Hey everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I'm trying to understand more about the correspondence between classical and quantum physics, in particular the relation between classical mechanics in Hamilton-Jacobi form and the Schrödinger equation. The former can be obtained from quantum theory by plugging

[tex]\psi=\psi_0 e^{\frac{i}{\hbar}S}[/tex]

into the Schrödinger equation, which yields

[tex]\frac{1}{2m}(\nabla S)^2 + \frac{\partial S}{\partial t} - \frac{i \hbar}{2m}\Delta S=0,[/tex]

where the last term vanishes in the classical limit. My first question is: What is the interpretation of this quantum term? How is the classical theory affected by it?

Then, one can also start from the classical Hamilton-Jacobi equation and, with the help of the wave function [tex]\psi[/tex] from above, rewrite it as

[tex]\frac{\hbar^2}{2m\psi}(\nabla \psi)^2 + i\hbar\frac{\partial \psi}{\partial t} = 0. [/tex]

In this light, one obtains the quantum equation by "replacing the square of the derivative with the second derivative", citing a phrase I sometimes came across. So my second question is: How can this nonlinear variant of the Schrödinger equation be interpreted, how is its behaviour different from the full quantum theory?

In summary: What is the interpretation of the operation "replace square of derivative with second derivative" ?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Correspondence betw. Hamilton-Jacobi & Schrödinger eqns.

Loading...

Similar Threads - Correspondence betw Hamilton | Date |
---|---|

I Hamilton-Jacobi equation | Sep 5, 2017 |

I What does the Hadamard correspond to in hardware? | Apr 5, 2017 |

Correspondence principle and its use | Nov 22, 2015 |

Standing waves corresponding to a confined particle | Feb 24, 2015 |

Action corresponding to photon emission | Nov 7, 2014 |

**Physics Forums - The Fusion of Science and Community**