LC oscillator - is it quantised?

[Confused2]

LC oscillator - is it quantised?

If yes and the oscillation is maintained by a transistor then what is the transistor amplifying?

 

[vanesch]

LC oscillator - is it quantised?

If yes and the oscillation is maintained by a transistor then what is the transistor amplifying?

I would say that a pure LC oscillator is quantized, in the sense that it has stationary states and so on (but probably, the best description is using coherent states). However, from the moment you have losses and a transistor amplifier, this becomes an interacting system. You can try to treat it perturbatively if the losses (= interactions) are weak.
Now, as an exercise, you should calculate what are the classical amplitudes corresponding to a particular LC, say, with a frequency of 100 MHz. You'll find them to be VERY VERY small. I would guess that thermal noise is already much much bigger.

Let's do the calculation:
the energy quantum of a 100 MHz oscillator is h x nu ~ 6.6 10^(-26) J
On a 1pF capacitor, for instance, that corresponds to (E = C V^2/2)
0.36 microvolt. So the ground state of this oscillator will be such that the maximum voltage on the capacitor equals 0.36 microvolt.
Now, the thermal energy that should be present in this oscillator, at room temperature, is 1/2 kT = 0.5 1.38 10^(-23) 300 = 2.1 10^(-21) J, which is already much higher than the lowest quantum. Even at liquid helium temperature (4K) we have 2.7 10^(-23) J of thermal noise. It corresponds to 7.5 microvolts of thermal noise on the 1 pF capacitor.

So the quantization of "classical" electronic circuits is usually completely swamped by thermal noise considerations.

 

[Confused2]

I would say that a pure LC oscillator is quantized, in the sense that it has stationary states and so on (but probably, the best description is using coherent states). However, from the moment you have losses and a transistor amplifier, this becomes an interacting system. You can try to treat it perturbatively if the losses (= interactions) are weak.

Hi Vanesch ..
Many many thanks. The overall 'drift' of this question is in the direction of analysing a transmitter at (say) 100Mhz entirely in terms of quantum theory.. from transmitter to receiver. I fear this is as much an exercise in what is 'acceptable' as it is in theory.
Having accepted the initial premise as far as the LC. .
Would it be fair to say that the transistor is now making good lost photons ( energy approx 6.6 10^-26 J) .. transistors aren't normally analysed in this way .. any thoughts or pointers as to where such an analysis might be found, please..?
-C2.

 

[vanesch]

Would it be fair to say that the transistor is now making good lost photons ( energy approx 6.6 10^-26 J) .. transistors aren't normally analysed in this way

Yes, that's a way to see it. But what you attempt to do is damn difficult! Wouldn't really know where to start, myself.

.. any thoughts or pointers as to where such an analysis might be found, please..?
-C2.

I think the closest you can come, is the quantum description of a laser amplifier, if you consider your transistor amplifier as a coherent stimulated emitter. Look up the "bible" of Mandl and Wolf for that, on quantum optics. However, I think that the quantum-optical setting is "simpler" than your transistor! Nevertheless, you could find inspiration to write down a phenomenological quantum model of your circuit by looking at how a laser amplifier works.

 

[Confused2]

Many thanks vanesch, both for your wise comment and suggestion.. I agree it looks laserish .. but if transistors work that way in this circuit then they should always work that way..

This has the potential to be a trip into my own private Crackpot Theoryland. A little handwaving would be most welcome (preferably not of the good-bye type)..

My own hand waving explanation looks rather like drowning. Clearly we maintain the total energy of the LC circuit at a level far above thermal noise.. but how does the transistor know that's what we want it to do? I've always treated transistors as a current driven device and they have always responded well to this type of treatment.. holes, bandgaps etc. . I've never seen one fed with a quant.. I can't even imagine how to get one in there, let alone get more out.

The appearance (waves hands wildly) is that it will not be possible to treat the transistor 'in isolation'..

Hand waving most welcome..