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

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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 (4,295 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

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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 (12,425 points) [ no revision ]

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