What is the shape of a band electron in momentum space?

Question

Band electrons occupy adjacent sharply defined momentum states that in xyz space take the form of a spectrum of wave functions. These wave functions span the entire xyz volume of any compact unit of metal, e.g. a 1 cm diameter perfect crystal of silver.

For any selected xyz space axis crossing such a unit of metal, the wave function of a particular band electron will have the form of two sharply defined momentum states, one for "leftward electron travel," and one for "rightward electron travel." There will also be lower momentum components created by the reflection points at the xyz edges of the metal unit, but these will be very low in amplitude compared to the left and right momentum components.

Now, my point is just this: When folks talk about "Fermi seas" and "Fermi liquids," I've always tended to think vaguely about them as literal droplets occupying momentum space, with cold electrons swarming in the middle and the hottest electrons racing quickly around the Fermi surface.

Alas, that image seems to be wrong in the following sense: By Fourier transformation of the momentum spectra, the amplitudes that define "where" an electron is in momentum space seem to form hollow shells. This is in sharp contrast to the strong tendency in xyz space for particle wave functions to form compact, particle-like wave packets centered around the last location at which they were "observed."

If so, the correct visualization of individual electron wave functions in a Fermi fluid is a much more intriguing: It is a highly stable Russian Doll model in which each higher-energy electron forms a shell that completely encloses all lower-energy electrons.

That kind of complex and very un-particle-like distribution of amplitudes just does happen in xyz space, where the slightest perturbation of the wave function causes it to collapse to a singular xyz location.

So, after all of that, two brief questions:

(1) Is the Russian Doll model correct?

(2) Somewhere in past readings of the literature on electron bands for specific metals, I've seen lovely figures of complex, sharply-pointed 3D surfaces that I did not how to interpret at the time. What are these figures called, so I can look them up online? Also, are these figures equivalent topologically to the Russian Doll wave functions of electrons in momentum space?

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2016-05-29.07:25 EST - Wrap-up notes

(a) The "Russian nesting dolls" are also called matryoshka dolls. [thanks @LubosMotl]

(b) The phrase I was missing in my question 2 was, ironically, one that I overlooked because I already knew it very well from a slightly different context. To me "Fermi surface" [thanks @garyp and @LubosMotl] was just a quick way to refer to the highest momentum electrons in a band, that is, the top of a Fermi sea.

These are the same and only electrons that in a metal perform such marvelous tricks as reflecting photons back in a mirror. Given the tendency in physics to name unique representation forms after specific people (e.g. Brillouin zones), it did not occur to me that this simple phrase would also be the name for the specific and delightful 3D figures that show the topological implications of these highest-energy states.

(c) Queue up philosophical discussion... (Translation: All real physicists I beg you, stop reading now!)

Sometimes I think folks give up a bit too too easily on exploring and marveling at the beauty of simple relationships in both math and physics.

For example, the very existence of the Planck-centered mathematical symmetry between xyz space and momentum space strikes me as one of the most remarkable and unexpected relationships in all of physics, and thus one that merits a bit of exploration using as many of our powerful perceptual tools as possible.

One way to do that kind of exploration is to go where the math takes you, while trying hard not to make judgment calls that are instead based on what is most familiar.

For example, is momentum space a "real" space? Both because it is not the space we see in everyday life, and because it can always interpreted as nothing more than a Fourier transform of what we see in xyz space, the deeply biological reflex is to say "no, of course momentum space is not 'real' ").

The situation is not quite so clear if for example you decided to analyze it using a machine intelligence, which by design would only follow the math to decide what is real and what is not. In the case of xyz space and momentum space, the extraordinary simple mathematical symmetry between them would result in distinction or prioritization between the two spaces. Machine intelligence in that case would assess situation as one were the biological intelligences that preceded it explored the familiar side of the equation very thoroughly, and the other side rather incompletely due to a tendency to assume that any difficulties or inconsistencies that popped up were a reflection of the artificial nature of the momentum space construction.

Yet both mathematics and physics are replete with examples where simply following the math is what leads to some of the most interesting insights. Dirac for example discovered antimatter by finally taking the bizarre implications of a simple equation as a prediction rather than as an anomaly. In math, it literally took centuries of harumphing and pontificating before imaginary numbers were promoted to being sufficiently "real" to use in serious mathematical work. In both cases, the inclination to ignore the math because it didn't fit biases dominated over the simpler story told by the math itself.

So, please forgive me if for the sheer joy of exploration I sometimes like looking at something like momentum space as a real space. Such silly thinking forces me to look at seeming failures of that model not as mathematical flaws, but as examples where my understanding of how to use that space is incomplete.

Trying to understand the less-explored side of this simple Fourier transform relationship is not going to lead to any new physics. But that is fine with me, since my only goal in this case is to gain a conceptual understanding of a particularly lovely symmetry that sits at the very heart of quantum mechanics: The Planck-centered transformational symmetry between xyz space and momentum space.

Answer 1

First, to imagine "individual electrons" as having well-defined values of $\vec p$ is in no way necessary. The electrons collectively occupy the whole band – which may be described as a subspace of the one-electron Hilbert space. But this subspace of the Hilbert space may be described by many different bases. For any basis, one may say that 1 electron is created into each basis vector. The basis of momentum $\vec p$ eigenstates is just one (although natural) possibility among infinitely many.

Second, more importantly, there is absolutely no sense in which the sentence

This is in sharp contrast to the strong tendency in xyz space for particle wave functions to form compact, particle-like wave packets centered around the last location at which they were "observed."

is correct. The wave functions in no way want to clump. On the contrary, they typically evolve so that they get diluted in the whole space that is available.

Also, the wave function for the momentum $\vec p$ eigenstates has $\Delta \vec p=0$, and by the uncertainty principle $\Delta x \cdot \Delta p \geq \hbar/2$, it unavoidably follows that $\Delta x = \infty$. The wave function of a momentum eigenstate is unavoidably spread in the whole space. In fact, it must have the same probability for the electron to be at any point of the space (well, inside the material).

Electrons are detected at particular points. But the correct explanation of this fact isn't the assumption that wave functions want to get clumped. They don't want to clump at all. The correct explanation, as undergraduate students of physics learn already in the first lectures of quantum mechanics, is that the (squared absolute value of the) wave function describes the probability density that the electron will be found in the vicinity of a given point.

This point is no detail. It's a totally fundamental fact about the wave function and it's also what Max Born got his well-deserved Nobel prize for. It makes absolutely no sense to use the term "wave function" if the speaker doesn't understand that its squared absolute value is interpreted as the probability density.

Even if the wave function for an electron is absolutely delocalized and uniformly covers the whole 1-ton metallic object, the measurement of the electron's position will produce a sharp number. The role of the wave function is that we may predict where the electron is more likely or less likely to be detected. If the wave function is uniform in the whole metal (up to the phase), and it is in the case of a momentum eigenstate, it means that the probability to detect the electron at any point of the metal is the same.

But the wave function doesn't describe the shape (i.e. some internal property) of the electron. The electron is point-like (even in the Standard Model; one needs to go to string theory to revise this assumption). Instead, the wave function probabilistically describes the location of the point-like electron (i.e. an external property). To describe the internal properties of a particle, e.g. a hydrogen atom, one needs a wave function that depends on at least two vectors $\vec r_1,\vec r_2$, i.e. the positions of the nucleus and an electron (or, equivalently, the position of the center-of-mass and the relative position of the electron relatively to the nucleus).

The metal – the state of the electrons in it – is of course stable in the ground state. It is an eigenstate of the Hamiltonian. One may talk about points in the momentum space and say that some shells encircle others (like Russian dolls around other Matryoshkas). But this is just a way to use the language for the geometric (mathematical) relationships between points and surfaces in the momentum space. One must realize that the momentum space is in no way the real space. Physics isn't local in the momentum space (although it is local in the position space). So the encircling of some regions of the momentum space by other regions is much less "tight" than it is in a normal space. Particles may tunnel through the "walls" in the momentum space, e.g. by emitting other particles such as photons.

1. So whether the electrons in a metal are Matryoshkas depends on which properties of the Matryoshka you find important. Physics of metals is clearly not being described in terms of Russian dolls so there can't be any textbook answer to a question whether electrons in metals are Matryoshkas.

2. There are extremely many complicated Fermi surfaces for various materials, see e.g. the Google Images search for Fermi surfaces. The topologies may be many things, there may be sharp peaks on the surface, holes, and many other things. If you want to talk about bands separated by a gap, that's possible. But for conductors, it's really a big gap between the conduction band and the valence band which is basically assumed to be impenetrable, so only the conduction band is relevant for the dynamics of the metals. You need to talk about semiconductors if the probability to get through the gap.