Hmm. How about this:
Following (Fourier transform) holds for any wave packet
\psi(x,t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \phi(k) e^{i(kx-wt)} dk
Let us assume that matter is a kind of wave, so obeys this equation. Now, to relate the wave with quantities we all know and love, we assume (as an axiom) deBroglie hypotesis, and state following two:
E = \hbar w
p = \hbar k
and re-write the wave equation.
\psi(x,t) = \frac{1}{\sqrt{2 \pi} \hbar} \int_{-\infty}^{\infty} \phi(p) e^{\frac{i}{\hbar}(px-Et)} dp
By merely taking partial derivatives of both sides
\frac{\partial \psi(x,t)}{\partial x} = \frac{1}{\sqrt{2 \pi} \hbar} \int_{-\infty}^{\infty} (\frac{\partial}{\partial x}) \phi(p) e^{\frac{i}{\hbar}(px-Et)} dp
We get
\frac{\partial \psi(x,t)}{\partial x} = \frac{i}{\hbar} p \psi(x,t)
or
\frac{\hbar}{i} \frac{\partial \psi(x,t)}{\partial x} = p \psi(x,t)
Doing the same steps with partial time derivative, we can derive this
-\frac{\hbar}{i} \frac{\partial \psi(x,t)}{\partial t} = E \psi(x,t)
And this's it. This's how we can
extract energy and momentum from wave function. As we all know, it's the common way to abstract these equations from wave functions, and name rest
operators and
eigenvalues.
\frac{\hbar}{i} \frac{\partial }{\partial x} \Rightarrow \hat{p}
-\frac{\hbar}{i} \frac{\partial }{\partial t} \Rightarrow \hat{E}
Writing Schrödinger, or Klein-Gordon equation is straightforward here. For classical mechanics, we have
\frac{p^2}{2m} + V = E
Since we have to
extract momentum and energy from wave function, we'd rather write
\frac{\hat{p}^2}{2m}\psi + V\psi = \hat{E}\psi
which is actually
\frac{p^2}{2m}\psi + V\psi = E\psi
\frac{p^2}{2m} + V = E
If we preferred the relativistic case, we'd have
(E^2 - {(pc)}^2 - {(m_0 c^2)}^2) = 0
and by our usual substitution
(\hat{E}^2 - {(\hat{p}c)}^2 - {(m_0 c^2)}^2)\psi(x,t) = 0
Like it?
But I think, this doesn't justifiy we can derieve Schrödinger equation. It turns out that the equations we borrowed from classical mechanics are a consequence of quantum theory, and actualy
they are derived from QM.
