Classical limit of quantum mechanics

Question

I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\hbar$ goes to zero, the position wavefunction reduces to a delta function and that the Schrodinger equation/Feynman path integral reduces to the Newtonian/Lagrangian/Hamiltonian equations of motion.)

Answer 1

The short answer: No, classical mechanics is not recovered in the $\hbar \rightarrow 0$ limit of quantum mechanics.

The paper What is the limit $\hbar \rightarrow 0$ of quantum theory? (Accepted for publication in the American Journal of Physics) found that

Our final result is then that NM cannot be obtained from QT, at least by means of a mathematical limiting process $\hbar \rightarrow 0$ [...] we have mathematically shown that Eq. (2) does not follow from Eq. (1).

"NM" means Newtonian Mechanics and "QT" quantum theory. Their "Eq. (1)" is Schrödinger equation and "Eq. (2)" are Hamilton equations. Page 9 of this more recent article (by myself) precisely deals with the question of why no wavefunction in the Hilbert space can give a classical delta function probability.

Answer 2

The classical limit is achieved only if the quantum system is in nice enough states. The states in which the classical limit makes sense are called coherent states. For example, the Glauber coherent states of an anharmonic oscillator are labelled by the classical phase space coordinates $(q,p)$.

If the system is in a coherent state then for every observable $A$ with $\langle A\rangle = \overline A$, the variance $\langle (A- \overline A)^2 \rangle$ tens to zero in the limit $\hbar\to 0$. Moreover, the dynamics reduces in the limit to that of a classical system on the corresponding phase space.

For Glauber coherent state this is related to the classical limit of Wigner functions; see, e.g.,
http://www.iucaa.ernet.in:8080/xmlui/bitstream/handle/11007/206/256_2009.PDF?sequence=1
But the above statement can be proved for many classes of coherent states and many phase spaces, not only for Glauber coherent states. For the case of coherent states related to Berezin quantization, see, e.g., https://arxiv.org/abs/math-ph/0405065

There is a large number of highly cited papers on the classical limit, e.g.,
https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-71/issue-3/The-classical-limit-of-quantum-partition-functions/cmp/1103907536.full
http://projecteuclid.org/euclid.cmp/1103859623

Answer 3

If one is really interested this essay by Lubos Motl explains how classical fields emerge from quantum mechanical ones.

I quote the introduction:

After a very short summary of the rules of quantum mechanics, I present the widely taught "mathematical limit" based on the smallness of Planck's constant. However, that doesn't really fully explain why the world seems classical to us. I will discuss two somewhat different situations which however cover almost every example of a classical logic emerging from the quantum starting point:

1. Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)

2. Decoherence which makes us interpret absorbed particles as point-like objects and which makes generic superpositions of macroscopic objects unfit for well-defined questions about classical facts

Answer 4

Qmechanics' link gives a good illustration of the classical limit of the Schrodinger equation.

Re your question about position: it's possible to get the position to arbitrary accuracy in quantum mechanics. The problem is that the uncertainty principle means the momentum becomes infinitely uncertain. As $\hbar \rightarrow 0$ it becomes possible to get both the position and momentum to abitrary accuracy.

Answer 5

no, the wavefunction doesn't approach the delta function in the classical limit. the wavefunction packet still spreads, and gets entangled with the environment. What textbooks don't tell you is you need decoherence to get to the classical limit.

Answer 6

If somebody has told you that classical physics is the ℏ→0 limit of quantum mechanics that person is wrong. One problem with trying to understand the spread of the wavefunction in the classical limit by taking ℏ≈0 or taking the limit ℏ→0 is that in reality ℏ≠0. Taking the real non-zero value of ℏ there are cases of perfectly ordinary objects for which Ehrenfest's theorem doesn't imply anything like classical behaviour. Rather the wavefunction of that object spreads out a lot over time. See Zurek and Paz's paper on the quantum mechanical theory of the orbit of Saturn's moon Hyperion:

https://arxiv.org/abs/quant-ph/9612037

The classical limit is actually a result of decoherence and information being copied out of 'classical' objects into the environment:

https://arxiv.org/abs/quant-ph/0703160