- #1

Tertius

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- Homework Statement
- I am discretizing the Schrodinger equation and solving for a quantum tunneling probability

- Relevant Equations
- Schrodinger equation

I thought I solved the problem in answering my own post a few days ago, but the tunneling probability vs. energy trend is clearly wrong. I've remade the post because I have totally changed my approach and need a better understanding of the boundary setup.

$$-\frac{\hbar^{2}}{2m} \frac{d^{2}}{dx^{2}} \psi(x) +V(x)\psi(x)=E\psi(x)$$

$$H=-\frac{\hbar^{2}}{2m}+V(x)$$

The tri-diagonal matrix setup after applying finite derivates is ##\frac{h^2}{2ma^2}+V_i## for diagonals and ##\frac{-h^2}{2ma^2}## for the off-diagonals.

I have not included any boundary conditions explicitly. The standard setup for this problem is $$H\psi=E\psi$$ where H is the Hamiltonian and E are the energy eigenvalues. I don't need to compute the eigenvalues, however because this is an unbounded system. The energy eigenvalue for this system should be the kinetic energy of the incoming particle. This allows me to set it up as $$(H-IE)\psi=b$$ The vector b is used for the right side as an empty vector. ##I## is the identity matrix.

When the vector ##b## is actually all 0, the solution is always trivial. I have to include a very small, non-zero value somewhere in this vector for it to produce a solution that looks like the attached picture. However, when I recompute for different energies, the tunneling probablity changes in a weird, oscillatory way as shown in the other attached graph.

What is wrong with the assumptions of this setup? Do I need to include boundaries specifically, even though the solutions are plane waves?

**Overall description:**a plane wave approaches a barrier, tunnels, and a plane wave exits. I am doing a full 1D simulation so I can eventually use non-standard potentials, but I'm starting with a square potential.**Setup of Solution:**$$-\frac{\hbar^{2}}{2m} \frac{d^{2}}{dx^{2}} \psi(x) +V(x)\psi(x)=E\psi(x)$$

$$H=-\frac{\hbar^{2}}{2m}+V(x)$$

The tri-diagonal matrix setup after applying finite derivates is ##\frac{h^2}{2ma^2}+V_i## for diagonals and ##\frac{-h^2}{2ma^2}## for the off-diagonals.

I have not included any boundary conditions explicitly. The standard setup for this problem is $$H\psi=E\psi$$ where H is the Hamiltonian and E are the energy eigenvalues. I don't need to compute the eigenvalues, however because this is an unbounded system. The energy eigenvalue for this system should be the kinetic energy of the incoming particle. This allows me to set it up as $$(H-IE)\psi=b$$ The vector b is used for the right side as an empty vector. ##I## is the identity matrix.

When the vector ##b## is actually all 0, the solution is always trivial. I have to include a very small, non-zero value somewhere in this vector for it to produce a solution that looks like the attached picture. However, when I recompute for different energies, the tunneling probablity changes in a weird, oscillatory way as shown in the other attached graph.

**Question:**What is wrong with the assumptions of this setup? Do I need to include boundaries specifically, even though the solutions are plane waves?

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