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Normalization of kets to unity

  1. Jul 17, 2007 #1
    1. The problem statement, all variables and given/known data

    I understand how you normalize vectors to unity, but how would you normalize a function to unity?

    For example, how would you show that the function (2/L)^(1/2) sin(n*pi*x/L) is normalized to unity? You cannot just just take its modulus and set it to one since because x is variable...right?

    2. Relevant equations

    3. The attempt at a solution
    Last edited: Jul 17, 2007
  2. jcsd
  3. Jul 17, 2007 #2


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    Define the norm. Looks like QM, so you want L^2. The integral over the range of x of psi(x)*conjugate(psi(x))=1 defines a normalized function.
  4. Jul 17, 2007 #3


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    It's important to realize that the modulus squared (not just the modulus), [itex] | \psi(x)|^2 [/itex] of a wavefunction is a probability density , not a probability. This means that the quantity [itex] | \psi(x)|^2 dx [/itex] is aprobability, which represents the probability of finding the particle in the infinitesimal interval (x,x+dx). What must be normalized to one is therefore

    [tex] \int_{- \infty}^{\infty} | \psi(x)|^2 dx =1 [/tex]

    It seems as if you are considering an infinite squqre well located from x=0 to x=L. In that case, the wavefunction is zero outside of the well and the above condition reduces to

    [tex] \int_{0}^{L} | \psi(x)|^2 dx =1 [/tex]

    hope this helps

  5. Jul 17, 2007 #4
    I see. Thanks.
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