What is quantum coherence?

Often when people write about the "weirdness" of quantum mechanics, they talk about the difference between the interesting, often counter-intuitive properties of matter at the microscopic level (single electrons or single atoms) and the response of matter at the macroscopic level.  That is, they point out how on the one hand we can have quantum interference physics where electrons (or atoms or small molecules) seem to act like waves that are, in some sense, in multiple places at once; but on the other hand we can't seem to make a baseball act like this, or have a cat act like it's in a superposition of being both alive and dead.  Somehow, as system size (whatever that means) increases, matter acts more like classical physics would suggest, and quantum effects (except in very particular situations) become negligibly small.  How does that work, exactly?   

Rather than comparing the properties of one atom vs. 10²⁵ atoms, we can gain some insights by thinking about one electron "by itself" vs. one electron in a more complicated environment.   We learn in high school chemistry that we need quantum mechanics to understand how electrons arrange themselves in single atoms. The 1s orbital of a hydrogen atom is a puffy spherical shape; the 2p orbitals look like two-lobed blobs that just touch at the position of the proton; the higher d and f orbitals look even more complicated.  Later on, if you actually take quantum mechanics, you learn that these shapes are basically standing waves - the spatial state of the electron is described by a (complex, in the sense of complex numbers) wavefunction \(\psi(\mathbf{r})\) that obeys the Schroedinger equation, and if you have the electron feeling the spherically symmetric \(1/r\) attractive potential from the proton, then there are certain discrete allowed shapes for \(\psi(\mathbf{r})\).  These funny shapes are the result of "self interference", in the same way that the allowed vibrational modes of a drumhead are the result of self-interfering (and thus standing) waves of the drumhead.

In quantum mechanics, we also learn that, if you were able to do some measurement that tries to locate the electron (e.g., you decide to shoot gamma rays at the atom to do some scattering experiment to deduce where the electron is), and you looked at a big ensemble of such identically prepared atoms, each measurement would give you a different result for the location.  However, if you asked, what is the probability of finding the electron in some small region around a location \(\mathbf{r}\), the answer is \(|\psi(\mathbf{r})|^2\).  The wavefunction gives you the complex amplitude for finding the particle in a location, and the probability of that outcome of a measurement is proportional to the magnitude squared of that amplitude.  The complex nature of the quantum amplitudes, combined with the idea that you have to square amplitudes to get probabilities, is where quantum interference effects originate.   

This is all well and good, but when you worry about the electrons flowing in your house wiring, or even your computer or mobile device, you basically never worry about these quantum interference effects.  Why not?

The answer is rooted in the idea of quantum coherence, in this case of the spatial state of the electron.  Think of the electron as a wave with some wavelength and some particular phase - some arrangement of peaks and troughs that passes through zero at spatially periodic locations (say at x = 0, 1, 2, 3.... nanometers in some coordinate system).   If an electron propagates along in vacuum, this just continues ad infinitum.

If an electron scatters off some static obstacle, that can reset where the zeros are (say, now at x = 0.2, 1.2, 2.2, .... nm after the scattering).  A given static obstacle would always shift those zeros the same way.   Interference between waves (summing the complex wave amplitudes and squaring to find the probabilities) with a well-defined phase difference is what gives the fringes seen in the famous two-slit experiment linked above.

If an electron scatters off some dynamic obstacle (this could be another electron, or some other degree of freedom whose state can be, in turn, altered by the electron), then the phase of the electron wave can be shifted in a more complicated way.  For example, maybe the scatterer ends up in state S1, and that corresponds to the electron wave having zeros at x=0.2, 1.2, 2.2, .....; maybe the scatterer ends up in state S2, and that goes with the electron wave having zeros at x=0.3, 1.3, 2.3, ....  If the electron loses energy to the scatterer, then the spacing between the zeros can change (x=0.2, 1.3, 2.4, ....).  If we don't keep track of the quantum state of the scatterer as well, and we only look at the electron, it looks like the electron's phase is no longer well-defined after the scattering event.  That means if we try to do an interference measurement with that electron, the interference effects are comparatively suppressed.

In your house wiring, there are many many allowed states for the conduction electrons that are close by in energy, and there are many many dynamical things (other electrons, lattice vibrations) that can scatter the electrons.  The consequence of this is that the phase of the electron's wavefunction only remains well defined for a really short time, like 10^-15 seconds.    Conversely, in a single hydrogen atom, the electron has no states available close in energy, and in the absence of some really invasive probe, doesn't have any dynamical things off which to scatter.

I'll try to write more about this soon, and may come back to make a figure or two to illustrate this post.