Yes. Kirchhoff's laws are based on current conservation, which also holds in the quantum world, as long as the current is carried by conserved particles - as electrons.

Thank you for the answer.

Kirchoff's laws are based on indtroduction of the circuit as a model. This means that we say in some, however quite common conditions, description based on Maxwell's equation can be reduced to Kirchoff's laws (just to circuit currents and node potentials instead of fields and charges). So Kirchoff's laws are always there by definition - the only question is whether circuit-based description is an adequte model.

In general, no, Kirchoff's laws do not hold in quantum transport (though I met examples where they hold approximately quite well), since voltage drops cannot always be defined (e.g. what is the voltage drop across a portion of ballistic conductor? In balistic systems resistance does not scale with length.). Only when the elements are connected through incoherent regions (nods) can the voltage drops be well defined. In ballistic systems, a way around this problem is to introduce a fictitious Sharvin resistance in the resistace addition laws. In general there is no solution, you have to use circuit theory in terms of Green's functions or use master equations, depending what is better suited for your system. Look into a book on quantum transport.

Just to add to the previous.

In Fig. 8a of the paper below the resistance addition rules happen to be exact,

whereas in Fig. 8b they are only approximately correct (producing an error mostly in Coulomb blockade valleys).

http://link.aps.org/doi/10.1103/PhysRevB.83.075401

For the modification of the resistance addition law (using Sharvin resistance) in ballistic transport, see, e.g., the discussion below Eq.~(4) in this paper:

http://link.aps.org/doi/10.1103/PhysRevLett.103.066602

Dear Vitaly, thank you for the answer and usefull links. Indeed, I was mainly wondering how to define the voltage drop for the ballistic transport. Another point is whether charge is really conserves in the system where the electrical current is very small and electrons passing one by one. What is the probability that single electron can get in trap in silicon?

On average the current (dc) should be conserved in the stationary situation, regardless of the traps. You just have to measure it for a longer time to average over the trap switches. If you do not see conservation of the current, averaged on a long time scale (on the time scale that you see switches in the signal), then you probably have a leaking device. Often it is related to a leaking gate that was not well isolated.

By the way, the traps probably do not alter the current conservation by any measurable value. They induce most likely just an electrostatic gating effect, which pinches off your conducting channel or switches on another channel that was not supposed to conduct. As a result the conductance shows a telegraph noise or, in the extreme case of many traps, it shows an 1/f noise.

What is the characteristic recoil time for a single trap? I do not know exactly, something like seconds, or maybe even minutes. Maybe someone who studied it can help?

First Law is a consequence of the conservation of charge while the second is the consequence of the conservation of energy.The conservation of energy and charge

are one of the fundamentals in Physics. So both of them should be fulfilled.

Kirchhof'f's laws for electric circuits are consequences of the laws of conservation of charge (current law) and energy (voltage law). These two conservation laws are true in both classical and quantum domains. A rigorous justification of these laws starting from the microscopic descriptions in terms of the Maxwell's equations and quantum physics is also possible (the work of de Groot and Mazur is of relevance).

Of course, Kirchhof'f's laws are consistent with the energy and charge conservation laws, like most of physical laws. My point is that Kirchhoff's laws imply something more than the conservation laws. First thing is a presence of well-defined circuit elements (resistors, capacitance etc) connected at nodes. Second, one should define voltage drop and currents in these nodes. Even in classical physics, it is very hard sometimes to find accurate equivalent circuit model. When we deal with an element characterized quantum conductance, how could we define voltage drop? Does one electron at finite temperature contribute to voltage drop between points separated by a distance being slightly larger than the mean free path?

Some things are quite well explained in Chapters 2.4 and 2.5 of the book by Nazarov & Blanter, "Quantum Transport: Introduction to Nanoscience" (ISBN-13 978-0-511-54024-0). It is mostly within the semiclassical approximation, though.

To elaborate on Vitaly's answer a bit: The node voltage is a measure of Fermi level (or Helmholtz free energy per charge for electrons). It is a property of systems that contain enough electrons to act as effective reservoirs of electrons. (And reservoir implies an absence of quantum coherence through that region.) The circuit abstraction fails to apply unless you can identify regions with adequate numbers of mobile electrons in local quasi-equilibrium to consider them to be electron reservoirs. Something like 100 or more atoms of a metal in the normal (non superconducting) state will generally fit the requirements of a circuit node.

The question was addressed (some time ago) by Wim Magnus and me. I recommend to read the following sections in "Handbook of Numerical Analysis Volume 8" Special Volume: Numerical Methods in Electromagnetism" Editor P.G. Ciarlet , Guest Editors W.H.A. Schilders and E.J.W. ter Maten. I specifically refer to chapter 1, sections 6 and 9. For further reading you can find more references there.

I do not have quick access to Schoenmaker's reference, but its forum prompts the following general observation: Circuit theory is defined by the physical quantities which it references: node voltage and electrical current. There exists a very large body of literature which is flawed by the mistaken identification of node voltage with the electromagnetic scalar potential. This includes virtually all textbooks on electromagnetism, and works on electronic devices, where the distinction is critical, usually muddle the issue. Such an identification is correct only in a universe which consists only of vacuum and conductors of a single chemical composition. When we have two chemically different conductors, a contact between them can produce a contact (emag scalar) potential with zero voltage drop.

Every 20 years or so, there appears in Am. J. Phys. an article on "what does a voltmeter measure," followed by a series of comments. The authors never pose nor answer the question within its proper context, which is the statistical physics of condensed matter. To those who would disagree with this assertion, I invite them to demonstrate the operation of a purely photonic voltmeter.

I looked into Schoenmaker's reference this morning. It is a nicely written text on electromagnetism (using notions of differential geometry), mostly at the level of Maxwell equations, but touching also slightly on quantum mechanics. The Kirchhoff’s laws are derived within the classical theory of electromagnetism. In the Outlook, there is a subsection on Quantum Circuit Theory, where some qualitative QED picture is given in terms of quantized displacement and magnetic fluxes threading a cross section of the circuit (something like an effective-variables theory). In general, the problems one encounters in the classical electromagnetic theory are mainly due to the idealizations made upon introducing circuit elements. For example, a realistic capacitor behaves only approximately like an ideal capacitor. But this is of course! Any experimental setup has to be eventually somehow approximated before it can be modelled on a piece of paper. As I understand, the question of this thread is more like: Can one break a quantum transport system into constituent pieces, for which he knows the two-terminal correlators (e.g. conductance), and then write Kirchhoff's laws for the whole circuit?

I recall another example when something like Kirchhoff's laws is possible. In the case of quasiclassical theory of superconductivity, an analogue of Kirchhoff's laws can be written in terms of matrix Green's functions, with different levels of sophistication, depending on whether you need the transport to be energy resolved or integrated. But this works only in the diffusive limit as I know, which is kind of natural since conductance of diffusive materials scales linearly with length and superconductivity comes only at the expense of a matrix structure for the Green's functions. Without superconductivity it is a trivial theory (I guess reducing to Drude + classical Kirchhoff's laws, or something like that). The original paper is by Nazarov, see page 787 in Beenakker's review for it and other relevant references on this subject:

http://link.aps.org/doi/10.1103/RevModPhys.69.731

and see also the energy resolved version:

http://arxiv.org/pdf/cond-mat/9811155.pdf

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https://www.google.com/search?q=Numerical+Methods+in+Electromagnetics+Volume+13+Special+Volume+libgen

I am not certain about Kirchoff's laws, but generally we know that classical conductance models are incorrect since e.g. Anderson localization: standard diffusion predicts nearly uniform probability distribution for electron in defected lattice of semiconductor, while from experiment and QM we know that they should be localized.

This discrepancy can be understood and repaired by realizing that standard diffusion models are only approximation of the (Jaynes) maximal entropy principle which should be expected from statistical physics models - doing it right ( https://en.wikipedia.org/wiki/Maximal_Entropy_Random_Walk ) we get diffusion with stationary probability distribution exactly like predicted by quantum ground state.

We can also apply it to conductance: instead on nearly uniform electron flow in standard models, it leads to localized flow (with kind of short-circuits) - here is simple simulator: http://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/

It should allow to understand electron flow in microscopic situations, single molecules: providing useful approximation of complete quantum considerations - I was trying to get some "quantum Kirchoff's laws" this way, but it was too complicated for simple generalization.

                                                   Dear Pan Duda!

The first Kirchhoff law is a consequence of the law of conservation of charge, which is valid in the classical and quantum cases. The second law of Kirchhoff is a consequence of the law of conservation of energy, which is also true in the classical and quantum cases.

Perhaps in the quantum case, from the laws of conservation of charge and energy, it is necessary to obtain Kirchhoff's laws more correctly.

Sincerely, AA Sobko.

Dear Pan Sobko,

Imagine a large chemical molecule as an electric circuit with atoms as nodes, attached to two electrodes. Kirchoff's laws suggest we should be directly able to find all currents, while it is not that simple, we need QM - but you are right: the problem is not on Kirchoff's level, but rather e.g. Ohm's: the potential-flow dependency becomes much more complex, even nonlocal: dependent on the entire system.

Sincerely, Jarek Duda

It often helps to keep some fundamentals in mind.  Ohm's law, and its variation expressed as conductance, is pure Aristotian physics.  (A constant force [voltage] yields a constant velocity [current].)  Any electronic component that can be described by a steady-state (dc) I(V) characteristic is completely dissipative, and no quantum coherence can project beyond the device terminals.  If you want to use any variation of Kirchoff's laws, you have to partition your system into components which share no mutual electron coherence, and can thus be considered classical components.  

There is another circuit paradigm which is far more relevant to quantum systems: high frequency design using primarily reactive components and typically conducted using the Smith Chart.  It uses choices of reflection coefficients and location of reflecting interfaces to achieve desired transfer functions.  Of course in quantum analyses we are accustomed to use transfer matrices, and that is the place to look for a pure-state circuit theory.  The results of such a formulation will hardly be surprising: A closed system like an isolated molecule will show zero current circulating within it, because the wavefunctions in the ground states can always be expressed as purely standing waves.  You will have to make electrical contact, thus opening the system, to see anything interesting.  When you open the system you necessarily make it irreversible.

Now, some may be wondering how conductance can be such a classical effect when it is quantized in terms of fundamental units including \hbar?  It is not quantized, it is bounded, and that only in a very restricted class of systems.  The Landaur conductance formula is an invitation to unclear thinking about electron transport.  If you can repeatably measure a conductance, the system you are measuring is dissipative.  Ignore the small-signal argument that energy dissipation is 2nd order and therefore negligible. You do have to apply a real voltage in order to make a measurement.  Ask where do the ballistic electrons that are coming through the system actually lose their energy?  In the leads of course.  There is a perfect physical analogy in the old technology of vacuum thermionic devices ("tubes" or "valves").  Electrons moving through the space of such devices will suffer no inelastic collisions, but the devices had very well-defined I(V) curves and could end up dissipating many watts.  Where did this dissipation occur?  In the metal electrode that the electrons hit in order to be collected and conducted out of the device, usually called the "plate".  Precisely the same thing happens in the Landauer model.  (I once had to remind Rolf Landauer that this analogy applied to his theory.  He seemed to be much more focused on the scattering centers within the device.)  Thus the leads in an experimental device have to be able to absorb all excess energy of any collected electrons, restore them to a thermal distribution, and in the process destroy any remaining quantum coherence.  This allows us to draw a boundary around the system and view it as a classical component of an electronic circuit.

So it is fairly straightforward to treat completely incoherent systems, and also to treat purely coherent systems (out to the leads).  What about systems that do not preserve perfect quantum coherence?  Those are the domain of true quantum transport theories, which must necessarily be based upon a representation which describes partial coherence: density matrices, Wigner distribution functions, or Keldysh G< functions.  

I hope this helps.

- Bill Frensley

Dear Dr. Frensley, thank you for great answer giving an inspiration for further thinking and research. When I was asking the question I held in mind exactly those systems where coherence and some degree of decoherence and dissipation are equally important and the currents are such small that even a single electron counts.