[NOTES/QM-13006] Boundary condition at a rigid wall

Boundary condition at a rigid wall

We will first derive boundary condition at a rigid wall. A rigid wall is a boundary such that the potential on one side of the wall is finite and on the other side it is infinite. We will consider a rigid wall as $V_0 \to \infty $ limiting case of a potential step given by \begin{equation} V(x) = \begin{cases} 0, & x < 0 \\ V_0 & x > 0. \end{cases} \end{equation} Considering energy $0 < E< V_0$, the solution of the Schrodinger equation \begin{equation} -\frac{\hbar^2}{2m}\DD[u(x)]{x} + (V(x)- E) u(x) =0 \label{eq01}. \end{equation} in the region $x< 0$, with $ V(x) = 0$ is \begin{equation} u_\text{I}(x) = A \cos k x + B \sin kx,\qquad k^2 = \frac{2mE}{\hbar^2}. \label{eq02} \end{equation} For $x>0$ it is given by \begin{equation} u_\text{II}(x) = C \exp(-\alpha x) + D \exp(\alpha x). \qquad \alpha ^2 = \frac{2m(V_0-E)}{\hbar^2}.\label{eq03} \end{equation} We note that the solution must remain finite as $x\to \infty$, this gives $D=$. Demanding continuity of $u(x)$ and its derivative $u^\prime(x)$ at $x=0$ gives \begin{eqnarray} u_\text{I}(0) =u_\text{II}(0) \Rightarrow A= C+D\label{eq04} \\ u_\text{I}^\prime(0) =u_\text{II}^\prime(0) \Rightarrow kB=-\alpha(C-D)\label{eq05}. \end{eqnarray} Using $D=0$ the two conditions become \begin{equation} A =C , \qquad\qquad k B = -\alpha C. \end{equation} When $V_0$ tends to infinity $\alpha \to \infty$, $C$ must go to zero so that $B$ is finite. This gives $C=0$, but $kB$ which is the value of the derivative at $x=0$, becomes indeterminate product $0\times \infty$. {\it To summarise, we have shown that the solution must vanish for $x>0$, it must be continuous at $x=0$ and that no condition is forced on the derivative of the solution at $x=0$.} We leave it for the reader to show that the same conclusion holds if region where potential is infinite extends over finite range of $x$. The boundary condition here was derived here assuming a constant potential, equal too 0, on one side of the wall. However, the boundary conditions apply when the potential is a non constant function of $x$ next to the rigid wall.