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[[Saito Group (Open)]]
Started by Thomas.2013.09.02
This is Thomas's logbook for September of 2013.
#contents
* 1st week [#y8d4828f]
-09.02
--The Hamiltonian for the electron-photon interaction is given by $H = \frac{1}{2m} \left(\mathbf{p}-\frac{q}{c}\mathbf{A}\right)\left(\mathbf{p}-\frac{q}{c}\mathbf{A}\right) +q\phi$, in this equation $\mathbf{p}$ is the linear momentum, $q$ is the charge of the particle, $\mathbf{A}$ is the vector potential, $m$ is the particle's mass, $c$ is the speed of light and $\phi$ is the scalar potential. It is possible to check this by using the canonical equations from Hamilton, $\dot{\mathbf{r}}=\frac{\partial H}{\partial \mathbf{p}}$ and $\dot{\mathbf{p}}=-\nabla H$.
From those last equations together with the following identities $ \frac{d \mathbf{A}}{d t} = \frac{\partial \mathbf{A}}{\partial t} + \left(\frac{d \mathbf{r}}{d t} \cdot \nabla \right)\mathbf{A}=\frac{\partial \mathbf{A}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{A}$ and $(\mathbf{v} \cdot \nabla) \mathbf{A} = -\mathbf{v} \times (\nabla \times \mathbf{A}) + \nabla(\mathbf{v}\cdot \mathbf{A})$
From those last equations together with the following identities $ \frac{d \mathbf{A}}{d t} = \frac{\partial \mathbf{A}}{\partial t} + \left(\frac{d \mathbf{r}}{d t} \cdot \nabla \right)\mathbf{A}=\frac{\partial \mathbf{A}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{A}$ and $(\mathbf{v} \cdot \nabla) \mathbf{A} = -\mathbf{v} \times (\nabla \times \mathbf{A}) + \nabla(\mathbf{v}\cdot \mathbf{A})$ With all this considered it was possible to arrive at the equation of motion of a particle under an electromagnetic field, $m \ddot{\mathbf{r}}=\frac{q}{c}(\mathbf{v} \times \mathbf{B}) + q \mathbf{E}$.
Next steps:
Consider the wave functions from quantum well, hydrogen atom, and solve with the electromagnetic(electron-photon interaction) perturbation.