Time Evolution Operator & Classical Hamiltonian: Relating the Two

[jfizzix]

I have been studying the theoretical framework of quantum mechanics in an attempt to have a working understanding of the subject, if not a comprehensive one, and I have hit upon the following stumbling block.

Now, given that the orthogonality of states is preserved with time, it is easily shown that any time evolution operator has to be unitary, and with a clever choice of notation, we can in effect, derive the abstract form of the Schrödinger equation, where

i*hbar*(d/dt)Psi = H*Psi

and H is some hermitian operator.

My question then, is how to show that this observable H is in fact the quantum mechanical analogue of the classical Hamiltonian.

There are compelling connections to classical mechanics:

(d/dt)F = {F,H}, where {F,H} is the Poisson bracket of F(x,p) with the classical Hamiltonian H(x.p)

(d/dt)<F> = -(i/hbar)<[F,H]>, where <[F,H]> is the expectation value of the commutator of the quantum mechanical analogue F of F(x,p) with the observable H.


Though these equations are very similar, they don't actually show that H is the quantum mechanical analogue of the Hamiltonian.


Any help with this would be greatly appreciated.

-James

 

[meopemuk]

Hi James,

welcome to the Forum!

If you take some quantum-mechanical Hamiltonian H(p,x) and use it to study the time evolution of localized wave packets, you may find that centers of those wave packets closely follow trajectories predicted by classical Hamilton's equations of motions with the (classical) Hamiltonian H(p,x). So, you may decide to replace the wave packets with classical points and switch to usual classical mechanics. In this (classical) limit the quantum-mechanical Hamilton operator H(p,x) transforms to the classical function H(p,x) in the phase space.

 

[Hans de Vries]

Though these equations are very similar, they don't actually show that H is the quantum mechanical analogue of the Hamiltonian.

Any help with this would be greatly appreciated. -James

The path from the classical Hamiltonian to the time evolution operator of the
Schrödinger equation takes a number of steps:1) Start with the relativistic classical Hamiltonian (section 22.1 of ref 1)

2) Derive the classical Hamiltonian density (section 22.3 22.5 and 22.6 of ref 1)

3) Substituting derivatives into the classical version gives the Hamiltonian density
of the Klein Gordon field. (section 22.6 of ref 1)

4) The same substitution transforms the classical relativistic Lagrangian density
into the Lagrangian density of the Klein Gordon field. (section 22.5 of ref 1)

5) The Euler Lagrange formula is used to derive Klein Gordon equation (of motion)
from the Lagrangian Density. (section 22.4 and 22.5 of ref 1)

6) The Klein Gordon equation contains the quadratic form of time evolution operator.

7) Taking the first order approximation of the square root of this expression gives the
time operator of the non relativistic Schrödinger equation (section 13.4 of ref 2)
[ref 1] http://physics-quest.org/Book_Chapter_Lagrangian.pdf
[ref 2] http://physics-quest.org/Book_Chapter_Klein_Gordon_real_propagators.pdfRegards, Hans

 

[Civilized]

It's because the Hamiltonian is the generator of time translations. In classical mechanics the Hamiltonian H is a scalar function, dH is a 1-form on phase space, and the symplectic structure (which the phase space manifold inherits by virtue of it's being the cotangent bundle of the configuration space manifold) induces a canonical isomorphism from the space of 1-forms to the tangent space of the configuration manifold, i.e. with every Hamiltonian function we associate a Hamiltonian vector field. But it is a basic fact in ordinary differential equations that every smooth vector field on a manifold induces a smooth 1-parameter group of automorphisms on the manifold, and in classical mechanics this is called the phase flow generated by that particular hamiltonian. Hamilton's equations are a system of ODEs where the independent variable is time t, so in the sense given above the Hamiltonian is the generator of time translations in phase space.

As you say the (one-parameter, smooth) group of time translations must be unitary, and from this it follows that U = exp{i t H} for some hermitian H which is belongs to the lie algebra i.e. is a generator of the group. In classical mechanics, the Hamiltonian function induces a Hamiltonian vector field, and this vector field generates time translations. In quantum mechanics, an infinite-dimensional version of the same thing happens, a Hamiltonian function induces a Hamiltonian operator (i.e. a vector field on an infinite dimensional manifold) and this operator generates time translations.