While it’s often said that the Schrödinger Equation cannot be derived and “came from the mind of Schrödinger”, a quick google search led me to Schrödinger’s original paper where he introduced the equation. In it, he appears to be describing having “derived” it from the Hamilton-Jacobi Equation. Wikipedia also tells me: “The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave.”

Reading through the paper, much of Schrödinger’s thought process is lost on me, so if anybody could explain it in perhaps a more easily digestible way that would be much appreciated. Looking at the equation as is, It’s not immediately obvious to me that it is a wave equation, especially one that could be reduced to the form of Schrödingers.


Let's take a simple system as an example: a single particle (with mass $m$) in a potential $V(\vec{r})$.

The Hamilton-Jacobi equation of this system is $$ \frac{1}{2m}(\vec{\nabla}S)^2 + V(\vec{r})= - \frac{\partial S}{\partial t}. \tag{1}$$

Schrödinger's equation for the same system is: $$ -\frac{\hbar^2}{2m}\vec{\nabla}^2\psi + V(\vec{r})\psi = i\hbar\frac{\partial\psi}{\partial t}. \tag{2}$$

To solve (2) you can make the approach $$ \psi(\vec{r},t) = A(\vec{r},t) e^{iS(\vec{r},t)/\hbar} \quad \text{with some slowly varying } A(\vec{r},t) \tag{3}$$ Obviously here $A$ is the amplitude, and $S/\hbar$ is the phase of the wave function $\psi$.

When inserting (3) into (2), then it can be shown that the Hamilton-Jacobi equation (1) is approximately true (exactly true in the limit $\hbar \to 0$).
For details see also "How do you get quantum Hamilton-Jacobi equation from Schrödinger equation?".

It is true what you have heard: Schrödinger's equation cannot rigorously be derived from classical physics. Therefore Schrödinger had to guess it with a great amount of intuition. As shown above it is actually the other way round: The Hamilton-Jacobi equation (1) can be derived from Schrödinger's equation (2).

The HJE describes classical trajectories of a particle, but not a wave. Therefore I don't agree with Wikipedia's statement "The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave."
However, the solution $S(\vec{r},t)$ of the HJE is closely related to a wave. The surfaces $S(\vec{r},t)=\text{const}$ are the wave-fronts (the surfaces of constant phase) of a Schrödinger wave-function $\psi(\vec{r},t)$.

The intuition leading Schrödinger to his equation were these requirements:

  • It should be a linear differential equation (in order to produce the experimentally observed superposition phenomenons).
  • It should be a second-order differential equation in space (in order to produce the experimentally observed waves).
  • It should somehow lead to the classical HJE, at least as an approximation.
  • $\begingroup$ Thank you very much for your answer. I think what i'm really wondering though is where schrodinger got the intuition to make a wave equation that could be used to re-obtain the hamilton-jacobi equation. Basically, as wikipedia states, how can the hamilton-jacobi equation (by itself) represent the motion of a particle as a wave? $\endgroup$ – Thatpotatoisaspy Dec 11 '19 at 2:48
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    $\begingroup$ @Thatpotatoisaspy See my edit trying to answer your questions. $\endgroup$ – Thomas Fritsch Dec 11 '19 at 15:44
  • $\begingroup$ Fritsche Thank you a lot! That really helps. $\endgroup$ – Thatpotatoisaspy Dec 11 '19 at 23:44

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