the expectation values <E>, <x>, and <p>.
b) Show that <x> and <p> satisfy the relation <p>=m*d<x>/dt

2. Homework Equations

<E>=Int[Psi_star*(i*h_bar*d/dt)*Psi, x=-a/2..a/2]
<x>=Int[Psi_star*(-i*h_bar*d/dx)*Psi, x=-a/2..a/2]
<p>=Int[Psi_star*x*Psi, x=-a/2..a/2]

3. The Attempt at a Solution

a) <x>=Int[1/a*[sin(2*pi*x/a)*e^(i*E2*t/h_bar) + cos(3*pi*x/a)*e^
(i*E3*t/h_bar)]*x*[sin(2*pi*x/a)*e^(-i*E2*t/h_bar) + cos
(3*pi*x/a)*e^(-i*E3*t/h_bar)], x=-a/2..a/2]

After integration and simplification I get:

<x>=(-24*a/25*pi^2) * (e^(i(E3 - E2)t/h_bar)) + e^(i(E2-E3)t/h_bar)

And for <p>: <p>=Int[-i*h_bar/pi (sin(2*pi*x/a) * e^(i*E2*t/h_bar) +
cos(3*pi*x/a)*e^(i*E3*t/h_bar))*d/dx(sin(2*pi*x/a) * e^
(-i*E2*t/h_bar) + cos(3*pi*x/a)*e^(-i*E3*t/h_bar)),
x=-a/2..a/2]

After integration and simplification I get:

<p>=-i*h_bar/pi * [-4*a/(15*pi)*e^(i(E2-E3)t/h_bar) +
=3*a/(5*pi) * e^(i(E3-E2)t/h_bar)

After substituting E2=(2 * pi^2 * h_bar^2)/(m*a^2)
E3=9/2 * pi^2*h_bar^2/(m*a^2)

in <x> and <p> and applying m*d/dt to <x> I don't get
<p>=m*d<x>/dt

This is what I get for the RHS:

m*d<x>/dt=(-12/5)*i*h_bar/a * [e^(i((2.5*pi^2 * h_bar^2)/
(m*a^2)t/h_bar))-e^(-i((2.5*pi^2 * h_bar^2)/(m*a^2)
t/h_bar))

Can you please help. Thank you.
 

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