1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
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: Harmonic osilator energy using derivatives

  1. Dec 12, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that the energy of a simple harmonic oscillator in the n = 2 state is 5Planck constantω/2 by substituting the wave function ψ2 = A(2αx2- 1)e-αx2/2 directly into the Schroedinger equation, as
    broken down in the following steps.

    First, calculate dψ2/dx, using A for A, x for x, and a for α.

    Second, calculate d2ψ2/dx2.

    3. The attempt at a solution
    so I got the first derivative correct, it was

    A((4*a*x)*exp((-a*x^2)/2) +(2*a*x^2 -1)*(-a*x*exp((-a*x^2)/2)))

    but i can seem to calculate the second derivative correctly I'm getting
    [itex] A((4a)(exp(( - ax^2 ) / 2)) + (4ax) * ( - ax * exp(( - ax^2) /2)) + (4ax) * ( - a xexp(( - ax^2 ) / 2)) + (2ax - 1) (a^2x^2 *exp(( - ax^2 ) / 2))) [/itex]

    but this incorrect, am I missing something?
    Last edited: Dec 12, 2013
  2. jcsd
  3. Dec 12, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    $$\psi_2(x)=A(2\alpha x^2-1)e^{-\frac{1}{2}\alpha x^2}$$
    $$\frac{d}{dx}\psi_2(x)=A\left[4\alpha x e^{-\frac{1}{2}\alpha x^2} - (2\alpha x^2 - 1)\alpha x e^{-\frac{1}{2}\alpha x^2}\right]$$... is pretty messy so it will be easier to make mistakes: simplify this expression first. Then try the second derivative.

    You can bring an ##\alpha x \exp(-\frac{1}{2}\alpha x^2)## outside the brackets ... then deal with the terms inside the brackets.

    After that I suspect it is something you can do.

    note: use the "quote" button below this post to see how I got the equations to typeset like that ;)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted