Question

Particles incident from the left are confronted with a step in potential energy shown in Figure $mathrm{P} 28.62 .$ Located at $x=0,$ the step has a height $U .$ The particles have energy $E>U.$ Classically, we would expect all the particles to continue on, although with reduced speed. According to quantum mechanics, a fraction of the particles are reflected at the barrier. (a) Prove that the reflection coefficient $R$ for this case is $$R=frac{left(k_{1}-k_{2} ight)^{2}}{left(k_{1}+k_{2} ight)^{2}}$$ where $k_{1}=2 pi / lambda_{1}$ and $k_{2}=2 pi / lambda_{2}$ are the wave numbers for the incident and transmitted particles. Proceed as follows. Show that the wave function $psi_{1}=A e^{i k_{1} x}+B e^{-i k_{1} x}$ satisfies the Schrödinger equation in region 1 , for $x < 0$. Here $A e^{i k_{1} x}$ represents the incident beam and $B e^{-i k_{1} x}$ represents the reflected particles. Show that $psi_{2}=C e^{i k_{2} x}$ satisfies the Schrödinger equation in region $2,$ for $x > 0$. Impose the boundary conditions $psi_{1}=psi_{2}$ and $d psi_{1} / d x=$ $d psi_{2} / d x$ at $x=0$ to find the relationship between $B$ and $A$. Then evaluate $R=B^{2} / A^{2}$. (b) A particle that has kinetic energy $E=7.00 mathrm{eV}$ is incident from a region where the potential energy is zero onto one in which $U=5.00 mathrm{eV}$. Find its probability of being reflected and its probability of being transmitted.

          Particles incident from the left are confronted with a step in potential energy shown in Figure $mathrm{P} 28.62 .$ Located at $x=0,$ the step has a height $U .$ The particles have energy $E>U.$
Classically, we would expect all the particles to continue on, although with reduced speed. According to quantum mechanics, a fraction of the particles are reflected at the barrier. (a) Prove that the reflection coefficient $R$ for this case is
$$R=frac{left(k_{1}-k_{2}
ight)^{2}}{left(k_{1}+k_{2}
ight)^{2}}$$
where $k_{1}=2 pi / lambda_{1}$ and $k_{2}=2 pi / lambda_{2}$ are the wave numbers for the incident and transmitted particles. Proceed as follows. Show that the wave function $psi_{1}=A e^{i k_{1} x}+B e^{-i k_{1} x}$ satisfies the Schrödinger equation in region 1 , for $x < 0$. Here $A e^{i k_{1} x}$ represents the incident beam and $B e^{-i k_{1} x}$ represents the reflected particles. Show that $psi_{2}=C e^{i k_{2} x}$ satisfies the Schrödinger equation in region $2,$ for $x > 0$.
Impose the boundary conditions $psi_{1}=psi_{2}$ and $d psi_{1} / d x=$ $d psi_{2} / d x$ at $x=0$ to find the relationship between $B$ and $A$. Then evaluate $R=B^{2} / A^{2}$. (b) A particle that has kinetic energy $E=7.00 mathrm{eV}$ is incident from a region where the potential energy is zero onto one in which $U=5.00 mathrm{eV}$. Find its probability of being reflected and its probability of being transmitted.

Added by Mary C.

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