I've been reading through Griffiths QM book, and the only thing bugging me is they never fully described what $\Psi^* $ should be for any given function. I know it's the complex conjugate at the same time I think I just need concrete examples to solidify it in my head.
What is the corresponding $\Psi^*$ for
\begin{align}
\Psi_n(x,t) =& \sqrt{2\over a} \sin{n\pi x\over a} e^{-iE_nt} \qquad \text{(Infinite square well)} \\
\Psi_0(x,t) = &{m\omega\over{\pi \hbar}}^{1/4} e^{-{m\omega\over{2\hbar}}x^2-iE_0t} \qquad \text{(Simple Harmonic Oscilator)}\\
\Psi_k(x,t) =& Ae^{i(kx-{hk^2\over{2m}}t)} \qquad \text{(Free Particle)}
\end{align}
I think the part that is bugging me is that for the two prior cases the conjugate only alters the time term, but in the last equation, we are also altering the position term. How exactly should I rationalize this and come up with a good generalized concept of what $\Psi^*$ is?
Best Answer
For every $x$ and $t$, $\Psi(x,t)$ is a complex number. $\Psi^*$ is the conjugate of that number, no more, no less. The reason it seems like sometimes it's only the $t$ part that gets conjugated is simply that often it is the only part of the wavefunction that is complex. Let's use your examples:
$\Psi = \sqrt{\frac{2}{a}} \sin(\frac{n\pi x}{a})e^{-iE_n t}$. We want to calculate $\Psi^*$. Well, since the conjugate of the product of two numbers is the product of their conjugates (that is, $(zw)^* = z^* w^*$), let's do it step by step.
First we need to conjugate $\sqrt{\frac{2}{a}}$, but since it's a real number, it is equal to its conjugate. So we leave it alone and move on. Now we need to conjugate $\sin(\frac{n\pi x}{a})$, but again, this is a real number, because $\sin x$ is real whenever $x$ is real. The last part is $e^{-iE_n t}$. This is actually complex, so we need to conjugate it, and its conjugate is $e^{iE_n t}$. So putting it all together, we have $\Psi^* = \sqrt{\frac{2}{a}} \sin(\frac{n\pi x}{a})e^{iE_n t}$.
Notice how at no point did I say something like "$\sin(\frac{n\pi x}{a})$ depends on $x$ so it shouldn't be conjugated". This is because I don't care what $x$ and $t$ are; all I care about is whether something is real or complex; it just so happens that in your first two examples, only the part that depends on $t$ is complex. But in the free particle wavefunction, everything is complex, so you need to conjugate everything.