Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  What are the limitations of the analogy between a complex classical oscillator with imaginary energy and a quantum system?

+ 2 like - 0 dislike
698 views

In this video lecture about mathematical physics (after 1:20:00) Carl Bender explains that allowing the energy of a complex classical oscillator to become imaginary can some kind of mimic quantum behavior of the system.

To explain this, he uses the example of a classical particle with the potential energy given by

\(V(x) = x^4 -x^2\)

If the energy of the particle  is larger than the energy of the ground state (but smaller than the barrier between the two potential wells) $E_1 > E_0$ it can oscillate in one of the two potential wells centereed around the two minima of $V(x)$, but not travel between them.

However, allowing the energy of the particle to become imaginary, say

\(E = E_0 + \varepsilon i\)

the trajectories of the particle in the complex plane are no longer closed and it can some kind of "tunnel" between the two potential wells. The "tunneling time"  $T$ is related to the imaginary part of the energy as

\(T \sim \frac{1}{\varepsilon}\)

In addition, Carl Bender mentions that taking the limit $\varepsilon \rightarrow 0$ of such a complex oscillator system some kind of mimics the process of taking the classical limit of a quantum system.

What are the limitations of this analogy between a complex oscillator with imaginary energy and a "real" quantum system (no pun intended ;-) ...)?

An aside: the jumping between the two potential wells of a complex oscillator with imaginary energy reminded me of a Lorenz Attractor ...

asked Apr 18, 2014 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
edited Apr 18, 2014 by Dilaton

Not related to your question, just want to say Carl Bender did show some surprising stuff in these lectures, I remember in lecture 3(?) he claimed that the system with repulsive potential $V(x)=-x^4$ has discrete energy levels. I'm still puzzled by this even today, maybe I should post it as another question sometime.

1 Answer

+ 3 like - 0 dislike

The true analogy is between a classical stochastic oscillator (which is always given by a complex frequency) and a quantum system.

Classically, the ''tunneling'' is visualized as climbing over the barrier (which I believe is also the correct way to view the qunatum tunneling).

The deterministic limit of a classical stochastic system, where the noise (hence the imaginary part) goes to zero, is analogous to the deterministic limit of a quantum system, where Planck's constant goes to zero. 

Indeed, classical stochastic systems and quantum systems are analogous from many points of view. They both have a probabilistic interpretation in terms of densities (functions in the classical case, matrices/operators in the quantum case), and they both can be handled with path integrals. The main difference is that stochastic path integrals are well-defined as they do not have the factor i in the exponent that causes severe oscillations in the quantum case.

Chaotic systems are different; they are classical _and_ deterministic! In particular, the jump between the two sides of a Lorentz attractor happens frequently, while tunneling = jumps over barriers is quite rare (unless the barriers are very low). 

answered Apr 18, 2014 by Arnold Neumaier (15,787 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar\varnothing$sicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...