UPDATE March 14, 2022: Our reasoning about destructive interference of scattered waves from random collisions was erroneous.
If we sum random variables Xⱼ, then the variance of the sum is the sum of variances of each Xⱼ.
Let Eⱼ be the electric field strength of an electromagnetic wave at a certain time t at a location x. The expected value of Eⱼ is 0. The variance is defined as the expected value of Eⱼ². The power density of the wave is linearly proportional to the variance.
If we have random waves Eⱼ, then the power density of the sum of all Eⱼ is the sum of the power densities of each Eⱼ. Destructive interference does not reduce the power density of the scattered wave at all.
On the other hand, if scattering happens from an ordered structure, then destructive interference can wipe out almost all the power from the scattered wave. We will write more about this.
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But what is the classical model which best describes the phenomenon? What is the relation of Cooper pairs to this?
If electrons move in a random way, like molecules in a gas, then the electrons certainly will collide with nuclei and lose energy. Electrons in a superconductor presumably must have low energy and must be ordered into some kind of a "flowing crystal" to avoid collisions with nuclei.
When we raise the temperature of a superconductor, the "flowing crystal" melts. The specific heat of the superconductor must be high at this transition temperature, because it is a phase transition of the free electron system.
The BCS theory and Cooper pairs
The Bardeen-Cooper-Schrieffer theory (1957) explains superconductivity.
Electrons are supposed to form Cooper pairs which can "condense into the same state" like bosons can.
The electron speed and the electron drift speed in a metal
The link contains an example of a copper wire, where the speed of electrons at the top of the Fermi sea is 1,570 km/s, and the drift speed of electrons is only 4 mm/s.
The drift speed is minuscule. It is not that individual electrons flow in the wire, but the "cloud" of electrons flows.
The Fermi surface and Cooper pairs
Joe Polchinski (1992) considers the surface of the Fermi sea under the weak interaction (~ 1 meV) which is assumed to form Cooper pairs.
How can Cooper pairs survive if the sea of electrons has much larger energies (~ 7 eV)?
An electron should do a Brownian motion under the bombardment of other electrons. How can a weak ~ 1 meV interaction make any difference?
Joe Polchinski suggests that Cooper pairs form in an effective field theory where we can ignore individual particles of the Fermi sea. It is like a wave in water: we can ignore water molecules and study waves in a higher-level description of the liquid.
A wave in a material with lots of particles
Let us consider a wave of visible light which propagates inside a block of glass. Each atom of glass individually scatters the wave. Is there destructive interference which removes almost all of the scattering?
Question. What is the long-wavelength electron wave? Is it an individual electron or a collective phenomenon?
The water wave analogy suggests that the wave is not an individual electron. A hypothesis is that it is a "dressed" electron, with interactions included in the quasi-particle.
The double-slit experiment
In the standard double-slit experiment there is the following problem: the photon passing through a slit interacts with atoms in the edges of the slit. Can we from this interaction somehow find out the the slit through which the photon passed?
If we can determine the slit through which the photon passed, then there should be no interference pattern on the screen, according to rules of quantum mechanics.
A possible solution: look at the behavior of a classical wave of light. If the classical wave forms an interference pattern, then the quantum wave must do the same because otherwise the classical limit of quantum mechanics is broken.
Conclusions
In a material where particles are dense, there can be almost total destructive interference of scattered waves. This may explain why glass is transparent and why we can ignore the repulsion of individual electrons in a Fermi gas.
We have to think more about this. Does the behavior of classical waves determine quantum mechanics, just like in the double-slit experiminent? What is the precise role of the quantum then?
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