A Modern Approach to Quantum Mechanics · Problem 6.24

The One-Dimensional Harmonic Oscillator — John S. Townsend

Physics — Quantum Mechanics · scattering-square-well-ramsauer

Problem

(a) Show that the transmission coefficient for scattering from the potential energy well V(x)={0x<0−V00<x<a0x>aV(x) = \begin{cases} 0 & x < 0 \\ -V_0 & 0 < x < a \\ 0 & x > a \end{cases} is given by T=[1+sin⁡22mℏ2(E+V0) a4EV0(E+V0)V0]−1T = \left[1 + \frac{\sin^2\sqrt{\dfrac{2m}{\hbar^2}(E+V_0)\,}a}{4\dfrac{E}{V_0}\dfrac{(E+V_0)}{V_0}}\right]^{-1} Suggestion: What is the transcription required to change the wave function (6.142) into the one appropriate for this problem? What happens to the transmission coefficient (6.144) under this transcription? (b) Show that for certain incident energies there is 100 percent transmission. Suppose that we model an atom as a one-dimensional square well with a width of 11 \AA\ and that an electron with 0.7 eV of kinetic energy encounters the well. What must the depth of the well be for 100 percent transmission? This absence of scattering is observed when the target atoms are composed of noble gases such as krypton.

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