This page presupposes a certain level of comprehension regarding the wave function and the postulates of quantum mechanics. If you lack familiarity with these topics, we recommend reviewing the basics of Wave and the fundamentals of quantum mechanics before proceeding further with this page.
Given the wavefunction, these are all the predictions that can be made about the system at a given moment in time. So what happens at a later time? We have seen that translations are tied to time derivatives, so the real question is, what is $\dfrac{\partial \psi(x,t)}{\partial t}$?
From basic plane waves, we have
$$ \psi(x,t;k)=e^{i(kx-\omega t)}. $$
Just remind you that
$$ E=\hslash \omega. $$
If we separate the exponential part and treat $e^{ikx}$ as a eigenfunction $\phi(x)$, it yields
$$ \psi(x,t)=e^{-i\omega t}\phi(x) $$
for any wave function. We can also see that
$$ i\hslash \dfrac{\partial \psi(x,t)}{\partial t} = \hslash \omega \psi(x,t) $$
in such a case. This seems to suggest that the energy operator $\hat{E}$ is tied to the operator $i\hslash \dfrac{\partial}{\partial t}$. Along with this, we need that translation in time respect superposition and that the total probability
$$ \int^{\infty}{-\infty}\mathfrak{p}(x,t)\mathrm{d}x=\int{-\infty}^{\infty}\psi^{*}(x,t)\psi(x,t) \mathrm{d}x =1 $$
be conserved. This means that the time derivative, which is linear, acting on a wavefunction should be equal to a linear operator acting on the wavefunction.
<aside> 📌 SUMMARY: We introduce the time dimension into the system, but it's essential to recognize that time possesses unique characteristics. In reality, time cannot be reversed, and notably, other observables like $x$ can not influence the progression of time. The $-\infty$ and $\infty$ signify that it considers every potential position $x$ for any possible $t$. Therefore, integration with respect to $x$ (or rest of the observables apart of $t$ ) is 1.
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