Transition from quantum to classical mechanics

Question

As I understand it, if $S \gg h$ then we are in the classical realm, whereas if $S \leq h$ we are in the quantum realm. My question is what happens somewhere in between those 2 limits? Are we quantum and classical at the same time?

Answer 1

The heuristic that compares the action $S$ to Planck's constant is vaguely useful as an initial criterion, but the limit from quantum to classical mechanics is rather more subtle, in ways that make the simplistic comparison close to useless in practice.

As a pair of counter-examples:

• If you prepare a harmonic oscillator in a coherent state, then in practice it will be indistinguishable from something you could model as a classical harmonic oscillator with some added shot noise, and this happens regardless of the mean number of excitation states or of the ratio $S/h$.

• On the other hand, it is technologically challenging but in-principle possible to prepare an $n$-photon Fock state with an arbitrarily high but well-defined photon number $n$, and this will exhibit clearly quantum behaviour even for arbitrarily large $S/h$.

Thus the limit from quantum to classical mechanics needs to be done more carefully, and a simple heuristic will never suffice beyond serving as a fuzzy qualifier.

Answer 2

In Feynman path integral formalism, which is the context of your question as I understand it, the classical domain is recovered for $S \gg \hbar$. If $S$ is on the contrary of the order of $\hbar$, then the system will exhibit quantum behaviours of one sort or another, regardless of whether $S \ge \hbar$ or $S \le \hbar$. Now as @Countto10 and @Emilio Pisanty correctly stated, the devil is in the details and this statement of mine is full of caveats. But I guess you just wanted the gist of it.

Answer 3

There are two different ways in which this question can be asked:

1) ISOLATED SYSTEM: If you are asking about an isolated system then recall that the classical equations of motion are derived by minimizing the action. If $S>> \hbar$ then the saddle point approximation to the path integral

$$ \int [dx] e^{i S(x)/\hbar} \approx e^{i S(x_{cl}) /\hbar} $$

is a good approximation as fluctuations around the classical path cancel out. This implies that all the correlators will be peaked around their classical values.

Let me give a trivial example. Suppose you are interested in the transition amplitude from $x_0$ at time $t_0$ to $x_1$ at $t_1$. This is give by

$$ \mathcal A = \langle x_1| e^{-i H (t_1-t_0)}|x_0 \rangle $$

Using the Hamiltonian for a free particle we get

$$ \mathcal A = \int_{-\infty}^\infty dp e^{- i (\frac{p^2}{2m} \delta t - p \delta x)} $$

Being a Gaussian integral its trivial to solve but solving it will defeat the purpose. What we want to observe is that if $\delta x,\delta t \gg 1$ ($\hbar$ in natural units) we can approximate the integral by the value at the saddle point $p= m \frac{\delta x}{\delta t}$. You'll recognize this as the classical definition of momentum.

This method is often called 'sum over all paths' but the different paths are coming only because the initial and final states are not eigenstates of the Hamiltonian. For instance if one were to take a Harmonic oscillator in the $n^{th}$ state we would get

$$ \langle m,t| n,0 \rangle = e^{-i n t} \delta_{mn} $$

that is unless the final state is exactly the same as the initial the amplitude is 0. One can also start off the harmonic oscillator in a position eigenstate or coherent state and see how it leads to sum over all paths and can also play with why the evolution of a coherent state seems to track a classical path even when $S\sim \hbar$ as mentioned in another answer above.

2) OPEN SYSTEM: If you are interested in an open system then decoherence from interaction with the environment puts the system in a impure state or density matrix where the density matrix is diagonal in the so called pointer basis (environmental superselection). For large enough ''classical objects'' this basis is usually position and for small enough ''quantum objects'' this basis is usually energy. However, in a lab setting this decoherence can be controlled by tuning the interaction with the environment to be something else. For instance a very trivial example is making the pointer basis up-down or left-right trajectory of a beam of electrons by rotating the Stern-Gerlach apparatus. [ref: http://www.springer.com/gp/book/9783540357735]