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: Normalizing a wave function and calculating probability of position

  1. Apr 8, 2014 #1
    Forgive me if this goes in elementary physics, but I think since it's an upper level undergrad class
    1. The problem statement, all variables and given/known data

    A state of a particle bounded by infinite potential walls at x=0 and x=L is described by a wave function [itex]\psi = 1\phi_1 + 2\phi_2 [/itex] where [itex]\phi_i[/itex] are the stationary states.
    a) Normalize the wave function.
    b) What is the probability to find the particle between x=L/4 and x=3L/4?

    2. Relevant equations

    OK so was looking at how to approach this. One thought was to start with Schrödinger's equation

    [itex]i \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V(x)\psi(x,t)[/itex]

    3. The attempt at a solution

    We have a situation where V(x) = 0 0 < x < L and V(x) = infinity outside of that. SO the V(x) term for inside the well disappears (it's zero).

    And Energy, [itex]E = \frac{\hbar^2 k^2}{2m}[/itex] where k is a constant, in this case 1.

    That leaves us with a partial differential equation

    [itex]i \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}[/itex]

    which, moving it around a little,

    [itex]i \hbar \frac{\partial \psi}{\partial t} - \frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} = 0 [/itex]

    and that's a 2nd order differential equation. The solutions are

    [itex]\psi(x,t) = [A \sin(kx) + B \cos(kx)] e^{-i\omega t}[/itex]

    k is a wavenumber and for this to work -- for it to be zero when x is greater than L or less than 0, and some value anywhere else, the cosine term has to go, and since the wavenumber is limited to nπ/L (that's the only way you get an integral number of waves) So I should end up with

    [itex]\psi(x,t) = [A \sin(k_nx)] e^{-i\omega_n t}[/itex]

    But I am not entirely sure why the cos term has to go; I feel like I am missing a step. Either way, the probability of the particle being at any point from 0 to L is 1. So I need to integrate the wave function squared over that interval:

    [itex]\int^L_0 |[A \sin(k_nx)] e^{-i\omega_n t}|^2 dx[/itex]

    but that's the thing, I feel like I have lost the plot with this.

    ANother way to approach it was to assume that the wave function given can be

    [itex]\psi = 1\phi_1 + 2\phi_2 [/itex]
    [itex]\psi = (1\phi_1 + 2\phi_2)(1\phi_1^* + 2\phi_2^*)[/itex]

    and then multiply this out
    [itex]\psi = (1\phi_1^* \phi_1 + 2\phi_1 \phi_2^* + 2\phi_1^* \phi_2 + 4\phi_2^*\phi_2)[/itex]

    SInce the phi functions are eigenvalues, the ones on the diagonal of the matrix are the only ones not zero. So we get
    [itex]\psi = (1\phi_1^* \phi_1 + 4\phi_2^*\phi_2) = (1 + 4) [/itex]

    because the complex conjugate of a function multiplied by a function is 1.

    Tht makes the whole thing add up to five. and since the probability of finding the particle on the interval 0 to x is

    [tex]\int^L_0 |\psi|^2 dx = 1 \rightarrow \int^L_0 |5|^2 dx = 1 \rightarrow 25x = 1[/tex]

    so x = 1/25 for the whole interval, so normalizing the wave function I should get

    [itex]\psi = \frac{1}{25}\phi_1 + \frac{2}{25}\phi_2 [/itex]

    and for the probability that the particle is at L/4 and 3/4 L

    (25/4)^2 and (75/4)^2

    Now, if someone could tell me where I am getting lot and doing this completely wrong :-)

    thanks in advance.
    Last edited: Apr 8, 2014
  2. jcsd
  3. Apr 8, 2014 #2
    In the original question you state that the wave function is "=11+22, where are the stationary states" are those boxes a mistake?
  4. Apr 8, 2014 #3
    try it now...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted