Time derivative of complex conjugate wave function [duplicate]

Question

We have $$\frac{\partial \Psi}{\partial t} = \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$$$\frac{\partial \Psi^*}{\partial t} = -\frac{i\hbar}{2m} \frac{\partial^2 \Psi^*}{\partial x^2} + \frac{i}{\hbar}V\Psi^*.$$

My question is essentially the same as this question, however what I'm struggling to understand is why you can't simply plug in $\psi^*$ into the first relation. Is it not a wave function? Is that what goes wrong? I get that conjugating both sides of the first equation gives the second one, but what is it that is preventing me from just using $\psi^*$ in it? In the other question this is just callously stated in a comment without further information, and for some reason I don't seem to be able to find anything anywhere else.

Answer 1

Is $\Psi^*$ a wavefunction? Depends on what you mean by that: $\Psi^*(x,t)$ is certainly a function, it has the property that $\int dx\, |\Psi^*|^2 = 1$ for all $t$ (assuming that $\Psi$ does), so you could interpret it as the wavefunction of a particle.

However, a priori we don't know if $\Psi^*$ satisfies the Schrödinger equation with our given potential. In this context, $\Psi$ is supposed to be a fixed function - or at least, it's a member of the set of solutions of the equation. Therefore, $\Psi^*$ can't just stand any function, and it's meaningful to ask whether it's a solution to the SE. And in fact, in general it isn't. We have to reverse the time if we want to conjugate: If $\Psi(x,t)$ is a given solution, then $\Psi_\text{rev}(x,t) \equiv \Psi^*(x,-t)$ is also a solution, but $\Psi^*(x,t)$ isn't.

Answer 2

Call your first displayed equation $E(\Psi)$ and your second $F(\Psi^*)$.

Theorem: Suppose $E(\Psi)$ is true. Then $E(\Psi^*)$ is true if and only if ${\partial \Psi/\partial t}=0$.

Proof: If $E(\Psi^*)$ is true, add it to $F(\Psi^*)$ to get $2 \partial \Psi^*/\partial t=0$, then divide by $2$ and take the complex conjugate. Conversely, if ${\partial \Psi/\partial t}=0$, take the complex conjugate of $E(\Psi)$ to get $E(\Psi^*)$. $\blacksquare$

We want to be able to study wave functions that vary in time. So we don't want to assume that $\partial \Psi/\partial t=0$, which (by the theorem) is the same thing as saying we don't want to assume that $E(\Psi^*)$ is true.