Quantum mechanical interpretation of resistance and voltage drop

Question

It is generally said that Ohm's law isn't a real law since it is only a fine model found through experiments to explain the nature of circuits. And resistance is defined as the ratio of the voltage difference between two poles and the current flow through the circuit. Often resistance is explained as electrons having collision with the atoms of the conductor and is some kind of drag (?) through which electrons loose energy. This explanation doesn't seem to be exact case.

What is actually going on if we zoom into the quantum level? How are the concepts of resistance and voltage drop interpreted then?

Answer 1

To me, the most interesting characteristics of electrical resistance is the orbital positions, which feeds common quantum calculations. That is the most electrical conductivity, thereby the lowest electrical resistance, comes in the middle of the periodic chart. It is the column, 29-Cu Copper, 47-Ag Silver, and 79-Au Gold.

The challenge for thinking of these using the quantum model is that the electrons are in the toward the max of the current d-shell. These are all d-shell (Xd9) which you get the quantum number from:

What is the right quantum numbers of Copper?

That reference is important because it states that the last electron quantum number for Copper is the same as the last electron quantum number of Zinc, which has very different - low - electrical conductivity -- the opposite of Copper. As such, one would not expect the expression with the same last electron to yield such opposite if the quantum number is the same. One would expect that the last electron is the critical one for electrical conductivity - being the most removed (lowest energy to release).

I believe the answer lies in other research about 'transition metals' which exhibit a number of contradictory. Specifically, at:

https://en.wikipedia.org/wiki/Transition_metal

It states that d-Shell transition provides a) magnetic, b) when in solution color, and c) oxidation, but not electrical resistance.

So, to me the linkage of electrostatic conductivity to current quantum theory remains inadequate. Quantum theory is good at only certain properties.

Answer 2

When I started learning QM in the first lecture we tried a "semi classical" way to imagine these things. Imagine a metallic cube. Let's assume that some current $I$ is flowing through it. The current through an $A$ cross section will be : $$I=qnvA.$$ where $q$ is the unit charge, $v$ is the average speed of the electrons and $n$ is the volumetric concentrations. We can assume that the speed is proportional to the potential energy $E$ $$v=\mu E$$ Where $\mu$ represents the ability of the electrons to move. Then $$I=qn\mu EA.$$ And the current density $j$ is $$j=qn\mu E.$$ Hence the conductivity $\gamma$ is $\gamma=\frac{j}{E}$ then it is $$\gamma=qn\mu$$ And the resistance is $$\rho=\frac{1}{\gamma}=\frac{1}{qn\mu}$$ And where quantum mechanics comes in? It is neccessary when we want knowledge about $\mu$.At first assume that we are working with a metal which has a perfect crystalline structure, then the electrons won't scatter much on the atoms so a perfect crystal would not have any resistance. The resistance is from the defects of the crystal where electrons scatter. So the resistance is due to scattering, not collision, is an electron hits a nucleus it is called electron capture which is another interesting topic. Since $\mu$ is not a number I should better call it $\mu(x,y,z)$ and $n$ is really $n(x,y,z)$ and they are both dependent on the shape of the metal piece and the position too. so it won't be a simple equation to use(And we did not even calculated with the electrons effects on each other!). The voltage drop sould be easy to calculate for you then.I hope my answer was good enough :)

Answer 3

I can't be sure about the voltage drop issue, but I can certainly provide an explanation of resistance from a quantum-level perspective. Classically, Drude's theory describes resistance in terms of collisions, which is somewhat "simplistic." In quantum mechanics, collisions are better described as scattering, which occurs due to the difference in potential between the incident particle (in this case, the conduction electron) and the finite potential source (the ion cores).

Materials are composed of a periodic lattice structure, where ion cores reside at the lattice points, and free electrons move through the structure. The motion of these electrons isn't entirely "free" because they experience a potential from the ion cores. This potential is periodic, meaning that

V(r)=V(r+a), where $\mathbf{a}$ is the lattice translation vector.

To understand the behavior of electrons in this periodic potential, we solve the time-independent Schrödinger equation:

$\hat{H}$ψ(r)=Eψ(r), Due to the periodicity of the potential, the solutions to this equation are Bloch functions, which take the form:

ψ(r)=$u_{\mathbf{k}}(\mathbf{r})$$e^{i \mathbf{k} \cdot \mathbf{r}}$

where $u_{\mathbf{k}}(\mathbf{r})$ is a function with the same periodicity as the lattice, i.e., $u_{\mathbf{k}}(\mathbf{r})$=$u_{\mathbf{k}}(\mathbf{r+a})$ and $e^{i \mathbf{k} \cdot \mathbf{r}}$ is a plane wave component. This form of the wavefunction is known as a Bloch wave, and it reflects the fact that electrons in a periodic potential retain some plane-wave-like behavior, but with modulation due to the periodic lattice.

Since the potential is periodic, there is no overall potential difference within an ideal lattice, and thus, quantum mechanical scattering within such a perfect structure does not occur. As a result, no resistance would be present in this idealized situation.

At this point, you might be wondering: if this description is accurate, why do metals exhibit resistance? The answer lies in the fact that no real material is perfectly periodic. There can be defects, such as vacant or distorted lattice points, which break the periodicity. Additionally, dynamical resistance can arise due to the vibrations of ion cores (electron-phonon scattering). These imperfections and vibrations disrupt the periodic potential, leading to quantum mechanical scattering and, consequently, resistance in metals, semiconductors, and insulators. I hope you understand it