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Commit 5c72f7d2 authored by Anton Akhmerov's avatar Anton Akhmerov
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Merge branch 'Physics00-master-patch-61441' into 'master'

Update 11_nearly_free_electron_model.md

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......@@ -93,7 +93,7 @@ In this figure, the orange curves represent the nearly-free electron dispersion,
### Analyzing the avoided crossings
*Remark: An avoided crossing is an important concept in quantum mechanics that can be analyzed using **perturbation theory**. You will only learn this theory later in QMIII, so we will need to postulate some important facts here.*
_Remark: An avoided crossing is an important concept in quantum mechanics that can be analyzed using **perturbation theory**. You will only learn this theory later in QMIII, so we will need to postulate some important facts here._
To analyze what happens near the crossings, we first neglect the lattice potential and consider the free-electron dispersion near the crossing at $k=\pi/a$ in 1D. Near this crossing, we see that two copies of the dispersion come together (one copy centered at $k=0$, the other at $k=2\pi/a$). We call the corresponding plane-wave eigenfunctions $|k\rangle$ and $|k'\rangle =|k-2\pi/a\rangle$. We now express the wavefunction near this crossing as a linear superposition $|\psi\rangle = \alpha |k\rangle + \beta |k'\rangle$. Note that this wave function is very similar to that used in the LCAO model, except there we used linear combinations of the orbitals $|1\rangle$ and $|2\rangle$ instead of the plane waves $|k\rangle$ and $|k'\rangle$.
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