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Radial Eigenfunction; Differential Equation

  1. Mar 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that the radial eigenfunction unr,l is a solution of the differential equation:
    ħ2/2me×d2unr,l/dr2+[l(l+1)ħ2/2mer2 - e2/4πε0r]unr,l=Enr,lunr,l


    2. Relevant equations
    The radial function is R(r)=u(r)/r, so that the expression on the RHS is E×u.
    3. The attempt at a solution
    I know that this equation is the radial Schrodinger equation, and that the particle with angular momentum L behaves like a particle in one-dimensional effective potential.But, to prove it, i don't know where to start.
    I tried comparing this potential with the analogous effective potential and set the first term, l(l+1)ħ2/2mr2, but with no luck. Any ideas, please?
     
  2. jcsd
  3. Mar 9, 2015 #2

    BvU

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    I had to look it up in Griffiths (here) and the problem statement seems OK. I do expect, either in the problem statement, or in the relevant equations, an expression for the radial eigenfunction ##u(n, r, l)##. Because the exercise wants you to show that it's a solution.

    Your relevant equation R = u/r isn't of interrest, I'm afraid. They want you to differentiate u twice.

    Embark on that task and post the work (in reasonable detail) if you still get stuck.

    You may want to adopt the notation (4.55) in the link to avoid excessive amounts of ##\hbar\over 2m## etc. -- but then you need your u also in terms of ##\rho##.
     
  4. Mar 10, 2015 #3
    well, i had to look at the soln on the back of the book. Never mind, i got it anyway.
     
  5. Mar 10, 2015 #4

    BvU

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    Was it something like "differentiate u twice and verify that it satisfies the equation" or was it different ?
     
  6. Mar 11, 2015 #5
    nope, it was different. The book u r referring to, is the one i have already read. This is a different book, and a quite old one too---- It doesn't have a proper cover.
     
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