Re: Quantum mechanics and operators

From: lunatic (holdkooros_at_mailbox.hu)
Date: 11/17/04 

Date: Wed, 17 Nov 2004 02:33:03 +0100

> cfgauss@u.washington.edu (cfgauss) wrote in message
> news:<9e2e7039.0411142331.6c0a8309@posting.google.com>...
>> 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* (-i hbar
>> d/dt)
>> 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?
>>
>> 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?
>>
>> 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?
>>
>> I've tried to look through several quantum mechanics books for answers to
>> these questions, but I haven't seen any argument that doesn't seem
>> circular
>> to me.
>>
>> Thanks very much!
>> - Jeremy

Why operators?

I'd like to answer this question if I may in two ways. Firstly in my normal
voice and then in a kind of silly high-pitched whine...... ...no, sorry.
Firstly from a historical viewpoint and then from a modern axiomatic way.

(historically)
This is a 25 year long story beginning in 1900 so I cannot even begin to go
into details here.

Using insight and results from names no less than Planck, Rutherford, Bohr,
Einstein, Ehrenfest and Kramers, it was Werner Heisenberg who became
responsible for getting operators into business. When working on the then
evergreen problem of (atomic) spectral lines and their intensities, he tried
to keep the known dynamical equation of motion x''+f(x) = 0 but to reject
the interpretation of x as simply position as a function of time.

Now normally x would be expanded into a Fourier series, when thinking of an
electron of an atom interacting with light, absorbing and emitting at
certain freqencies: x(t) = sum a_n exp(inwt) /read w as omega/ a_n are the
Fourier coefficients. However, instead of a_n he made use of a(n,n-m) , a
quantity depending on two indices, to introduce 'transition amplitudes' into
his formula (electron jumping from energy level n to level n-m). This seems
quite bold, and yes it was, but there were good reasons to do this.
Transition amplitudes were already introduced by Einsteins paper from 1916 -
he had A_nm and such everywhere, where m and n denoted energy levels of the
bound electron. You really need to see preceding papers on dispersion theory
and the correspondence principle which was guiding all these 'guesses'.

Heisenberg very strongly felt that the reason for classical dynamics not
working in the atomic domain was to be found in the underlying kinematics.
He clearly expressed this in the introduction for his paper. So he tried to
change that.

As I see this was the very turning point, as now suddenly all kinds of
kinematical quantities got two indices with the formulae remaining the same!
Multiplication of these quantities obeyed the rules of multiplications of
matrices, as a consequence of the Fourier analogy. Bohr and Jordan
formalized these results very neatly the same year, so that no one could
deny any more that there were matrices (operators) in work. They worked out
pq-qp=h/2pi.i (By another sideline, Shrödingers work, wave equations, were
shown to be equivalent with the matrix method at a later time.)

So this was how it all began....

I find it very strange that all the phenomena involved in the calculations
which led to the new formalism is related to either black body radiation,
discrete spectra or dispersion theory - so in the end related to the
interaction of a photon with a bound electron. Matrices came because of
discrete energy levels and transitons between them! Noone seemed to care
about how a photon or free electron moves or what really happens when they
don't interact. Yet the new formalism was applied to these, later, without
the least of hesitation.

(axiomatically)
The logic of a mechanical system is an orthomodular lattice. (OK, logic is
the way how all the events of a given phyisical system relate to each
other). Physical quantities are mappings so that every statement about a
physical quatity is associated with an element of the logic, an 'event'. We
assume that such statements constitute a set algebra.

1 Now if our system is quantum-mechanical, its logic is (isomorphic to) a
lattice of projections of a complex Hilbert space.

2 Components of coordinate and momentum in a given frame are real
quantities, so statements regarding them correspond to Borel sets of R
(reals), constituting the set algebra in this case.

Because of 1 and 2 our physical quantities are now mappings from the Borel
set to the lattice of projections. These are called projection-valued
measures. For every such measure there is a unique self-adjoint operator on
the Hilbert space whose spectral extension is the given projection-valued
measure, that is physical quantity! Backwards: this operator can be
constructed by integrating the identity function on the reals with respect
to the projection-valued measure. So in this sense the physical quatity can
be identified with this self-adjoint operator.

Fun, isn't it?

-lunatic-

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