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4 finally inserted { } into eq. of motion

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 $q=x_0$.

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.

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.

3 deleted 1 characters in body

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 that 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 $q=x_0$.

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.

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.

2 deleted 1 characters in body

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 that 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: $d/dt \Psi = H\ Psi$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$. Consider sumanifold defined by the equations$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$which is not close with respect to Poisson bracket we must close it - so we get coisotropic submanifold. Lagrangian - just restiction on the dimension - that it should be of minimal possible dimension - so after quantization we may expect finite dimensional subspace (in the best case 1-dimensional). Exercise: derive from this general prescription an example above. Current research It is fruitful line of research to think about the correspondence between quantum and classical realms. Not just the limit, but sometimes one may hope to go in opposite direction and to completely describe quantum objects in terms of classical one. Sometimes it is subject of deepest and fascinating conjectures: http://arxiv.org/abs/math/0512169 Automorphisms of the Weyl algebra Alexei Belov-Kanel, Maxim Kontsevich Abstract: We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. ... So the conjecture says that classical object - automorphisms of Poisson algebra${ 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.

1