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I Does this wavefunction make sense?

  1. Apr 30, 2016 #1
    Hi all!
    Consider a wavefunction, where ##\left| \psi (x) \right|^2 = e^{-ax^2+1}+e^{-bx^2-1}## where a and b are real, positive numbers that satisfy normalization (they are purpously inside the exponent). Even if it is normalized, there are still 2 spots that ##\left| \psi \right|^2 > 1## which makes no sense. What is going on here?
    Thanks!
     
  2. jcsd
  3. Apr 30, 2016 #2
    Recall that ##|\psi^2|## is a probability density. The probability for finding a particle in an interval ##[a,b]## is ##\int\limits_a^b|\psi^2|\mathrm{d}x##. This integral is what must be ##\le 1## (where it equals one when you integrate over all of space). To convince yourself, take one of the Gaussian functions you wrote and normalize it, evaluate at ##x=0## and start increasing ##a##. Despite still being normalized ##|\psi^2|## can be made arbitrarily large. Why is that alright?
     
  4. Apr 30, 2016 #3
    Ah, I get that. I was confused as I thought of it as a probability and not a probability density.
     
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