Finding constants for 3 wave functions

[terp.asessed]

Homework Statement

The ground, 1st excited state and 2nd excited state wave functions for a vibrating molecules are:

Ψ₀(x) = a e^-α2/2
Ψ₁(x) = b (x+d) e^-α2/2
Ψ₂(x) = c (x² + ex + f) e^-α2/2

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.

Homework Equations

Given above

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 + 2e/α² - 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!

 

[mfb]

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

 

[terp.asessed]

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

There is another orthogonality you can use.

Do you mean ∫ψ₁ψ₂dx?

 

[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?

 

[ehild]

Is the function x³e^-αx2 odd or even?

 

[mfb]

Do you mean ∫ψ₁ψ₂dx?

Sure, what else?