Can a quantum formalism exist without a Hamiltonian Formalism

[elduderino]

I was thinking about this. In every problem I have worked, we suppose a hamiltonian exists which can describe the system. There are obviously Hamiltonians which are not possible classically, such as in the 1-D ising model of paramagnetism, where the Hamiltonian contains terms of $$s_i. s_j$$ where both s_i and s_j are discrete variables with values $$\pm 1$$. However the Hamiltonian is a classical concept. So does that mean that a quantum formalism cannot exist on its own without presupposing that an established classical theory exists?

 

[A. Neumaier]

I was thinking about this. In every problem I have worked, we suppose a hamiltonian exists which can describe the system. There are obviously Hamiltonians which are not possible classically, such as in the 1-D ising model of paramagnetism, where the Hamiltonian contains terms of $$s_i. s_j$$ where both s_i and s_j are discrete variables with values $$\pm 1$$. However the Hamiltonian is a classical concept. So does that mean that a quantum formalism cannot exist on its own without presupposing that an established classical theory exists?

The Hamiltonian is both a classical and a quantum concept, with a slightly different interpretation in the two cases.

In quantum mechanics, the Hamiltonian is the infinitesimal generator of the time translations. This is true no matter whether or not there is an associated classical system.

But this does not answer the question in the title, where the answer is no for conservative systems. A conservative quantum system whose state depends on time
is necessarily described by the Schroedinger equation, which can be viewed as a definition of the Hamiltonian.

Dissipative quantum systems, however, are characterized by a Lindblad operator in place of a Hamiltonian, and the latter not always has a Hamiltonian part. However, this is again analogous to the classical case: A damped harmonic oscillator cannot be described by a Hamiltonian (at least not without giving up the interpretation of the Hamiltonian as energy).

 

[elduderino]

A conservative quantum system whose state depends on time
is necessarily described by the Schroedinger equation, which can be viewed as a definition of the Hamiltonian.

If you make the transformation $$\psi (r,t) = e^{i \frac{S(r,t}{\hbar}}$$ then the Schrödinger equation goes to the Hamilton Jacobi equation of classical mechanics. Mathematically that is fine, but I don't quiet understand how this correspondence can be understood physically.

I can vaguely convince myself saying that the very definition of the Schrödinger equation includes the components necessary to construct a hamiltonian and can hence do the same things that the Hamiltonian can do (within the framework of the wave mechanics, which brings along uncertaintly relations and other queer things, which make commutators connect between canonical conjugates rather than poisson brackets, etc.)

Dissipative quantum systems, however, are characterized by a Lindblad operator in place of a Hamiltonian, and the latter not always has a Hamiltonian part. However, this is again analogous to the classical case: A damped harmonic oscillator cannot be described by a Hamiltonian (at least not without giving up the interpretation of the Hamiltonian as energy).

I did not know this. I will read into it. Thanks.

 

[tom.stoer]

I think that how strange a conservative quantum system may be, you always need an Hamiltonian; and strictly speaking even the path integral formalism (which does not contain H) is derived via the canonical formalism based on H.

 

[element4]

There are obviously Hamiltonians which are not possible classically, such as in the 1-D ising model of paramagnetism, where the Hamiltonian contains terms of $$s_i. s_j$$ where both s_i and s_j are discrete variables with values $$\pm 1$$.

Actually the Ising model is pretty much a classical model, it contains no quantum fluctuations. The Quantum Heisenberg model would be a better example.

 

[arkajad]

Think of a spin in a magnetic field. You have your Schrödinger equations for 2x2 matrices, no Hamilton-Jacobi equation, no" classical Hamiltonian" (unless you want to adapt your methods to your goal). Pure matrix algebra. And yet it is a quantum theory.

 

[tom.stoer]

But still you have a Hamiltonian in your Schrödinger equation

 

[arkajad]

But it does not come from quantization of a classical Hamiltonian of the rigid spinning top. It is a pure quantum mechanical case. Of course you can have "analogies" with classical physics, but they are not necessary. It's all group theory. You have a unitary representation od SU(2), so you have its generators. You have $\hbar$ and the magnetic field (perhaps time dependent), you use them to cook up the generator of time translations $U(t_1,t_0)$.

 

[tom.stoer]

But it does not come from quantization of a classical Hamiltonian ...

That was not my intention.

... you use them to cook up the generator of time translations $U(t_1,t_0)$.

Exactly.

Even a PI can be constructed (in principle) w/o having a classical action or a classical H. You simply take the Hilbert space, H and U and apply Feynmans construction.

 

[arkajad]

Even a PI can be constructed (in principle) w/o having a classical action or a classical H. You simply take the Hilbert space, H and U and apply Feynmans construction.

That would be rather unusual approach and certainly not the simplest one.
Spin IS peculiar in quantum mechanics. Even Bohmian's have some navigational problems with it.

 

[tom.stoer]

But wo/o having a classical action you can't do anything else :-)