(Quantum Mechanics) Infinite Square Well

[emol1414]

Hi, I'm stuck in this Griffiths' Introduction to QM problem (#2.8)

Homework Statement

A particle in the infinite square well has the initial wave function

[itex]\Psi(x,0) = Ax(a-x)[/itex]

Normalize [itex]\Psi(x,0)[/itex]

Homework Equations

[itex] \int_{0}^{a} |\Psi(x)|^2 dx = 1 [/itex]

The Attempt at a Solution

Haha, this is supposed to be the least of my problems but... doing

[itex] A^2 \int_{0}^{a} x^2 (a-x)^2 dx = 1 [/itex]
gives us [itex] A = \sqrt{\frac{30}{a^5}} [/itex].

When the correct answer is [itex] A = \sqrt{\frac{2}{a}} [/itex]. I have no clue what I did wrong...

 

[kuruman]

The normalization constant A = sqrt(2/a) is appropriate for ψ(x) = A sin(nπx/a). Here you have ψ(x) = Ax(a-x). Different wavefunctions have different normalization constants.

 

[emol1414]

Kuruman... indeed [itex]A = \sqrt{\frac{2}{a}}[/itex] is appropriate for [itex]\psi[/itex] as a sin function. But when checking the result i got, i found this solution for the exact same problem:

http://img714.imageshack.us/img714/6186/64732846.jpg

The integral limit is set to 0,a/2... I'm not sure why, and if this is correct. For now i'll stick with my solution

_____________________________________________________________________________
By the way... (a correlated problem) trying to prove that: [itex] \int_{-\infty}^{\infty} \psi_n^*(x) \psi_m (x) dx = \delta_{n,m}[/itex] (Kronecker delta).

For n = m, and [itex]\psi_n (x) = \sqrt{\frac{2}{a}} sin(\frac{n \pi x}{a}),\in \Re[/itex], I'm coming up with

[itex] \int_{-\infty}^{\infty} \psi_{n}^{2}(x) dx [/itex] = [itex]\frac{-1}{a} cos(\frac{n\pi x}{a}) sin(\frac{n\pi x}{a})|_{-\infty}^{\infty} + \frac{1}{a} \int_{-\infty}^{\infty} dx[/itex]

The first term goes to 0, but the second would result in 1 for like... [itex] \int_{0}^{a}[/itex], not [itex]\int_{-\infty}^{\infty} [/itex]. Again... where am I messing up?

 

[kuruman]

Kuruman... indeed [itex]A = \sqrt{\frac{2}{a}}[/itex] is appropriate for [itex]\psi[/itex] as a sin function. But when checking the result i got, i found this solution for the exact same problem:

http://img714.imageshack.us/img714/6186/64732846.jpg

The integral limit is set to 0,a/2... I'm not sure why, and if this is correct. For now i'll stick with my solution

It is not the exact same problem. The problem above has yet a third wavefunction that is constant for 0≤x≤a/2 and zero everywhere else. That is the reason why the upper limit is set at a/2. If you extend the integral beyond a/2, you will be adding a whole bunch of zeroes since ψ(x) is zero, so you might as well use a/2 for the upper limit.

By the way... (a correlated problem) trying to prove that: [itex] \int_{-\infty}^{\infty} \psi_n^*(x) \psi_m (x) dx = \delta_{n,m}[/itex] (Kronecker delta).

For n = m, and [itex]\psi_n (x) = \sqrt{\frac{2}{a}} sin(\frac{n \pi x}{a}),\in \Re[/itex], I'm coming up with

[itex] \int_{-\infty}^{\infty} \psi_{n}^{2}(x) dx [/itex] = [itex]\frac{-1}{a} cos(\frac{n\pi x}{a}) sin(\frac{n\pi x}{a})|_{-\infty}^{\infty} + \frac{1}{a} \int_{-\infty}^{\infty} dx[/itex]

The first term goes to 0, but the second would result in 1 for like... [itex] \int_{0}^{a}[/itex], not [itex]\int_{-\infty}^{\infty} [/itex]. Again... where am I messing up?

Here I assume that you are using particle-in-the-box wavefunctions. All of them are zero for x≤0 and x≥a. What do you think your limits of integration ought to be?

 

[emol1414]

Hmm, ok! So it's a different problem

It makes sense now, I forgot the conditions... You're right, it is [itex]\int_{0}^{a}[/itex]!
Thank you