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Usually, you have to write your 2d lattice in 1-d array, and the $\exp(iHt/\hbar)$ a 2d matrix. For example, I have 2d lattice $2 \times 2$ $$ \begin{matrix} (1,1) & (1, 2) \\ (2, 1) & (2 …

Question: can the term $\phi_N T_N \phi_e$ be neglected? $${T_N}| \phi_N \phi_e \rangle = \phi _e T_N \phi_N + \phi_N T_N \phi_e - \hbar^2 \sum_\alpha {1 \over M_\alpha}{\vec \nabla_\alpha} \cdot \{ \ …

Let's first recall the process of emergence of the momentum schrödinger equation, strating from the time-independent schrödinger equation. We make a Fourier transformation in time, then obtain a spati …

The relation of the first derivative of the wave function is meant to keep the flux of propability constant across the interface. Therefore, if the effectice masses are different across a heterojuncti …

The properties of a physical system depends on the boundary condition as well as the governing equations. All eigen vlaues are results from a giving boundary condition. Therefore, it is to be kept in …

Let's write wave functions in these three regions for a level of energy $-E, E>0$. The corresponding exponent of wavefunction is $q = \sqrt{\frac{2 m E}{\hbar^2}}$: Region 1: $x < -a$ $$ \Psi_1(x) …

Starting from your equation: $${\Psi^*(x)}\,(d^2\Psi(x)/dx^2)-(d^2{\Psi^*(x)}/dx^2)\,\Psi(x)=0$$ You are just one-step away from the answer: \begin{align} 0 &= {\Psi^*(x)}\,(d^2\Psi(x)/dx^2)-(d^2{\Psi …