Semiclassical Regime: What Is It & When to Use It

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In summary, a semiclassical regime is when a system is treated as a combination of classical and quantum behaviors. This is typically done through the WKB approximation, where quantum fluctuations around the classical trajectory are considered small corrections. The path integral and free energy can also be calculated in this regime, allowing for a description of one part of the system as semiclassical while the other remains fully quantum. This approach is often used in cases such as electrons in a classical electromagnetic field.
  • #1
xylai
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Many papers say "we are treating a semiclassical regime".
I don't know what is a semiclassical regime and when a system can be treated as a semiclassical regime?
Thank you!
 
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  • #2
This term is usually used quite loosely.
The idea is that you look at what the system is doing classically, and then treat quantum fluctuations around the classical trajectory as being small corrections.

I would suggest you take a look at the WKB approximation (in say, Griffiths, or some other QM book; Wikipedia's explanation is terrible)

Ok, so here's the money:

In quantum mechanics we're often interested in the path integral, which is of course a sum over paths weighted by their classical action:

[tex]Z = \int\!\mathcal{D}q\, e^{i S[q]}[/tex]

Let's Wick rotate to make things convergent

[tex]Z = \int\!\mathcal{D}q\, e^{-S[q]}[/tex]

The assumption is that S is quadratic around it's minimum, and that it is very steep (compared to [tex]\hbar[/tex] which I have set to 1), meaning that the paths that veer slightly from the classical minimum get heavily suppressed by the exponential. Thus, we may use a saddle-point approximation:

[tex]S[q] \approx S[q_0] + \frac{1}{2!}(q-q0) S''[q_0] (q-q0)[/tex]

where [tex]q_0[/tex] is the classical solution, and [tex]q-q0[/tex] is a small fluctuation.

[tex]Z \approx e^{-S[q_0]} \int\!\mathcal{D}q\, e^{-(q-q0)S''[q](q-q0)}[/tex]

or

[tex]Z \approx e^{-S[q_0]} \sqrt{ \det S''[q_0] } [/tex]

Even more interesting than the path integral or partition function is the logarithm of it, or the free energy, which generates connected Feynman diagrams. Thus

[tex] W = \ln Z = -S[q_0] - \ln \det S''[q_0][/tex]

Why would we want to do such a thing? Well, sometimes you can treat one part of the system semiclassically and leave the other part fully quantum mechanically. Then you'd still have to integrate over some other dynamical degree of freedom, say, [tex]x[/tex], but you have an effective action description of [tex]q[/tex]. An example is to have electrons in a "background" or "classical" electromagnetic field where you treat the field's quantum corrections on a first order basis.
 
  • #3
many thanks.
You have said about this problem so detailedly.
 

1. What is the semiclassical regime?

The semiclassical regime is a concept in quantum mechanics that combines elements of both classical and quantum mechanics. It is used to describe systems where the classical description is insufficient, but the full quantum mechanical description is not necessary.

2. How is the semiclassical regime different from classical and quantum regimes?

In the classical regime, the behavior of a system can be described using Newton's laws of motion. In the quantum regime, the behavior is described using wavefunctions and probabilities. In the semiclassical regime, both of these descriptions are used simultaneously.

3. When should the semiclassical regime be used?

The semiclassical regime should be used when the system being studied has both classical and quantum aspects that are important to its behavior. This is often the case for systems with both macroscopic and microscopic components.

4. What are some common examples of systems in the semiclassical regime?

Some common examples of systems in the semiclassical regime include atoms and molecules, where the electrons are described quantum mechanically but the nuclei can be described classically, and quantum dots, where the electrons are confined to a small region but the motion of the entire dot can be described classically.

5. How is the semiclassical regime related to the correspondence principle?

The semiclassical regime is related to the correspondence principle, which states that the predictions of quantum mechanics should approach those of classical mechanics in the limit of large quantum numbers. In the semiclassical regime, this principle is applied to systems with both classical and quantum aspects, allowing for a more accurate description of their behavior.

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