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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

1 vote
2 answers
712 views

Rotation which diagonalizes the Hamiltonian

I stumbled upon the following question: Given the Hamiltonian of a spin-$1/2$ particle $$\hat{H}=\epsilon\begin{pmatrix} 0 & -e^{i\pi/4}\\ -e^{-i\pi/4} & 0 \end{pmatrix} = \frac{2\epsilon}{\hbar} \ …
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5 votes
1 answer
115 views

Zener breakdown - a quantum mechanical derivation

In §6.8 of Ziman's "Principles of the Theory of Solids" he derives the imaginary component of the wave vector of an electron inside an energy gap (due to action of an electric field). He starts by wri …
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2 votes

Zener breakdown - a quantum mechanical derivation

I think I figured this out. Substituting $$ \mathcal{E}^{0}=\frac{\hbar^{2}}{2m}\left(\frac{1}{2}G\right)^{2}=\frac{G^{2}\hbar^{2}}{8m} $$ into $$ \frac{\hbar^{4}}{4m^{2}}\kappa^{4}-\left(\frac{\hbar^ …
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10 votes
2 answers
2k views

Flat bands and localization in real space

When reading this paper I came across the following claim: From a band-theory point of view, the flat bands should have localized wavefunction profiles in real space Is there a rigorous proof of thi …
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1 vote
1 answer
564 views

How the matrix representation of a Hamiltonian affects the eigenvalues?

Suppose we're given the following Hamiltonian: $$\hat{H}=\frac{\omega}{\hbar} \left(\hat{S}_+^2+\hat{S}_-^2\right)$$ Suppose also that we measure $\vec{S}^2$ and get $6\hbar^2$, i.e. reduced to the $s …
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1 vote
0 answers
78 views

Fourier decomposition range in field quantization procedure

Consider the complex Klein-Gordon field (in finite volume $V$), which can be expanded in terms of plane waves as: $$ \phi\left(\mathbf{x},t\right)=\frac{1}{\sqrt{V}}\sum_{\mathbf{k}}\left(A_{\mathbf{k …
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3 votes
1 answer
1k views

Propagator of harmonic oscillator at specific times

It is well known that the propagator (kernel) of a simple harmonic oscillator is given by $$ U\left(x_{b},T;x_{a},0\right)=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\left\{ \frac{im\omega}{2\h …
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4 votes
1 answer
2k views

Representing a rotation around an arbitrary axis using Wigner $D$-matrix

It is known that an arbitrary rotation can be expressed in terms of three consecutive rotations called the Euler rotations. So instead of expressing the rotation operator as $\hat{R}(\hat{n},\phi) = \ …
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