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Shankar One-Dimensional Problem

  1. Oct 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Exercise 5.2.2 (b.)
    Prove the following theorem: Every attractive potential in one dimension has at least one bound state. Hint: Since [tex]V[/tex] is attractive, if we define [tex]V(\infty)=0[/tex], it follows that [tex]V(x)=-|V(x)|[/tex] for all [tex]x[/tex]. To show that there exists a bound state with [tex]E<0[/tex], consider
    [tex]\psi_{\alpha}(x)=\left(\frac{\alpha}{\pi}\right)^{1/4}\text{e}^{-\alpha x^{2}/2}[/tex]
    and calculate

    [tex]E(\alpha)=<\psi_{\alpha}|H|\psi_{\alpha}>,[/tex] [tex]H=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-|V(x)|[/tex].

    Show that [tex]E(\alpha)[/tex] can be made negative by suitable choice of [tex]\alpha[/tex]. The desired result follows from the application of the theorem approved above.

    2. Relevant equations

    3. The attempt at a solution
    I evaluated the expectation value using the given wave function and special Hamiltonian and received a simpler equation of [tex]E=\frac{\alpha\hbar^{2}}{4m}-\int_{-\infty}^{\infty}|V(x)|\psi_{\alpha}^{2}dx[/tex]. I have no idea where to go from here.
  2. jcsd
  3. Oct 16, 2009 #2


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    Homework Helper
    Gold Member

    Hmmm.... You might try using the mean value theorem for integration.
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