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advanced27 min studyLesson 502 in Physics

Open a gap by mixing degenerate plane waves

Use a two-state block at a Bragg plane and state the approximation’s limits.

Before you begin
On this page

The question for today

What you’ll understand

  • Identify the Fourier component coupling two plane waves.
  • Diagonalize a nearly degenerate Bragg pair.
  • Distinguish a local avoided crossing from a global insulating gap.

A weak periodic potential can have a large effect on states that were degenerate before the lattice was introduced. Near a Bragg plane, two free-electron waves differ by a reciprocal vector yet have nearly equal kinetic energies. Ordinary nondegenerate perturbation theory divides by their small energy separation and fails. Keeping both waves in a small Hamiltonian block resolves the failure and reveals the leading band gap, including the factor of two that is easily lost in Fourier conventions.

A Fourier coefficient is a scattering amplitude

Expand the periodic scalar potential in reciprocal harmonics, defining each coefficient as a cell average with a negative exponential. Matrix elements between normalized plane waves then connect wavevectors differing by a reciprocal vector. A real potential has conjugate coefficients at opposite reciprocal vectors, ensuring a Hermitian Hamiltonian. The zero coefficient shifts all relevant energies uniformly and cannot by itself open an avoided crossing between two otherwise free waves.

For a cosine potential, its quoted real-space amplitude is twice the magnitude of either nonzero complex Fourier coefficient. Confusing these two amplitudes doubles the predicted gap. The nearly-free approximation treats the nonzero Fourier coefficients as small relative to separations from remote free-electron states. It does not require them to be small compared with the energy difference of the resonant pair, because that pair will be diagonalized exactly within the retained subspace.

V(r)=∑GVGeiG⋅r,V−G=VG∗,V(x)=2Wcos⁡(Gx)⇒V±G=WV(\mathbf r)=\sum_{\mathbf G}V_{\mathbf G}e^{i\mathbf G\cdot\mathbf r},\qquad V_{-\mathbf G}=V_{\mathbf G}^{*},\qquad V(x)=2W\cos(Gx)\Rightarrow V_{\pm G}=W
V and VG are potential energies in eV or J, not electric potential in volts. G is an angular reciprocal wavevector. In the cosine example W is real and carries energy units.↔ Scroll wide equations horizontally.

Resolve the resonant block

At a one-dimensional zone boundary, the free states at k and k−G have equal kinetic energies when k=G/2. In their two-plane-wave basis, the diagonal entries are the two free energies plus the mean potential and the off-diagonal entry is the relevant Fourier coupling. Diagonalizing this Hermitian block gives two eigenvalues separated by twice the coupling magnitude at exact degeneracy. The phase of a single coupling changes the eigenvector phase convention but not that splitting.

Away from the crossing, the diagonal energy difference competes with the coupling. The resulting avoided-crossing formula smoothly approaches nondegenerate perturbation behavior when detuning is large. If several free states are simultaneously degenerate, a two-state truncation may be insufficient and a larger block is needed. Symmetry can also force a coupling to vanish; a zone boundary is therefore a location where a gap may open, not a guarantee that every branch is gapped.

H=(E1WW∗E2),E±=E1+E22±(E1−E22)2+∣W∣2H=\begin{pmatrix}E_1&W\\W^*&E_2\end{pmatrix},\qquad E_\pm=\frac{E_1+E_2}{2}\pm\sqrt{\left(\frac{E_1-E_2}{2}\right)^2+|W|^2}
E₁ and E₂ include the common mean-potential shift and W couples the retained plane waves. All entries have energy units. Remote states are omitted only when their coupling-induced corrections are small.↔ Scroll wide equations horizontally.

A local split does not decide the whole solid

For real positive coupling at degeneracy, the equal-weight sum and difference of the plane waves have different standing-wave charge patterns and different energies. Their density relative to the ionic potential explains the splitting physically. In higher dimensions a gap at one Bragg plane may coexist with states at the same energy elsewhere. To call a material a band insulator one must check the whole Brillouin zone, electron filling, spin structure, and any overlapping bands.

The quantum lab isolates the relative-phase distinction. Reset and set mixing angle to 90° and relative phase to 0°; predict equal basis probabilities and Bloch x=+1. Reset and set mixing angle to 90° and phase to 180°; predict the same probabilities but Bloch x=−1. Inspect time zero before playback, which adds coherent phase precession. This pure two-state geometry has no Hamiltonian-energy readout and does not calculate a crystal’s band gap.

E1=E2=E0⟹Δ=E+−E−=2∣W∣E_1=E_2=E_0\quad\Longrightarrow\quad \Delta=E_+-E_-=2|W|
Δ is the local two-state gap in the same energy units as W. The result is leading nearly-free-electron theory at a resonant Bragg pair, not a statement about a global indirect gap.↔ Scroll wide equations horizontally.

Work it through

Track a cosine amplitude

A weak periodic potential is V(x)=0.30 cos(Gx) eV. At a Bragg degeneracy the two unperturbed energies including their mean shift are both 2.00 eV. Find the Fourier coupling, split energies, and local gap.

  1. The complex exponential identity for cosine gives a coefficient 0.15 eV at +G and the same at −G. Thus the off-diagonal coupling is W=0.15 eV, not the full 0.30 eV real-space amplitude.

  2. At exact degeneracy the eigenvalues are 2.00±0.15 eV. They are 1.85 eV and 2.15 eV. For positive W, the symmetric basis sum is the upper state and the antisymmetric difference is the lower state in this chosen basis.

  3. The separation is 0.30 eV, twice the Fourier coefficient. This is a local result for the retained pair; possible states at other wavevectors must still be checked before claiming an insulating energy window.

The result

The coupling is 0.15 eV, the split energies are 1.85 and 2.15 eV, and the local gap is 0.30 eV.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Distinguish equal populations from equal phase

Reset and set θ=90°, φ=0°; predict basis probabilities 1/2 and Bloch x=+1. Reset and set θ=90°, φ=180°; predict the same probabilities and x=−1. Inspect the reset-time readouts.

Distinguish equal populations from equal phaseA normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Current readings: Probability |0⟩: 71.1%; Probability |1⟩: 28.9%; Bloch (x, y, z): 0.78, 0.45, 0.42. All parameters are adjustable with the labeled controls below.xyz|0⟩|1⟩Basis probabilitiesP(0)71.1%P(1)28.9%
Bloch vector / P(0)P(1)Phase latitude
Probability |0⟩
71.1%
Probability |1⟩
28.9%
Bloch (x, y, z)
0.78, 0.45, 0.42
t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe lab represents pure two-state geometry; Play adds relative-phase precession. It does not diagonalize a crystal Hamiltonian or display electronic energies.

Pause & reason

Check your understanding

For V(x)=2W cos(Gx), what is the leading local gap at a two-plane-wave Bragg degeneracy?

Your turn

Try it yourself

A retained Hermitian block has diagonal energies 1.7 and 2.3 eV and real off-diagonal coupling 0.4 eV. Find both eigenvalues and compare the splitting with the zero-detuning gap at the same coupling.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use the mean 2.0 eV and half-detuning 0.3 eV; the square root forms a 3–4–5 triangle.

Compare with the solution

The square root is √(0.3²+0.4²)=0.5 eV, so the eigenvalues are 1.5 and 2.5 eV. Their separation is 1.0 eV. At zero detuning the same coupling gives a minimum gap 0.8 eV. The larger off-resonant separation includes the original diagonal energy difference.

Keep the idea

Take it with you

  • A cosine amplitude and its complex Fourier coefficient differ by a factor of two.
  • Degenerate waves must be diagonalized together.
  • A local avoided crossing does not establish a global insulating gap.

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