Re: Quantum mechanics and operators?

From: Hendrik van Hees (hees_at_comp.tamu.edu)
Date: 11/17/04 

Date: Wed, 17 Nov 2004 17:13:37 +0000 (UTC)

Jeremy Price wrote:

>
>
> I'm trying to find out more about where the operators in quantum
> mechanics
> come from. From what I see, the justification for using them is that
> using them with the wave function, which is already known, gives you
> the expectation value for that operator, i.e., <E> = Integral[ phi*
> (-hbar/2m)
> phi*]. But then I see the justification for the wave function as that
> if you put the operators in the equation E = p^2/2m, you get the
> Schrodinger
> equation. Well, that's great, but that argument just goes in a big
> circle. If you didn't know the wave equation, the solution to the wave
> equation, or
> the operators, how would you be able to come up with them? And what
> is the
> justification for what the operators are, other than "because it
> works." Do they "come" from anywhere?

You must have read a quite old-fashioned "wave-mechanics" guide instead
of a modern quantum-theory textbook.

Indeed the operators come from symmetry principles. The only thing, one
has to accept, is the general structure of quantum theory which is an
abstract formalism to summarize the empirical facts about the behaviour
of particles.

A nice textbook on QT is L. Ballentine, Quantum Mechanics. There you
find the construction of the Hamiltonian of non-relativistic quantum
theory from the analysis of the group theory of Galilei-Newton space
time.

>
> Also, if we look at the relativistic correct wave equation, my quantum
> mechanics books tell me that it comes from putting the operators into
> the
> relativistic energy equation. How can you justify doing this? How do
> you
> know the operators are relativisticly correct? How do we know that we
> don't have anything like an operator for mass that we have to put into
> the equations?

There is no way to justify this, except Dirac's hole theory, which is
awfully complicated, compared to the more elegant and straight-forward
formalism of relativistic quantum-field theory. The point is that in
the realtivistic realm you can always create and destroy particles so
that a one-particle wave-function interpretation of relativistic
quantum theory breaks down sooner or later. Quantum field theory is the
natural description of this situation. Here, I recommend Weinbergs
marvelous books on the subject:

S. Weinberg, Quantum Theory of Fields, 3 Vols., Cam. Uni. Press.
>
> I'm also interested in the use of complex numbers to write the
> equation. My quantum mechanics book mentions that they are used to
> simplify the
> equations, and that it doesn't physically mean anything. It mentions
> that you can write, for example, the equations describing electricity
> and magnetism in a form like that by saying F = E + icB, then you can
> write Maxwell's equations in terms of this F instead of separate
> equations for E
> and for B. So, you could also write the wave equation as the sum of
> two
> other "things," phi = phi_1 + i phi_2. Is there any kind of
> representation
> for these "fields" phi_1 and phi_2? Are they physically meaningful?
> Also, is there a physically meaningful representation of the field F =
> E + icB?

The latter idea, you find in Jackson, Classical Electrodynamics. It
reflects the fact that the Lorentz goup is homomorphic to SO(3,C), the
special orthonormal group on the vector space C^3. Sometimes this can
be used to simplify calculations in electrodynamics. 

-- 
Hendrik van Hees                        Cyclotron Institute 
Phone: +1 979/845-1411                  Texas A&M University 
Fax:   +1 979/845-1899                  Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/          College Station, TX 77843-3366

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