# Finding constants for 3 wave functions

1. Oct 24, 2014

### terp.asessed

1. The problem statement, all variables and given/known data
The ground, 1st excited state and 2nd excited state wave functions for a vibrating molecules are:

Ψ0(x) = a e2/2
Ψ1(x) = b (x+d) e2/2
Ψ2(x) = c (x2 + ex + f) e2/2

respectively, with constants: a, b, c, d, e and f. Use the fact that these wave functions are part of an orthogonal set of functions to find out the constants, d, e and f ONLY.

2. Relevant equations
Given above

3. The attempt at a solution
I used these equations:
∫(from x = -infinite to +infinite) Ψ0 Ψ1 dx = 0 b/c of orthogonality--> from which I got d = 0
∫(from x = - infinite to + infinite) Ψ0 Ψ2 dx = 0 b/c of orthogonality--> from which I got f = -1/2α

...so, to solve for e, I decided to use ∫ (from x = -infinite to + infinite) Ψ2 Ψ2 dx = 1 (normalization)...except, I could not figure out how to get e:

c2∫(x2 + ex - 1/(2α))2e-αx2 dx = 1
∫(x2 + ex - 1/(2α))2e-αx2 dx = 1/c2
∫(x4 + 2ex3 -x2/α + e2x2 - ex/α +1/(4α2))e-αx2 = 1/c2
1/c2 = (3!!)(1/2α)2(π/α)½ + 2e2 - 1/(2α)(π/α3)½ +e2/2 (π/α3)½ + 1/(4α2)(π/α)½

...and I am not sure if I doing right? If someone could point out the mistake or clarify, I would appreciate it!

2. Oct 24, 2014

### Staff: Mentor

The normalization won't help you as this can be used to fix c only.
There is another orthogonality you can use.

3. Oct 24, 2014

### terp.asessed

Rly? No wonder I kept getting constant c value... Aside, could you expand on what you mean by:

Do you mean ∫ψ1ψ2dx?

4. Oct 24, 2014

### terp.asessed

I am trying to integrate ∫ψ1ψ2dx:

∫ψ1ψ2dx = bc ∫xe-αx2 (x2+ex-1/2α) dx = 0
∫x3e-αx2 + e x2e-αx2 - xe-αx2/(2α) dx = 0

...I am having trouble integrating ∫ (x = -∞ to +∞) x3e-αx2 dx part--is this 0 or 1/α2?

5. Oct 25, 2014

### ehild

Is the function x3e-αx2 odd or even?

6. Oct 25, 2014

### Staff: Mentor

Sure, what else?