# One-dimensional infinitely high walls pit

1. Jun 23, 2012

### marvinslug

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