Consider a potential well defined as U(x)=infty for x<0, U(x)=0 for 0<x<L, and U(x)=U0>0 for x>L

(a) For $0<x<L$, the potential energy $U(x)=0$. The Schrödinger equation in this region is given

Question

Consider a potential well defined as $U(x)=infty$ for $x<0, U(x)=0$ for $0<x<L,$ and $U(x)=U_{0}>0$ for $x>L$ (Fig. 40.27$)$ . Consider a particle with mass $m$ and kinetic energy $E<U_{0}$ that is trapped in the well. (a) The boundary condition at the infinite wall $(x=0)$ is $psi(0)=0 .$ What must the form of the function $psi(x)$ for $0<x<L$ be in order to satisfy both the Schrödinger equation and this boundary condition? (b) The wave function must remain finite as $x ightarrow infty$ . What must the form of the function $psi(x)$ for $x>L$ be in order to satisfy both the Schrödinger equation and this boundary condition at infinity? (c) Impose the boundary conditions that $psi$ and $d psi / d x$ are continuous at $x=L .$ Show that the energies of the allowed levels are obtained from solutions of the equation $k cot k L=-kappa,$ where $k=sqrt{2 m E / hbar}$ and $kappa=sqrt{2 m}left(U_{0}-E ight) / hbar$

          Consider a potential well defined as $U(x)=infty$ for $x<0, U(x)=0$ for $0<x<L,$ and $U(x)=U_{0}>0$ for $x>L$ (Fig. 40.27$)$ . Consider a particle with mass $m$ and kinetic energy $E<U_{0}$ that is trapped in the well. (a) The boundary condition at the infinite wall $(x=0)$ is $psi(0)=0 .$ What must the form of the function $psi(x)$ for $0<x<L$ be in order to satisfy both the Schrödinger equation and this boundary condition? (b) The wave function must remain finite as $x 
ightarrow infty$ . What must the form of the function $psi(x)$ for $x>L$ be in order to satisfy both the Schrödinger equation and this boundary condition at infinity? (c) Impose the boundary conditions that $psi$ and $d psi / d x$ are continuous at $x=L .$ Show that the energies of the allowed levels are obtained from solutions of the equation $k cot k L=-kappa,$ where $k=sqrt{2 m E / hbar}$ and $kappa=sqrt{2 m}left(U_{0}-E
ight) / hbar$

Added by Edward T.

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