# Does this wavefunction make sense?

• I
Gold Member
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!

## Answers and Replies

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?

• Isaac0427
Gold Member
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?
Ah, I get that. I was confused as I thought of it as a probability and not a probability density.