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One-dimensional infinitely high walls pit

  1. Jun 23, 2012 #1
    Hello. I have an assignment from my physics course. I am from Croatia so i google translated the assignment into English. Please note that I am sorry if this post is little messy, but for some reason I cant use toolbar for complex equations, my browser simply puts ??? in place of symbols.

    1. The problem statement, all variables and given/known data

    Consider the motion of two electrons (without interaction) in a closed nanotube length L=20*10^-9 m, as in the one-dimensional motion of an infinitely high potential pit (solid) walls. If the electrons are described in the single-particle states of the spatial wave functions:
    Y1(x)=Ax^2(x-L) *
    Y2(x)=By(y-L)^2 *

    determine:

    1) State wave function of electrons with total spin h, and spin projection equal to Oz +h. **
    2) The probability that both electrons are found in the area in this state
    3) Average energy of electrons in this state

    * Y = Psi
    * *reduced planck constant h/2*pi = 1.05457168*10^-34 Js

    (Note: The constants A and B determined from the conditions of standardization)

    2. Relevant equations

    The constants A and B are determined from the conditions of standardization
    P=integral(from 0 to L) |Y(x)dx|^2=1

    I got A=B=sqrt(105)/L^(7/2)


    3. The attempt at a solution

    1)
    I have to determine the spatial wave function. Does the spatial wave function of two electrons (or single) must be asymmetric or not? If it has to (and i have used that assumption), then the solution follows:

    Y(x,y)=Y1(x)*Y2(y)-Y1(y)*Y2(x)

    sqrt(105)/L^(7/2) * (x^3*y*L^2 - x^3*y^2*L - x^2*y*L^3 - x^2*y^3*L - x*y^3*L^2 +x*y^2*L^3)

    2) That probability is double integral dxdy from sqrt(105)/L^(7/2) * (x^3*y*L^2 - x^3*y^2*L - x^2*y*L^3 - x^2*y^3*L - x*y^3*L^2 +x*y^2*L^3) squared, with borders from L/4 to L.

    2) The equation is quite long, and i think not important right now..

    The problem

    The problem is that I got 0 as a solution to 2), and that isn't the right answer. That probability can't be 0, so i know i did something wrong here. If somebody can help me understand what did I do wrong, that would be great.

    Thank You in advance for help.

    edit: here is the image of all equations
     
    Last edited: Jun 23, 2012
  2. jcsd
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