Is there such a thing as charge density in quantum mechanics?

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?

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Nov 18 '16 at 0:51

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.

• So you're saying that the charge distribution in the filament is essentially stationary? Or what do you mean by higher-order fluctuations? – Marty Green Oct 5 '16 at 19:11
• Yeah, I would expect the probability distribution describing the positions of the charges to be stationary to a good approximation. But if you could actually measure the charge density, i.e. measure where all the electrons are at a given moment of time, you would of course find a different answer each time. These fluctuations would be quantified by higher-order moments such as $\langle n({\bf r})^2\rangle$. – Mark Mitchison Oct 5 '16 at 19:13
• If you have a hydrogen atom in the ground state and you"measure" the actual position of the electron, you also get a different answer each time. Is that the type of fluctuation you are talking about? Where the "charge cloud" remains essentially stationary? – Marty Green Oct 5 '16 at 19:16
• Yes, except that here the fluctuations will probably be mostly of thermal origin rather than quantum mechanical. I should also add: my assumption of steady-state conditions will only be approximately true for your light bulb, because voltage fluctuations etc. in the supply will cause even the average charge density to fluctuate (perhaps quite significantly). So what I have written is only true on a sufficiently coarse-grained time scale in reality. – Mark Mitchison Oct 5 '16 at 19:19
• By quantum fluctuations I mean those which occur even in a state of maximal knowledge (a pure state/wave function), due to non-commutativity of quantum observables. "Thermal" fluctuations (or perhaps "statistical" fluctuations would be better) means those which could theoretically be reduced if the observer performs some more measurements or otherwise is more careful about preparing the system. The charge fluctuations on a filament probably arise largely because the current source itself supplies electrons at random times, but perhaps there is also a significant quantum mechanical uncertainty – Mark Mitchison Oct 5 '16 at 19:34

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.

• Regarding fluctuation. The effect of current fluctuation in thermionic electron emission can be measured and is called shot noise. In addition to the flicker noise and Johnson noise. It is due to the discrete nature of the electron charges causing statistical fluctuations in the current. – freecharly Oct 5 '16 at 20:37

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.

• You have asked this question several times before, and the answer remains the same: the filament is entangled with the environment, the global quantum state is time dependent, but the entanglement makes all the local observables stationary. You have explicitly stated that you are uninterested in the answer. How exactly is this go-round any different? – Emilio Pisanty Nov 18 '16 at 0:28