Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Verifying an equation

  1. Feb 8, 2007 #1
    1. The problem statement, all variables and given/known data

    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. Relevant 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.
     
  2. jcsd
  3. Feb 9, 2007 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook