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Results tagged with quantum-mechanics
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user 23611
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
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2
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Rotation which diagonalizes the Hamiltonian
I stumbled upon the following question:
Given the Hamiltonian of a spin-$1/2$ particle
$$\hat{H}=\epsilon\begin{pmatrix} 0 & -e^{i\pi/4}\\
-e^{-i\pi/4} & 0 \end{pmatrix} = \frac{2\epsilon}{\hbar} \ …
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5
votes
1
answer
115
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Zener breakdown - a quantum mechanical derivation
In §6.8 of Ziman's "Principles of the Theory of Solids" he derives the imaginary component of the wave vector of an electron inside an energy gap (due to action of an electric field). He starts by wri …
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Zener breakdown - a quantum mechanical derivation
I think I figured this out. Substituting
$$
\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^ …
- 695
10
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2
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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
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1
answer
564
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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
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0
answers
78
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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:
$$
\phi\left(\mathbf{x},t\right)=\frac{1}{\sqrt{V}}\sum_{\mathbf{k}}\left(A_{\mathbf{k …
- 695
3
votes
1
answer
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Propagator of harmonic oscillator at specific times
It is well known that the propagator (kernel) of a simple harmonic oscillator is given by
$$
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 …
- 695
4
votes
1
answer
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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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