Need verification if this simple derivation of the Schrödinger equation is valid

Question

By 1924 it was well observed that matter (as well as light) has wave-particle duality (later named quantum), and the wavelength-momentum-energy relation of quanta $$\lambda=\frac{h}{p}\;\;\longleftrightarrow\;\;p=\hbar k\;\;\longleftrightarrow\;\;E=\hbar\omega$$ had been hypothesized (and experimentally validated) by de Broglie. Let us model a quantum as a periodic function $\Psi$ of spacial cooridnates and time, which can be expressed in the form of Fourier series $$\Psi=\sum_{n\in\mathbb{Z}}A_n\psi_n$$ where $$\psi_n=e^{i\left(\mathbf{k}_n\cdot\mathbf{x}-\omega_n t\right)}.$$ We want to get the information about the momentum and energy of such quantum, for now only its basis $\psi_n$. Consider the followings $$\begin{align*} \nabla^2\psi_n&=-k^2\psi_n& &\longleftrightarrow& -\hbar^2\nabla^2\psi_n&=p_n^2\psi_n\\ \frac{\partial}{\partial t}\psi_n&=-i\omega_n\psi_n& &\longleftrightarrow& i\hbar\frac{\partial}{\partial t}\psi_n&=E_n\psi_n. \end{align*}$$ which can be interpreted as eigenvalue problems with the operators $$\hat{p^2_n}:=-\hbar^2\nabla^2\quad\text{and}\quad\hat{E_n}:=i\hbar\frac{\partial}{\partial t}.$$ The total energy of the quantum is given $$\sum_{n\in\mathbb{Z}}E_n=\frac{1}{2m}\sum_{n\in\mathbb{Z}}p_n^2+V$$ and by superposition principle we can write down the following equation $$\hat{E_n}\psi_n=\frac{1}{2m}\hat{p_n^2}\psi_n+\hat{V}\psi_n$$ or equally $$i\hbar\frac{\partial}{\partial t}\psi_n=-\frac{\hbar^2}{2m}\nabla^2\psi_n+\hat{V}\psi_n.$$

I have never learned or seen this derivation. Other derivations were most of the time too advanced to me to follow (the math) or the equation itself was taken for granted in the first place. My question is that if this derivation makes sense and I'm doing right.

Answer 1

Sure, this is valid, subject to some hidden assumptions. The issue would be that those assumptions haven't been explicitly stated, and it isn't trivial to figure out the complete list of hidden assumptions. Some hidden assumptions:

1. That $E=\hbar \omega$. This is plausible on dimensional grounds and because of the relativistic analogy energy:momentum::time:space, but it gets cloudy when you remember that this is a derivation of the nonrelativistic Schrodinger equation. Einstein had already published this relation as an empirical one based on stuff like the photoelectric effect, but it wasn't obvious at the time how that fit into the structure of physics.

2. That wavefunctions live in the field of complex numbers, not, e.g., in the reals or the quaternions. This is basically needed if you want to get conservation of probability, but that would not be obvious if you hadn't already tried, e.g., making a real Schrodinger equation and seeing conservation of probability fail.

3. That the relevant degree of freedom is position, as opposed to something like spin.

4. That position is an observable. This is false in quantum field theory, and even in nonrelativistic field theory there is the issue that we don't have eigenstates of position unless we allow things like Dirac deltas.

5. That it's OK for phase and normalization to be unobservable.