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.)
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/17651/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Jul 15, 2012 at 22:08
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$\begingroup$ A delta function isn't some sort of classical limit of quantum mechanics, its just an eigenstate of the position operator. $\endgroup$– DJBunkCommented Jul 16, 2012 at 2:36
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$\begingroup$ What isyour background knowledge to which an answer should be adapted? $\endgroup$– Arnold NeumaierCommented Jul 16, 2012 at 7:41
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1$\begingroup$ Possible duplicate, but still quite an interesting alternative perspective. $\endgroup$– Emilio PisantyCommented Jul 17, 2012 at 0:15
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$\begingroup$ Cross-posted to mathoverflow.net/questions/102313 $\endgroup$– Qmechanic ♦Commented Jan 5, 2013 at 16:55
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6 Answers
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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.
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4$\begingroup$ Dear @juanrga, for your information, Physics.SE has a policy that it is OK to cite oneself, but it should be stated clearly and explicitly in the answer itself, not in attached links. Also it is frown upon to post nearly identical answers to similar posts. $\endgroup$– Qmechanic ♦Commented Oct 29, 2012 at 21:41
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$\begingroup$ Edited, although the policy only states "that you should disclose personal connections whenever you reference something you're involved with" and the nick makes the connection rather evident. Regarding nearly identical questions, it is waited that the answers will be very similar and the material cited reply both questions. How to answer in such case? $\endgroup$– juanrgaCommented Oct 30, 2012 at 11:38
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$\begingroup$ In such cases, it is often better to just flag/comment about duplicate questions, so they can get closed. $\endgroup$– Qmechanic ♦Commented Oct 30, 2012 at 12:05
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2$\begingroup$ So, is the answer 'yes' or 'no'? Can Planck's constant be considered a variable in the limit to zero or does this fail? Lots of people seem to believe that it works, and can generate any representation of classical mechanics. $\endgroup$ Commented Nov 28, 2019 at 2:49
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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
answered Nov 23, 2012 at 11:38
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1$\begingroup$ The iucaa.ernet.in link does not work. Link: doi.org/10.1007/s12045-009-0091-8 Is this the correct article? $\endgroup$ Commented Jul 1, 2021 at 21:56
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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:
Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)
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
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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.
answered Jul 16, 2012 at 9:01
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$\begingroup$ What I don't understand goes deeper than this. What you say makes sense to me for $\sigma_x \sigma_p = \hbar/2$. But that's not how QM works, we have to explain how $\sigma_x \sigma_p \ge \hbar/2$ limits to classical motion and interactions. If a particle travels for a light-year, then its position spread will be huge, and interferometry experiments can be done, even with a small(er) $\hbar$. It would seem very hard to get a non-statistical universe. $\endgroup$ Commented Oct 29, 2012 at 21:18
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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.
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$\begingroup$ "the packet still spreads" — not entirely. In the limit $\hbar\to 0$, coherent states concentrate in arbitrarily small areas of phase space, and $\hat{U}(t_2-t_1)$ transports a decohered wavepacket localized at $\mathbf{x}_1$ along its classical trajectory. $\endgroup$ Commented Sep 5, 2019 at 1:50
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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:
