Postulates of Quantum Mechanics

[Virtual R]

In my course of modern Physics one postulate i read is
"Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system"
A question arised in my mind is if the particle is moving in non conservative field then can't we associate a wave function with the particle? or if we can how is Schrödinger eq is modified for that?

 

[Demystifier]

For conservative fields one can introduce a Hamiltonian. This is the reason why one can introduce the Schrödinger equation. The Hamiltonian can be introduced even for some non-conservative fields, for example those decribed by an explicitly time-dependent potential. In such cases, the modification of the Schrödinger equation is trivial.

However, in general, for nonconservative systems the Hamiltonian (as well as the Lagrangian) cannot be defined. In such cases it is not known how to quantize such systems.

 

[Anonym]

Virtual R:” In my course of modern Physics one postulate i read is
"Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system"

It seems that it is better to choose another book. Such wording only confuse the reader. In QM the interactions are not represented by forces. Where the “postulate”? The writer want to say that the mathematical framework used is complex Hilbert space. Similarly, he may say that the classical mechanics must use the classical analysis. It is not a postulate, at least not the physical postulate. Also obviously his statement is wrong, since eigenvalues of the dynamical observables “determines everything that can be known about the system”.

 

[Truth Finder]

Fellows,

Of course... the concept of force is chaned in QM rather than classical mechanics.You are right anonym, but the wave function describes everything in that system even in the presence of many charges as it carries the probability for the particle to be at very small volume at some place. The eigenvalue is a parameter... may be you mean eigen function?

There is a way to solve QM for non-conservative fields.. that is by the use of the canonical momentum, not normal momentum, and the hamiltonian. This method was discovered very recently.

Virtual R... sq. of mod of wave function can be used to determine the behavior of the described system by usingthe probability known mathematical calcs.

Get care when building your concepts about quantum mechanics especially in the first time.

But another question here which is if there would be another more exact theory beyond QED. Gravity is a pain as is still away from quantum. Loop quant gravity is a good candidate for this. But, will other like String Theory solve the whole puzzle. A curiositic question that is now asked very oftenby scientists.


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Either to do the work as required, or to leave it.

 

[StatMechGuy]

There is a way to solve QM for non-conservative fields.. that is by the use of the canonical momentum, not normal momentum, and the hamiltonian. This method was discovered very recently.

Virtual R... sq. of mod of wave function can be used to determine the behavior of the described system by usingthe probability known mathematical calcs.

I'm curious about your statement: the hamiltonian MUST be formulated in terms of the canonical momenta of a system. I don't see how this is a recent development: in fact you have to use it in order to properly treat magnetic systems. This problem was solved by Landau fifty years ago. However, this does not necessarily treat nonconservative fields.

Of course, typically "nonconservative" interactions arise for such things as friction, which one could argue is purely a classical effect. It is possible to create a hamiltonian which does not conserve probability, and this is usually done by creating a potential that is not hermitian. It's an interesting exercise, but try working with an interaction for a two-level system given by
$$\mathcal{H} = \left( \begin{array}{c c} E & i\Delta \\ i\Delta & -E \end{array} \right)$$

 

[Anonym]

Truth Finder:” The eigenvalue is a parameter... may be you mean eigen function?”

I mean exactly what I said: the knowledge about the QM system is given in terms of the spectrum of the complete set of mutually commuting self-adjoint operators. The wave function describe the system state and is unobservable quantity. Don’t confuse the beginner if you are not familiar with quantum physics.

Virtual R, if you are really the beginner, I suggest for you the following steps:
1.S. Gasiorowicz “Quantum Physics”.
2.R. Feynman Lectures on Physics.
3.P.A.M. Dirac “The Principles of QM”.
4.E. Schrödinger original papers from 1926, just to see what human mind is able to do during one year when he understand what he is doing.
5.J. von Neumann “The Mathematical Foundations of QM”.

After that you may proceed in any direction you like, fantasies of QG included.

 

[Virtual R]

Thank u every noe and anonym i hope readind all this literature would help me