# Does this wavefunction make sense?

• I
Gold Member

## Main Question or Discussion Point

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!

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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.