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Evaluate the ground state energy using the variational method

  1. Aug 3, 2007 #1
    1. The problem statement, all variables and given/known data

    [tex]V(x) = k|x|, x \in [-a,a], V(x) = \infty, x \notin [-a,a][/tex]. Evaluate the ground state energy using the variational method.

    2. Relevant equations

    [tex]a = \infty[/tex] and [tex]\psi = \frac{A}{x^{2}+c^{2}}[/tex].

    3. The attempt at a solution

    [tex]1 = |A|^{2}\int_{-a}^{a}\frac{1}{(x^{2}+c^{2})^{2}}\,\text{d}x.[/tex] Is this a correct way to start? How can I calculate this? I used Mathematica, but it only gives some weird-looking answers.
     
    Last edited: Aug 3, 2007
  2. jcsd
  3. Aug 3, 2007 #2

    Gokul43201

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    Yes, that's how you normalize the trial function. And what do you mean by "weird"?

    Also, why do you say a=infty?
     
  4. Aug 3, 2007 #3
    This is what I put:
    In[2]:=
    \!\(∫\_\(-a\)\%a\( 1\/\((x\^2 + c\^2)\)\^2\) \[DifferentialD]x\)

    This is what I got:
    Out[2]=
    \!\(2\ a\ If[Im[c\/a] ≥ 1 || Im[c\/a] ≤ \(-1\) || Re[c\/a]
    ≠ 0, \(c\/\(a\^2 + c\^2\) + ArcTan[
    a\/c]\/a\)\/\(2\ c\^3\),
    Integrate[1\/\((c\^2 + \((a - 2\ a\ x)\)\^2)\)\^2, {x, 0, 1}, \
    Assumptions \[Rule] Re[c\/a] \[Equal] 0 && \(-1\) < Im[c\/a] < 1]]\)

    I do not understand. What are those imaginary things?

    In the paper, it says: "Assume that [tex]a=\infty[/tex] and use the trial [tex]\psi(x)=...[/tex]."
     
  5. Aug 3, 2007 #4

    Gokul43201

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    I can't read that easily, but I believe it allows for values of c that are not real. For the problem, you could chose to limit yourself to real c.

    Then please write this down as part of the question.Always write down the complete question. Do not summarize or reword in any way.
     
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