# Time Evolution of Quantum systems.

Philcorp
In non-relativistic quantum mechanics time evolution is given by the usual $$e^{\frac{-i\hat{H}t}{\hbar}}$$ (for non time dependent hamiltonians). How does one time evolve a quantum system in the context of relativity, where time and space have been placed on equal footing? We clearly cannot use the above expression since it is a result of the schrodinger equation which is not relativistically invariant...help?

Phil :!)

Homework Helper
Before I or someone else provide a serious attempt to clarify things, at least I need to be sure that you'd understand everything that may possibly be written.

"Time evolution of quantum systems" is an awfully imprecise statement. By virtue of the first two postulates of quantum mechanics, a quantum system is described through physical states and physical observables. So the time evolution is ascribed either to quantum physical states (Schrödinger picture), or to quantum (usually Hamiltonian) observables (Heisenberg picture), or to both states & observables (interaction/Dirac-Tomonaga-Schwinger picture).

So get a serious QM book and make sure postulates + formulations, descriptions & representations are crystal clear.

Daniel.

Philcorp
I assure you that I understand what is going on with respect to non-relativistic quantum mechanics, and am also familiar with non-relatvistic quantum field theory. However, I do not know where to begin for a relativistic setting. The dirac/klein gordon equations govern the evolution of systems in quantum mechanics; but as far as i can tell the dirac equation is used for fermi systems whereas the klein gordon equation is used for bosons. This poses some confusion to me since particle statistics are built into the commutator relations in our quantum field theory, so why do we have two different equations which govern time evolution?

/boggled

Phil

Homework Helper
Unfortunately things are not that simple. The KG equation is the field equation for the simplest possible classical field theory. No electric charge, no constraints, spin 0 boson. The full classical analysis of this simple model (free real scalar field) should be clear to you before you jump into quantization. The classical free Dirac field (actually, both models don't have a physical significance at classical level) is the most complicated one may have: two sorts of particles, II-nd class constraints, fermion 1/2 spin.

Assuming the classical analysis of the KG field is clear, which picture of the 3 in QM would you choose?

Daniel.

In relativistic field theory time and space are described by c#parameters. And, no matter what "picture" (Schrodinger, Heisenberg, Interaction) you use, the name of the game is dynamics, evolution in time, or whatever you choose to call it.

The answer to your question lies with the development of covariant perturbation theory. And, we are talking the basis of field theory, which can be found in any number of references -- there is no quick way to get to covariant dynamics. Weinberg's, Gross's texts on field theory, Schweber's QED and the Men Who Made It, Feynman's original papers, ...

Yes, done properly, the appropriate exponential of the interaction Hamiltonian is covariant, and forms the basis for developing the S-matrix.
Regards,
Reilly Atkinson

Staff Emeritus
Gold Member
reilly said:
Yes, done properly, the appropriate exponential of the interaction Hamiltonian is covariant, and forms the basis for developing the S-matrix.

In the limit of T-> infinity (the S-matrix). It's harder to talk about *finite* time evolution in QFT, although there may be techniques now that I don't know of.

cheers,
Patrick.

Juan R.
Philcorp said:
In non-relativistic quantum mechanics time evolution is given by the usual $$e^{\frac{-i\hat{H}t}{\hbar}}$$ (for non time dependent hamiltonians). How does one time evolve a quantum system in the context of relativity, where time and space have been placed on equal footing? We clearly cannot use the above expression since it is a result of the schrodinger equation which is not relativistically invariant...help?

Phil :!)

I see nobody offers you a simple and easy answer.

Of course, there is only a evolution of quantum systems. That is the Schödinger like picture.

The evolution (for fields) is the same in the context of relativistic quantum mechanics. See Weinberg The quantum theory of fields volume 1, p49.

Weinberg said:
First some good neew: quantum field theory is based on the same quantum mechanics that was invented by Schrödinguer, Heisemberg, Pauli, Born, and others in 1925-26, and has been used ever since in atomic, molecular, nuclear, and condensed mater physics.

There is not relativistic quantum mechanics. In fact, it cannot exist.

Philcorp
All of these replies seem to indicate to me that relativistic systems are evolved in the usual way (ie, shrodinger, hiesenberg or whatever picture). This has allowed me to rephrase my question: Does it not matter that the schrodinger equation is not relativistically invariant?

Phil :!)

James Jackson
Juan R. said:
There is not relativistic quantum mechanics. In fact, it cannot exist.

Errm, what? As a (simple) example, the Dirac equation is the result of making Schrodinger's Equation Lorentz invarient.

Philcorp
James Jackson said:
Errm, what? As a (simple) example, the Dirac equation is the result of making Schrodinger's Equation Lorentz invarient.

This was also my understanding of the matter, however the Dirac equation is only for spin-1/2 paticles, correct? Is there a more general equation which is Lorentz invariant?

Phil

Homework Helper
James Jackson said:
Errm, what? As a (simple) example, the Dirac equation is the result of making Schrodinger's Equation Lorentz invarient.

Nope. For a free particle, the Lorentz covariant form of the Schrödinger equation is the Klein-Gordon equation. For heaven's sake, don't you know that Schrödinger initially obtained the relativistic equation and just then considered the nonrelativistic case? Read the first chapter in Weinberg's book, first volume and the references therein.

Daniel.

Homework Helper
Philcorp said:
This was also my understanding of the matter, however the Dirac equation is only for spin-1/2 paticles, correct?

Yes. Massive, parity invariant , electrically charged 1/2 spin particles.

Philcorp said:
Is there a more general equation which is Lorentz invariant?

Nope, there's no general case. The equations of motion are simply the Lagrange field equation for each classical field corresponding to all finite dimensional nonunitary reps of the restricted Lorentz group.

Daniel.

Philcorp
So to study a relativistic system what can be done? Is the best I can do to start with a relativistic lagrangian and derive the propagator via the feymann path integral method using this relativistic lagrangian? Seems like the best idea I can think of?

Phil :!)

Staff Emeritus
Gold Member
dextercioby said:
The equations of motion are simply the Lagrange field equation for each classical field corresponding to all finite dimensional nonunitary reps of the restricted Lorentz group.

Another way of looking at the field equations for non-interacting fields is that they naturally characterize the infinite-dimensional unitary irreducible representations of the (universal cover of the) Poincare group.

Regards,
George

Juan R.
Philcorp said:
All of these replies seem to indicate to me that relativistic systems are evolved in the usual way (ie, shrodinger, hiesenberg or whatever picture). This has allowed me to rephrase my question: Does it not matter that the schrodinger equation is not relativistically invariant?

Phil :!)

It is the nonrelativistic Schrodinger equation that is not Lorentz invariant.

Juan R.
James Jackson said:
Juan R. said:
There is not relativistic quantum mechanics. In fact, it cannot exist.
Errm, what? As a (simple) example, the Dirac equation is the result of making Schrodinger's Equation Lorentz invarient.

No. It is not so simple. Dirac equation is incosistent and the developing of relativistic quantum mechanics is imposible. This is the reason of that Dirac equation was abandoned in relativistic quantum field theory, where the basic dinamical equation is a Schrodinger like one.

$$i \hbar \frac{\partial \Psi}{ \partial t} = H \Psi$$

Note: Dirac equation is not the result of making Schrodinger's Equation Lorentz invarient: that is the well-known relativistic Schrodinger equation.

Dirac equation was derived from the searching of a first order (in time) representation of Klein-Gordon equation more spin corrections for hidrogen atom.

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Juan R.
Philcorp said:
So to study a relativistic system what can be done? Is the best I can do to start with a relativistic lagrangian and derive the propagator via the feymann path integral method using this relativistic lagrangian? Seems like the best idea I can think of?

Phil :!)

Use the canonical formalism. Obtain the corresponding hamiltonian from the lagrangian and use it directly in the S-matrix.

The path integral can offer wrong answers.

Juan R.
dextercioby said:
Nope. For a free particle, the Lorentz covariant form of the Schrödinger equation is the Klein-Gordon equation. For heaven's sake, don't you know that Schrödinger initially obtained the relativistic equation and just then considered the nonrelativistic case? Read the first chapter in Weinberg's book, first volume and the references therein.

Daniel.

The Lorentzian version of the Schrödinger equation is the relativistic Schrödinger equation, which is not the Klein-Gordon equation. In fact, the relativistic Schrodinger equation has no problems with negative probabilities for example and is very used in semirelativistic quantum chemistry .