I often see both terms used in textbooks, but I am not sure whether I understand the difference between them. Both describe the state of a system, however, they seem different in some ways. From what I have found, what is important in the wavefunction is its direction. Wavevectors, on the other hand, require a magnitude and direction. What is their difference? And how are they related?
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$\begingroup$ Can you link to a specific example where both the terms are used ? Different authors can use different terms to mean the same thing. $\endgroup$– biryaniCommented Feb 26, 2016 at 8:15
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$\begingroup$ @biryani The author does not use both terms. I am asking this because I am studying Fermi surfaces. Only states that are close to the Fermi surface can get excited. In other words, only wavevectors that have a length similar to the Fermi wavevector can get excited. However, from what I know of quantum mechanics, states are represented by wavefunctions. And for wavefunctions, the length does not matter. The statement that "only states that are close to the Fermi surface can get excited" appears to imply that a state has to be specified by a magnitude and direction. $\endgroup$– CoffeeIsLifeCommented Feb 26, 2016 at 8:37
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1$\begingroup$ @user97554 From your post, it is obvious that wavevector denotes the ordinary three-dimensional vector that simply denotes in space what is the direction and magnitude of the electron momentum. $\endgroup$– dominecfCommented Feb 26, 2016 at 8:58
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1$\begingroup$ @dominecf I see. I was also interested in the question in general. Maybe you could elaborate your comments into an answer? I think that may be helpful. $\endgroup$– CoffeeIsLifeCommented Feb 26, 2016 at 9:07
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3 Answers
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Wave functions, $\psi(x)$, describe the state of a quantum mechanical system, say a valance electron in an atom, in the position basis. An important set of states in any system are the eigenstates of the system's Hamiltonian. Another way to say this, is that these states have a well-defined (as in "not probabilistic") energy associated with them.
In a confined system such as an atom these energy eigenstates are discrete -- these are the ubiquitous shells of Bohr's modell or the orbitals of chemistry. In the case the electron moves arbitrarily far away as time passes, such as the case of a free particle moving through space, the spectrum is continuous, i.e. the energies of 'neighboring states' are infinitely close together. (This may not be 100% sound mathematically.)
For convenience, we seek to label these energies and their associated states by some other quantities than just the energy. (One reason is, that sometimes energies are degenerate, i.e. there is more than one different state with the same energy and we'd like to discern them.) These labels are referred to as quantum numbers. Now, with a valance electron in an atom, the quantum numbers are $n, l, m$ and spin -- you may have seen these in chemistry class to label orbitals. They arise from the mathematical treatment of the Schrödinger equation. In contrast, for a free particle, the energy eigenstates can be labelled by the particles momentum, which is proportional to a three-dimensional vector, $\vec k$, the so-called wave vector: $\vec p=\hbar\vec k$. People tend to set $\hbar=1$ and use momentum and wave vector synonymously.
In a solid-state environment -- the case you are interested in -- the quantum numbers turn out to be the band index (which is typically the same as the atomic quantum numbers) and a momentum or wave vector. This arises from Bloch's theorem.
Long story short: the wave vector (+ the band index) uniquely identifies an energy eigenstate of the solid state system which is itself described by a wave function.
answered Feb 26, 2016 at 9:44
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The notion of vectors is not only useful in our common three-dimensional geometry (with the x-y-z basis), but it is also widely used in other more abstract areas. In quantum mechanics, where objects are described as complex wavefunctions, it is useful to express the wavefunction as a superposition of some well-chosen infinite set of basis functions. The respective coefficients then form an infinite-dimensional complex vector, which fully describes the wavefunction.
One should not get confused that the notion of vectors is used in completely different contexts. In your textbook, the Fermi wavevector is just a three-dimensional vector describing some direction and magnitude of the electron momentum.
For completeness, note that if you consider a harmonic, infinite plane wave in the form exp(ik.r), where r is the radius vector, then the three-dimensional k vector it also uniquely denotes one particular shape of the wave.
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A wave function can be thought to be analogous to a vector in the sense that the set of all functions on a domain forms a vector space. Therefore, each function can be thought of as a vector in the vector space of all continuous functions on that domain. The reason you might have seen these differently is that matrix formulation can be easier to deal with in certain applications such as quantum computing, where the state space is finite (and similarly other finite dimentional systems), you can think of the wave function in terms of a matrix called the densite matrix, and the operators acting on it as linear transforms on that vector space.
