Correspondence between Hamiltonian mechanics and QM

In summary, the dynamical variables in quantum mechanics correspond to linear Hermitian operators, while the vector fields generated by these variables in classical mechanics are associated with the propagator in QM. However, not all vector fields in classical mechanics can be directly associated with operators in QM, as they do not satisfy certain axioms of pre-quantization. A deeper understanding of this topic can be found in mathematical physics books under the topic of pre-quantization.
  • #1
guhan
43
1
To which entitity (operators, wavefunctions etc) in quantum mechanics do the dynamical variables and the hamiltonian vector fields that they generate (through Symplectic structure of classical mechanics) correspond to?
 
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  • #2
The dynamical variables of position and momenta correspond to (linear) Hermitian operators. In particular, the commutator of two QM operators is given by the imaginary unit times the poisson bracket of the corresponding classical variables. The phase space of a quantum system is in general an infinite-dimensional Hilbert space, in contrast with the symplectic manifold of classical phase space.
 
  • #3
Ok.

And to what (in QM) does the vector field, say [tex]\overline{df}[/tex], generated by a dynamical variable, say [tex]f[/tex] (in Classical mechanics) correspond to?

I think I can associate the propagator in QM to the vector field generated by the Hamiltonian in classical mech, but what about other vector fields?
 
  • #4
I thought I would give this thread a proper burial...

The formulation of QM is through a homomorphism of Lie algebras from the poisson bracket in classical hamiltonian mech to commutation relationships in QM. This is to handle the correspondence principle used in physics, in a more general way. The requirements are more than just the existence of this homomorphism (the kernel of which are the constant functions, i.e. [tex]f=constant[/tex]). Although we would be tempted to associate [tex]X_f[/tex] (the vector field generated by [tex]f[/tex] in classical Hamiltonian mechanics) with the operators in QM, it turns out that it does not satisfy some axioms of pre-quantization (specifically, the axiom that constant functions should be mapped to multiplication by that function). In the end, we can associate [tex]-iX_f + f + \theta (X_f)[/tex] (where [tex]d\theta = \Omega[/tex], the symplectic 2-form) to QM operators. Anyway, a full treatment of this is given in mathematical physics books under the topic of (geometric) 'pre-quantization'. As far as I understand, a closely related issue is proving to be a big hurdle in rigorously formulating topological QFT.

note: [tex]X_f = \overline{df}[/tex] = 1-vector field that is generated by [tex]f[/tex] and which corresponds to the 1-form [tex]df[/tex] (correspondence is through the isomorphism between the tangent and cotangent space that is induced by the symplectic 2-form [tex]\Omega[/tex] )
 
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Related to Correspondence between Hamiltonian mechanics and QM

What is the correspondence between Hamiltonian mechanics and quantum mechanics?

The correspondence between Hamiltonian mechanics and quantum mechanics is a mathematical framework that connects the classical mechanics of Hamiltonian systems to the quantum mechanics of wave functions. It allows for the translation of classical equations of motion into quantum operators, providing a bridge between the two theories.

How does Hamiltonian mechanics relate to the Schrödinger equation?

Hamiltonian mechanics provides a classical description of a system, while the Schrödinger equation describes the quantum behavior of the same system. By applying the correspondence principle, classical equations of motion can be translated into quantum operators, which can then be used in the Schrödinger equation to determine the time evolution of a quantum system.

What are the advantages of using the correspondence between Hamiltonian mechanics and QM?

The correspondence between Hamiltonian mechanics and QM allows for a better understanding of the relationship between classical and quantum systems. It also allows for the application of classical methods and techniques, such as the use of Hamiltonian and Lagrangian mechanics, to quantum systems.

What are some examples of systems that can be described using the correspondence between Hamiltonian mechanics and QM?

The correspondence between Hamiltonian mechanics and QM can be applied to a wide range of systems, including the harmonic oscillator, the hydrogen atom, and the study of molecular vibrations. It is also used in solid state physics, nuclear physics, and many other areas of quantum mechanics.

Are there any limitations to the correspondence between Hamiltonian mechanics and QM?

While the correspondence between Hamiltonian mechanics and QM is a powerful tool, it does have its limitations. It does not fully capture the probabilistic nature of quantum systems, and it cannot be applied to systems with a large number of particles. Additionally, it does not provide a complete understanding of the behavior of entangled systems.

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