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Finding constants for 3 wave functions

  1. Oct 24, 2014 #1
    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. jcsd
  3. Oct 24, 2014 #2

    mfb

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    The normalization won't help you as this can be used to fix c only.
    There is another orthogonality you can use.
     
  4. Oct 24, 2014 #3
    Rly? No wonder I kept getting constant c value... Aside, could you expand on what you mean by:

    Do you mean ∫ψ1ψ2dx?
     
  5. Oct 24, 2014 #4
    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?
     
  6. Oct 25, 2014 #5

    ehild

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    Is the function x3e-αx2 odd or even?
     
  7. Oct 25, 2014 #6

    mfb

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    Sure, what else?
     
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