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Is there such a thing as a time-varying charge density in quantum mechanics? I am looking at the tungsten filament in a light bulb. Am I allowed to say that there is a wave function which describes that filament, and that in some sense there is a time-varying charge density (which some people call a probability) associated with the square of that wave function? There are electrons which have a probability distribution, and there are nuclei which are vibrating within the lattice, but all in all is there, or is there not, a time-varying charge distribution associated with that glowing filament? And if not, why not?

EDIT: There seems to be some question as to just what kind of charge-density fluctuations I am talking about. Let me clarify: in a hydrogen atom in the ground state, I would say there are NO charge density fluctuations. But if there is a superposition of the ground state and a 2p state, I would say the charge distribution fluctuates sinusoidally at the difference frequency of the two states. THAT is the kind of charge density fluctuation I am asking about for in the tungsten filament. Does it or doesnt' it?

I hope this clarifies the intent of my question.

RE-EDIT: In a separate (and rather hard-fought) discussion, it appears that people who know more than me agree that there is indeed an oscillating charge distribution in the hydrogen superposition as described above (see Is there oscillating charge in a hydrogen atom? ).

So if there is an oscillating charge in the hydrogen atom, why wouldn't there be oscillating charge in a heated tungsten filament?

Community
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Marty Green
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3 Answers3

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In quantum mechanics there is a charge density of electrons when you describe a metal like your W filament, even when you consider the delocalized nature of an electron in the solid and use the single particle wave functions (Bloch functions) for the electrons, which are normalized to one electron per wave function, and their dispersion relations according to the E-k band structure. The single electron wave function gives you the probability of finding the electron and thus its elementary charge in a certain infinitesimal volume at a certain location, which is essentialy constant in the whole metal volume. Because you have a very large number of electrons in the metal per unit volume, you get the (mean) charge density in the metal by counting how many electrons you have per unit volume. Also, disregarding very small fluctuations, there is no time varying electron charge density in your glowing W filament emitting electrons because the electrons emitted are immediately resupplied by the leads.

freecharly
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The tungsten filament is an open quantum system, which is in contact with its environment. In particular, it is exchanging electrons with the wires which supply the power, in addition to continuously radiating and absorbing photons. Formally, this means that the tungsten filament cannot be described by a wave function, but rather its state is described by a density operator. Nevertheless, there is no problem with defining the charge density. Quantum mechanically, this will be described by an operator $\hat{n}({\bf r})$ depending on position $\bf r$. In the steady state, which is probably the situation you are currently looking at (neglecting fluctuations of the driving voltage etc. which may be important in reality), its expectation value will be constant in time, i.e. $\partial_t \langle \hat{n}({\bf r})\rangle = 0$ (but not spatially constant for a diffusive conductor). However, there will certainly be fluctuations of $\hat{n}({\bf r})$ as quantified by its higher-order moments.

Mark Mitchison
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Is there a fluctuating charge density in quantum mechanics? OF COURSE there is. A hot piece of metal has a vibrating lattice which is made of charged nucleii, and when those atoms vibrate the charge is vibrating. And the sea of electrons in the conduction band...the billions of pure eigenstates which represent the "solution" of the system are of course in superposition with each other, and those superpositions create oscillating charge densities exactly the same way you get oscillating charge in a hydrogen atom when you combine the 1s and 2p states.

None of this should be in controversial in the slightest. And yet none of the other correspondents who have participated in this discussion seem to acknowledge it. How is there not an oscillating charge density in quantum mechanics? Of course there is.

Marty Green
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