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Wavefunction normalisation

  1. Mar 11, 2014 #1
    1. The problem statement, all variables and given/known data
    Determine the constant λ in the wave equation

    [itex]\Psi(x) = C(2a^2 x^2 + \lambda)e^{-(a^2 x^2/2)}[/itex]

    where [itex]a=\sqrt{mω/\hbar}[/itex]

    2. Relevant equations

    Some standard integrals I guess

    3. The attempt at a solution

    So I believe the wave equation just needs to be normalised. Using the usual conditions for normalisation,

    [itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} | x^2 e^{-(a^2 x^2/2)} + e^{-(a^2 x^2/2)} |^2 dx =1[/itex]

    From there,

    [itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} |2x^2 e^{-(a^2 x^2/2)}|^2 dx =1[/itex]

    Then squaring the function inside the integral and moving the '4' outside the integral as it is a constant,

    [itex]4(C2a^2 + C\lambda)^2 \int^{∞}_{-∞} x^4 e^{-(a^2 x^2)} dx =1[/itex]

    Now that should be a standard integral but I don't know any involving an x term to the fourth power. Or perhaps I've done something else wrong?
     
  2. jcsd
  3. Mar 11, 2014 #2

    DrClaude

    User Avatar

    Staff: Mentor

    Since when is
    $$
    [(ab+c) d]^2 = (a+c)^2(bd+d)^2
    $$

    The next step is also completely wrong. Start again from ##|\psi|^2 = \psi^* \psi##. You should also allow ##C## and ##\lambda## to be complex.
     
  4. Mar 12, 2014 #3
    If the function doesn't contain any complex exponentials, then [itex]\psi^{*}[/itex] is the same as [itex]\psi[/itex], isn't it?
     
  5. Mar 12, 2014 #4

    DrClaude

    User Avatar

    Staff: Mentor

    As all physical observables depend ultimately on ##| \psi |^2##, the wave function of a physical system is only defined up to a complex phase. In other words, ##\psi## and ##\psi e^{i \delta}##, with ##\delta## real, decribe the same thing. Therefore, you can choose the normalization constant ##C## in
    $$
    \psi(x) = C f(x)
    $$
    to be real, because if it is complex, you can always do a rotation in the complex plane such that ##C' = C e^{i \delta}## is real.

    But you also have the ##\lambda## in there and, unless told otherwise, you can't assume that it is real. You should appraoch the problem without restricting ##C## or ##\lambda## to be real, and see what you get.
     
  6. Mar 16, 2014 #5
    use Maple to do the follow step:

    assume(a>0)

    int(C*(2*a^2*x^2+B)*exp(-(a^2*x^2)/2),x=-infinity..infinity)

    where B is your λ. then we can get the result is
    (B+2)*C*(2*Pi)^0.5/a

    I hope this can help you
     
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