1
$\begingroup$

I am reading "The undivided universe: an ontological interpretation of quantum theory" and cannot understand this derivation.

From the Schrödinger equation: $$ i\hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi $$ They derive the quantum Hamilton-Jacobi equation $$ \frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V + Q = 0 $$

using the polar form $\psi = Re^{iS/\hbar}$, but I don't understand what they do afterwards.

$\endgroup$

closed as off-topic by ACuriousMind, honeste_vivere, Gert, Cosmas Zachos, garyp Jul 28 '16 at 19:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, honeste_vivere, Gert, Cosmas Zachos, garyp
If this question can be reworded to fit the rules in the help center, please edit the question.

2
$\begingroup$

I don't know where you have trouble, but it's straightforward if you just assume $R(x,t)$ and $S(x,t)$ in the form given for $\psi$ and then expand out the operations, where:

$$\begin{align} \psi &\equiv R(x,t)e^{i S(x,t)\hbar} \\ \partial_t \psi &\rightarrow \Big(\frac{\partial_t R}{R} + \frac{i \partial_t S}{\hbar} \Big)\psi \\ \nabla \psi &\rightarrow \Big(\frac{\nabla R}{R} + \frac{i \nabla S}{\hbar} \Big)\psi \\ \nabla^2 \psi &\rightarrow \Big(\frac{\nabla^2 R}{R} + 2\frac{\nabla R}{R} \cdot \frac{i \nabla S}{\hbar}+ \left(\frac{i \nabla S}{\hbar}\right)^2 \Big)\psi \\ \end{align} $$ divide through by $\psi$ and you can separate the Schrödinger equation into real and imaginary parts to get two independent equations:

\begin{align}\partial_t S &= \frac{\hbar ^2 }{2m}\frac{\nabla^2 R}{R} - \frac{1}{2m}(\nabla S)^2 - V\\ \partial_t R &= -\frac{1}{m}\nabla R \cdot \nabla S -\frac{1}{2m}R \nabla^2 S\end{align}

The first equation is the one you seek, where I've written $Q$ explicitly.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.