What is the meaning of classical and quantum equations?

[ndung200790]

Please teach me this:
It is seem to me that the classical equation is an equation describing the relation between operators.But quantum equation describes the relation of expectation values of physical quantities.Then corresponding principle only implies the one-one coresponding between operators and physical quantities,not corresponding between classical equation of classical values of physical quantities and equation of corresponding operators.
Thank you very much in advanced.

 

[ndung200790]

I think it maybe in the passage from classical physics to quantum physics,with accidental happening,there are a similar in special classical equations of macro physics quantities and quantum equation of physical operators(e.g the similar of Poisson bracket and commutators).Then the classical Lagrange equation is the starting point for all quantum procedure,because there is a corresponding between macro physics Lagrange equation and Lagrange equation of operators in quantum physics.
Thank you very much for your kind heart to help me.

 

[ndung200790]

In the limit to classical mechanics of quantum mechanics,the phase of wave function is proportional to the action.But I do not know whether this action is the same the action of the system when we consider the movement of wave packet(small packet) in the classical trajectory.

 

[tom.stoer]

Instead of comparung the Lagrangian one could compare the Hamiltonian formulations of classical and quantum mechanics: the classical q, p and H (Hamiltonian) is replaced with operators - and usually the classical equations of motions (for q and p derived from H) remain valid exactly as operator equations for the Heisenberg operators.

 

[A. Neumaier]

Instead of comparung the Lagrangian one could compare the Hamiltonian formulations of classical and quantum mechanics: the classical q, p and H (Hamiltonian) is replaced with operators - and usually the classical equations of motions (for q and p derived from H) remain valid exactly as operator equations for the Heisenberg operators.

Apart from the problems of ordering the operators...

There is usually a well-defined classical limit of a quantum system (or several such limits), but there is no well-defined procedure to go from a classical system to a quantum system.

Tho answer the original question: It is most natural to compare quantum mechanics with classical mechanics with uncertain initial conditions, since then both are stochastic systems. in this case an equation is interpreted as an equation between random variabl;es in the classical case, and between operators in the quantum case. In both cases, one takes expectations (or more complex statistical features) to get information about measurable things.

 

[ndung200790]

Because there is no well-defined procedure to change from classical mechanics to quantum mechanics.I wonder whether it is ''reality'' in quantized procedure of starting from classical Lagrangian in Quantum Field Theory.

 

[A. Neumaier]

Because there is no well-defined procedure to change from classical mechanics to quantum mechanics.I wonder whether it is ''reality'' in quantized procedure of starting from classical Lagrangian in Quantum Field Theory.

Although not a werll-defined procedure, there is a lot of heuristics for quantizing a system. But the result depends in the canonical approach on the ordering of the operators (e.g., qp^2q and pq^2p are essentially different operators but the same classically), and in the functional integral approach on details of how the path integral is defined.

 

[ndung200790]

How ''reality'' in this procedure: considering Klein-Gordon and Dirac equation,then finding out the corresponding classical Lagrangians.At last,using canonical or functional integral formalism.I am suspecting at the finding the classical Lagrangian from the quantum equations.Is it true that the classical Lagrangian is a Lagrangian of operators?

 

[ndung200790]

If the Lagrangian is of operators,why call them are ''classical'' Lagrangian?

 

[A. Neumaier]

If the Lagrangian is of operators,why call them are ''classical'' Lagrangian?

In the functional integral approach for Boson fields, the Lagrangian is classical since one integrates over classical fields. However, the results computed by a functional intgral approach give properties of a corresponding quantum theory.

In the covariant approach, the Lagrangian is _not_ classical, though it often has the same form.