# How can the Hamilton-Jacobi equation represent the motion of a particle as a wave? [duplicate]

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.
• 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? – Thatpotatoisaspy Dec 11 '19 at 2:48
• @Thatpotatoisaspy See my edit trying to answer your questions. – Thomas Fritsch Dec 11 '19 at 15:44
• Fritsche Thank you a lot! That really helps. – Thatpotatoisaspy Dec 11 '19 at 23:44