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Hydrogen atom wave functions

  1. Apr 29, 2014 #1
    1. The problem statement, all variables and given/known data

    I solved the Schrödinger equation, obtaining a wave function in terms of Radial and the spherical harmonics as follows:

    $$Ψ(r,0)= AR_{10} Y_{00} + \sqrt{\frac23} R_{21} Y_{10} + \sqrt{\frac23} R_{21} Y_{11} - \sqrt{\frac23} R_{21} Y_{1,-1}$$


    2. Relevant equations


    3. The attempt at a solution
    The constant A is equal to i; is this result right or not?
     
  2. jcsd
  3. Apr 29, 2014 #2

    Simon Bridge

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    Please provide a problem statement.
    But I doubt the constant is going to be imaginary - what is your reasoning?
     
  4. Apr 29, 2014 #3
    The states is more to write but I make a print screen.

    http://www.gfxroad.com/print-wf [Broken]
     
    Last edited by a moderator: May 6, 2017
  5. Apr 29, 2014 #4

    Simon Bridge

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    I can see why you didn;t want to write that down ;)
    See the line below the equation where it says "where A is a real constant..."?
    Your question:
    ... is answered.

    You seem to be trying to answer part (b).
    What is the condition that must be satisfied for ##\psi(\vec r,0)## to be normalized?
     
  6. Apr 29, 2014 #5
    ∫ψψ*dτ=1
     
  7. Apr 29, 2014 #6

    Simon Bridge

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    Very good ... imagine you had ##\psi = a\psi_a + b\psi_b## ... where ##\psi_a## and ##\psi_b## are already normalized. In order for ##\psi## to be normalized, ##a## and ##b## need to satisfy a condition ... what is it?
     
  8. Apr 29, 2014 #7
    a2+b2=1
     
  9. Apr 29, 2014 #8

    Simon Bridge

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    Well done.
    Technically: ##a^*a+b^*b=1## in case you have complex coefficients.

    Now imagine you have:

    ##\qquad \psi = a\psi_a + \sqrt{\frac{2}{3}}\psi_b + \sqrt{\frac{2}{3}}\psi_c - \sqrt{\frac{2}{3}}\psi_d##

    ... now your problem is that to get ##|\psi|^2=1## it looks like you have ##a^2+2=1 \implies a=\sqrt{-1}##

    But you are told that ##a## is real so this is a contradiction.
    Anyway, if ##a=i##, then ##a^*a= (-i)i = -i^2=1## not the -1 you were looking for.

    In fact, is there even a solution for ##a^*a=-1##?

    Therefore - what does this tell you about your approach?
    Did you properly account for the R and Y functions?
    i.e. is ##\psi_{nlm}=R_{nl}Y_{lm}## normalized already?
    ... did you do part (a) correctly?
     
    Last edited: Apr 29, 2014
  10. Apr 30, 2014 #9
    Re: Hydrogen atom wave function/ part a

    Yes, I thought part a was done correctly like:
    http://www.gfxroad.com/print2 [Broken]
     
    Last edited by a moderator: May 6, 2017
  11. Apr 30, 2014 #10

    Simon Bridge

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    Yeah, I'm getting the same thing ... I have a nagging feeling there's a wrinkle here I'm missing but on the face of it the textbook problem has no solution.

    It may be that the text-book has a typo.
     
  12. Apr 30, 2014 #11
    What's about e branch, is there any starting point or equation for this, because I don't know where can I starting. The other branches solved correctly.
     
  13. Apr 30, 2014 #12

    Simon Bridge

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    For (e) ##\text H\psi_{nlm}=E_n\psi_{nlm}##

    What do you mean "correctly"? Do you have model answers?
     
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