Re: A Question About A Simple LRC circuit

["John S. Denker" <jsd@MONMOUTH.COM>]

I wrote:
>  Flux and charge are dynamically conjugate, like position and
>  momentum.  They obey the Heisenberg equation
>           delta Q delta Phi > hbar/2
>  whereas voltage and current don't even have the right units for this.


Then at 11:20 AM 3/6/01 -0500, Robert B Zannelli wrote:
> I was aware that postion/mometum & Energy/time & angular momentum/ angle
> were all conjugates ( noncommuting variables)

I 100% agree with [position,momentum] being dynamically conjugate
variables, but the other examples are not so wonderful.  Position is
unlimited in both directions, but alas energy is bounded below, and angle
is bounded (or periodic) at both ends.  This truncation of the would-be
coordinate messes up the formalism.  I'll agree that they don't commute,
but they don't qualify as canonical coordinates.

>  but I was not aware that this
> applied for charge and flux though since Quantum mechanics rules supreme in
> physics it should not be too surprising. This suggest that one could set up
> an eigen function for these terms that should look like:
>
>   - i*hbar*(d/dphi [Y>)=Q*[Y>           [(equation 1)]
> where d/dphi is the differential operator, [Y>
> is the probability amplitude vector (Eigenvector) and Q (Eigenvalue) is the
> charge. Does this make any sense?

Nice try.  Equation 1 is dimensionally correct, and the physics is almost
correct.  The problem is that the quantum RLC oscillator is never in an
eigenstate of the charge operator.  This is 100% analogous to the
mass-on-a-spring, which is never in an eigenstate of the momentum
operator.  Free particles are in momentum eigenstates, but particles on
springs are not.

The closest you can get to a charge eigenstate is a "squeezed state" which
is derived from a Glauber state:
  http://www.physics.harvard.edu/fac_staff/glauber.html
  http://www.deas.harvard.edu/~jones/ap216/lectures/ls_3/ls3_u3/ls3_unit_3.
http://www.deas.harvard.edu/~jones/ap216/lectures/ls_3/ls3_u3/ls3_unit_3.html

> Of course it does for the all the conjugates I mentioned above.

Which goes to show that there's more to physics than eigenvector/eigenvalue
equations.

> This would provide a Hilbert space model for charge.

To analyze the quantum harmonic oscillator, you can't start with the
eigenvector/eigenvalue equation, because you don't have one, and even if
you did have one it wouldn't tell you everything you need to know.

You *can* derive pretty much everything starting from the action principle
        psi = exp(i S / hbar)
where S is the action, which is the integral (dx dy dz dt) of the
Lagrangian density.  In this lumped-element case, S is just the integral
(dt) of the Lagrangian.

If we choose flux (phi) as the canonical coordinate then charge (Q) is the
dynamically-conjugate momentum, and
        Lagrangian = kinetic energy - potential energy
                   = Q^2 / (2 C)   -   L phi^2 / 2

You can then construct the 2nd-quantization ladder operators A_dagger and
A.  From them you can construct the aforementioned Glauber states and
squeezed states.

> This is very interesting.

Oh, yeah.  The first really important scientific talk I ever gave, back
when I was a grad student, was on this topic.  It was one of those
10-minute contributed talks at the APS meeting.  The speaker before me
didn't show up, so the moderator (in violation of the rules) gave me his
slot.  I could have finished after 10 minutes, but the moderator kept
taking questions from the audience for more than 25 minutes, taking up all
of my slot and (in violation of the rules) postponing the following speaker
by quite a bit.

In particular, in this talk we explained how to handle the R in the quantum
RLC circuit.  (This was joint work with Bernie Yurke, who was the brains of
the operation.)  As you might imagine, it's not super-obvious how to write
down the Hamiltonian for something that (at first glance) doesn't conserve
energy.  We also explained how you could use squeezed states to build a
"quantum nondemolition voltmeter" that could measure a voltage within
precision better than sqrt(hbar) --- which, according to all the textbooks,
was impossible.