A Bloch state psi_k(r) = e^{ik·r} u_k(r) is NOT a plane wave, even though it contains the factor e^{ik·r}. What makes it different?
AThe factor e^{ik·r} oscillates but u_k(r) is constant, so it is actually a plane wave
BThe function u_k(r) is periodic with the lattice periodicity, so the Bloch state is a plane wave modulated by a periodic function — it has the lattice symmetry built into it
CBloch states have discrete k values while plane waves have continuous k
DBloch states only exist in one dimension; plane waves exist in three dimensions
A pure plane wave e^{ik·r} has uniform amplitude everywhere. A Bloch state multiplies this by u_k(r), which oscillates with the periodicity of the crystal lattice — it is large near atomic cores and small between them (or vice versa). The result is a wave that propagates through the crystal but whose amplitude is modulated to reflect the atomic arrangement. This modulation is what produces energy bands and gaps: the electron 'knows' about the crystal structure through u_k(r).
Question 2 Multiple Choice
Crystal momentum ħk for a Bloch electron is not true momentum. What does it actually represent?
AThe kinetic energy of the electron divided by its velocity
BThe quantum number labeling translational symmetry of the crystal — it is conserved modulo reciprocal lattice vectors G, and determines how the electron responds to external fields via ħ dk/dt = F_ext
CThe average momentum of the electron, identical to ħk for a free particle
DA classical quantity that has no quantum mechanical significance
True momentum ħk would require a state that is an eigenstate of the momentum operator, which Bloch states are not (the periodic part u_k breaks pure translational symmetry). Crystal momentum is the quantum number associated with discrete translational symmetry — it is defined modulo G and is conserved in that sense. Its physical importance is in semiclassical dynamics: an external force changes crystal momentum as ħ dk/dt = F_ext, and the electron's velocity is v = (1/ħ) ∂E/∂k. These equations govern electrical transport.
Question 3 True / False
Bloch's theorem implies that an electron in a perfect crystal moves without scattering — the periodic potential alone does not cause resistance.
TTrue
FFalse
Answer: True
This is a profound consequence. A Bloch state is a stationary state of the periodic Hamiltonian, meaning the electron propagates indefinitely through the perfect lattice without scattering. Electrical resistance in real metals comes entirely from deviations from perfect periodicity: thermal vibrations (phonons), impurities, defects, and grain boundaries. A perfect crystal at zero temperature would have zero resistance even without superconductivity — it simply would have no mechanism to scatter Bloch electrons.
Question 4 Short Answer
Explain why Bloch's theorem reduces an infinite-crystal problem to a unit-cell problem.
Think about your answer, then reveal below.
Model answer: The Bloch form psi_k(r) = e^{ik·r} u_k(r) means that once you know u_k(r) within one unit cell, you know the wavefunction everywhere — the factor e^{ik·r} provides the phase relationship between cells. Substituting the Bloch form into the Schrodinger equation gives an equation for u_k(r) alone, with periodic boundary conditions on the unit cell. Instead of solving for psi in all of infinite space, you solve for u_k in a finite volume (one unit cell) for each k in the Brillouin zone. This is computationally tractable and is the basis of all band structure calculations.
This dimensional reduction is what makes solid-state physics possible. An infinite crystal has ~10^23 atoms, but Bloch's theorem says you only need to understand one unit cell (plus how k labels the states). The k-dependence of the eigenvalues E_n(k) gives the band structure.