Suppose that $\Psi(x,t)$ is normalized at time $t=0$. Show that this implies that $\Psi(x,t)$ is normalized at all other times.

I know that this makes intuitive sense, and we'd certainly want our models of reality to have this property - if the particle has a 100% chance to be found somewhere in space initially, it should ALWAYS have a 100% of being found. I just don't know how to prove this mathematically.

I can at least get started by saying that, since $\Psi(x,0)$ is normalized, we know $$\int_{-\infty}^{\infty}\Psi^*(x,0)\Psi(x,0) \mathrm{d}x=1$$

Can anyone give a hint about where to go from here?


1 Answer 1


I will elaborate on count_to_10's comment. Consider the following current


where $m$ is the mass of the particle and I am working in units where $\hbar=1$. This is the famous probability current of QM. It is related to the probability density $\rho(x,t)=\psi\psi^*$ via


it is easy to convince yourself of this. Just take the definition of $j$ and the Schrödiger equation and you will see that the equation above is satisfied. If you have taken an electromagnetism course you probably are guessing that we will have a charge associated with $\rho$ which is conserved. Let's prove it. Let's integrate



assuming that the wave function vanishes asymptotically


and we have that $\int_{-\infty}^{\infty}dx\,\rho$ is time independent. But since you know that it is 1 at time 0 (since you assume the wave function is normalized) you know that it will remain normalized.


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