Classical limit of quantum mechanics

Question

There is a well-known principle that one can recover classical mechanics from quantum mechanics in the limit as $\hbar$ goes to zero. I am looking for the strongest statement one can make concerning this principle. (Ideally, I would love to see something like: in the limit as $\hbar$ goes to zero, the position wavefunction reduces to a delta function and Schrödinger's equation / Feynman's path integral reduces to the Newtonian/Lagrangian/Hamiltonian equations of motion).

Answer 1

There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things out from this and then move to the Feynman integral. The other approach relies on substantial mathematical theorems due to Elliott Lieb and Barry Simon in the '70 and is essentially valid for many-body physics. These latter results make the limit $\hbar\rightarrow 0$ and $N\rightarrow\infty$ equivalent while the former is not really a physical limit due to the fact that Planck constant is never zero.

Starting from Feynman path integral, the standard formulation applies to a mechanial problem described from a Lagrangian $L$, normally $L=\frac{\dot x^2}{2}-V(x)$ but one can extend this to more general cases, and then the postulate is that, given a path $x(t)$, this must contribute to the full quantum mechanical amplitude of a particle going from the point $x_a$ to $x_b$ with a term $e^{\frac{i}{\hbar}S}$ being $S=\int_{t_a}^{t_b}dtL(\dot x,x,t)$ the action. All the possible paths contribute and so, the full amplitude will be given by the formal writing $$ A(x_a,x_b)\sim\int[dx(t)]e^{\frac{i}{\hbar}\int_{t_a}^{t_b}dtL(\dot x,x,t)}. $$ Be warned that this integral is not proved to exist yet, but the Wiener counterpart, that can be obtained changing $t\rightarrow it$, exists and describes Brownian motion. Now, if you take the formal limit $\hbar\rightarrow 0$ to this integral you will immediately recognize the conditions to apply the stationary phase method to it. This implies that the functional must have an extremum and this can be obtained by pretending that $$ \delta S=\delta \int_{t_a}^{t_b}dtL(\dot x,x,t)=0 $$ that is, the paths that give the greatest contribution are the classical ones and one recover the classical limit as a variational principle as learned from standard textbooks.

While this is a quite common approach, to extend what really happens to a macroscopic system that we can see to respect all the laws of classical mechanics, we have to turn our attention to the limit of a large number of particles $N\rightarrow\infty$. In this case one has more rigorous results. These are due to Lieb and Simon as already said above. They published two papers about

Lieb E. H. and Simon B. 1973 Phys. Rev. Lett. 31, 681.

Lieb E. H. and Simon B. 1977 Adv. in Math. 23, 22.

In the first paper, their theorem 4 states

Theorem: For $\lambda < Z$, let $E_N^0$ and $\rho_N^0(x)$ denote the ground-state energy and one-electron distribution function for N spin-$\frac{1}{2}$ electrons obeying the Pauli principle and interacting with $k$ nuclei as described above. Then (a) $N^{-\frac{7}{3}}E_N^0\rightarrow E_1$, as $N\rightarrow\infty$; (b) $N^{-2}\rho_N^0(N^{-\frac{1}{3}}x)\rightarrow\rho_1(x)$ as $N\rightarrow\infty$, where convergence in (b) means that for any domain $D\subset R^3$, the expected fraction of electrons in $N^{-\frac{1}{3}}D$ approaches $\int_D\rho_1 (x)d^3x$.

Where $\rho_1(x)$ and $E_1$ refer to the Thomas-Fermi distribution and the corresponding energy. This theorem states that the limit $N\rightarrow\infty$ for a quantum system, under some mild conditions, is the Thomas-Fermi distribution. A system with this distribution is a classical system. The fact that a system with a Thomas-Fermi distribution is a classical one can be seen through the following two references:

W. Thirring(Ed.), The Stability of Matter: From Atoms to Stars - Selecta of E. Lieb, Springer-Verlag (1997).

L. Hörmander, Comm. Pure. Appl. Math. 32, 359 (1979).

The second paper just gives the mathematical support to derive Thomas-Fermi approximation as the leading order of a classical expansion for $\hbar\rightarrow 0$ that I will not present here.

Answer 2

The question can be seen from two points: rather elementary one and hot research topic. Let me comment on both.

The elementary ones:

1) As it was mentioned by Jon and Chris by the stationary phase approximation you can immediately see that Feynman path integral for h->0 corresponds to extrema of the Lagrangian - which are precisely the classical equations of motion in Lagrange's description. (In particular case L=kineticEnery - PotentialEnergy you will get Newton's equation - this is subject of classical mechanics textbooks).

2) In the Heisenberg picture of QM one considers the equations of motion in the form

$d/dt \hat O = [\hat H, \hat O] $

The commutator of operators corresponds to Poisson brackets in the classical mechanics. So classical limit of this equation is:

$d/dt O = ${ $ H, O $ }

Which are Hamilton equations of motion in classical mechanics. So we see that quantum motion -> classical motion for h->0.

PS It might be worth to remind here the connection between Heisenberg picture and Schrodinger's: $d/dt \Psi = H\Psi$. This is purely linear algebra: if you consider the evolution on vector space V given by this equation then operators (i.e. V\otimes V^*) will evolve according to d/dt O = [H, O].

3) Now about what is the classical limit of wave function. The main surprise that it does not correspond to delta functions as naively expected. The first approximation is that classical limit corresponds to Lagrangian submanifolds.

Let me first give example: consider the wave function which is $\Psi(x)= \delta(x-x_0)$ (it is non-normalized, but still) naively it corresponds to particle which have coordinate equal to $x_0$. The classical limit of this would be a line on the phase space (p,q) which is $q=x_0$.

Let me repeat from Coherent states vs quantization of Lagrangian submanifold

Consider sumanifold defined by the equations $H_i=0$. Consider "corresponding" quantum hamiltonians $\hat H_i $, consider vector $\psi$ in the Hilbert space such that $\hat H_i \psi = 0$. This $\psi$ we are talking about. Why it is important "Lagrangian" ? It is easy. If $A \psi =0$ and $B\psi = 0$ then it is true for commutator $[A,B]\psi = 0$. In classical limit commutator correspond to Poisson bracket so we see that even if we start from $H_i$ which is not close with respect to Poisson bracket we must close it - so we get coisotropic submanifold. Lagrangian - just restiction on the dimension - that it should be of minimal possible dimension - so after quantization we may expect finite dimensional subspace (in the best case 1-dimensional).

Exercise: derive from this general prescription an example above.

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Current research

It is fruitful line of research to think about the correspondence between quantum and classical realms. Not just the limit, but sometimes one may hope to go in opposite direction and to completely describe quantum objects in terms of classical one. Sometimes it is subject of deepest and fascinating conjectures:

http://arxiv.org/abs/math/0512169

Automorphisms of the Weyl algebra

Alexei Belov-Kanel, Maxim Kontsevich

Abstract:

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. ...

So the conjecture says that classical object - automorphisms of Poisson algebra $\{ p_i, q_k \} =\delta_{ik}$ is exactly the same as quantum object - automorphism of Weyl algebra ($[p_i, q_k]=\delta_{ik}$)

It is related to famous Jacobian conjecture, see http://arxiv.org/abs/math/0512171 by the same authors.

Answer 3

Even though the $\hbar \rightarrow 0$ is commonly referred as the classical limit of quantum mechanics (QM), the truth is that QM doesn't have such a well-defined classical limit. Moreover, any argument on $\hbar \rightarrow 0$ is not valid as $\hbar$ is not a dimensionless parameter $^{*}$. For example, in special relativity $v/c \rightarrow 0$ represents the classical limit of the theory, and so does $r/r_c \rightarrow 0$ in General relativity ($r_c$ is the Schwarzschild radius), by which you can restore Newton's law of gravity. But the story is totally different for QM as there is no such a counterpart by which you can restore the classical limit.

($^{*}$To be clear, the fact that $\hbar$ isn't dimensionless means that depending on the units you choose you can make it arbitrarily small/large, thus rendering it invalid as a classical limit.)