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Eigenfunctions and eigenvalues

  1. Jul 11, 2005 #1
    This is probably a straight forward question, but can someone show me how to solve this problem:

    [tex]
    \frac {d^2} {d \phi^2} f(\phi) = q f(\phi)
    [/tex]

    I need to solve for f, and the solution indicates the answer is:
    [tex]
    f_{\substack{+\\-}} (\phi) = A e^{\substack{+\\-} \sqrt{q} \phi}
    [/tex]

    I know I've covered this before - just need a refresher on how to solve.
     
  2. jcsd
  3. Jul 11, 2005 #2
    One way to solve this problem is to "guess" a solution. Since the form of your equation is very basic, this guess is usually taught as a standard method.

    As your solution suggests, your guess should be the following:
    [tex]f(\phi)=e^{r\phi}[/tex]
    Simply replace every function f with that equation above and solve for the unknown, r. Then, once you find r, plug it back into the equation above. Note that the "A" in your answer is an arbitrary constant used to solve with initial conditions.
     
  4. Jul 11, 2005 #3
    thanks - knew it was something simple. I actually remembered the other approach too, where you replace f'' with r^2, f' with r, and f with 1, and then solve for what r is. But either approach gives the same result.

    Thanks again though.
     
  5. Jul 11, 2005 #4
    An additional question, somewhat related:

    When determining the eigenvalues, the problem indicates that
    [tex]
    f (\phi + 2\pi) = f (\phi)
    [/tex]

    Given the answer already shown, why would this periodic function require:
    [tex]
    2 \pi \sqrt{q} = 2 n \pi i
    [/tex]
     
  6. Jul 12, 2005 #5

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Take your given condition and substitute in the solution that has been found. Write the exponent of a sum as a product of exponents and the desired result follows automatically.

    Do it. If you have trouble, show us what you tried and what part is bothering you.
     
  7. Jul 12, 2005 #6
    Nevermind, I got it now - didn't realize the relation between 1 and e^(i2n(pi))....

    [tex]
    A e^{\sqrt{q} \phi} = A e^{\sqrt{q} \left( \phi + 2 \pi \right)}
    [/tex]
    [tex]
    e^{\sqrt{q} \phi} = e^{\sqrt{q} \phi} e^{\sqrt{q} 2 \pi}
    [/tex]
    [tex]
    1 = e^{\sqrt{q} 2 \pi}
    [/tex]
    [tex]
    e^{i 2 n \pi} = e^{\sqrt{q} 2 \pi}
    [/tex]
    [tex]
    i 2 n \pi = \sqrt{q} 2 \pi
    [/tex]
    [tex]
    i n = \sqrt{q}
    [/tex]
    [tex]
    q = -n^2 (n=0,1,2....)
    [/tex]
     
    Last edited: Jul 12, 2005
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