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Is the normalisation constant of a wavefunction real?

  1. May 20, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider the wavefunction ##\Psi (x, t) = c\ \psi (x) e^{-iEt/ \hbar}## such that ##\int | \Psi (x, t) |^{2} dx = 1##.

    I would like to prove to myself that the normalisation factor ##c## is a real number.

    2. Relevant equations

    3. The attempt at a solution

    ##\int | \Psi (x, t) |^{2} dx = 1##

    ##\int c\ \psi (x)\ e^{-iEt/ \hbar}\ c^{*}\ \psi^{*}(x)\ e^{iEt/ \hbar}\ dx = 1##

    ##\int c\ \psi (x)\ c^{*}\ \psi^{*}(x)\ dx = 1##

    ##\int |c|^{2}\ |\psi (x)|^{2}\ dx = 1##

    This isn't getting me anywhere, though! :frown:
     
  2. jcsd
  3. May 20, 2015 #2

    Orodruin

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    You cannot prove this, quantum mechanics is fine with redefining all phases by the same phase factor and the physics does not depend on the phase of a state. It only depends on the relative phase of interfering states. However, you may choose the normalisation constant to be real for this very reason.
     
  4. May 20, 2015 #3
    Thanks!

    I knew that the phase factors are tunable, but I was not able to see that this could account for the plausibility of a real normalisation constant.

    Now, it's all clear!
     
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