Quantum Theory (physical chemistry)

[schnek]

Homework Statement

Consider the one dimensional space in which a particle can experience one of three potentials depending on its position. They are: V=0 for -$$\infty$$<x$$\leq$$0, 0, V=V$$_{2}$$ for 0$$\leq$$x$$\leq$$L, and V=V$$_{3}$$ for L$$\leq$$x<$$\infty$$. The particle wavefunction is to have both a component e$$^{ik_{1}x}$$ that is incident upon the barrier V$$_{2}$$ and a reflected component e$$^{ik_{1}x}$$ in region 1 (-$$\infty$$<x$$\leq$$0). In region 3 the wavefunction has only a forward component, e$$^{ik_{3}x}$$, which represents a particle that has traversed the barrier. The energy of the particle, E, is somewhere in the range of the V$$_{2}$$>E>V$$_{3}$$. The transmission probability, T, is the ration of the square modulus of the region 3 amplitude to the square modulus of the incident amplitude.

Base your calculation on the continuity of the amplitudes and the slope of the wavefunction at the locations of the zone boundaries and derive a general equation for T.

Homework Equations

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The Attempt at a Solution

This is a question that my professor removed from an assignment because we hadn't covered the material. Everything else in the course has been thermodynamics more or less, and I honestly have little clue as to what this question is referring to, but I'm curious as to how one might go about solving it. This is probably material that I'm supposed to know for Phys. Chem II... I don't want to be caught off guard.

 

[vela]

The problem has to do with finding the solution to the Schrödinger equation for the given potential. The basic idea is to find solutions valid in each of the three regions and then joining them together smoothly at the boundaries.