This article is translated from a Chinese article on my Zhihu account. The original article was posted at 2021-01-13 15:33 +0800.

We try solving the even function solutions to the time-independent Schrödinger equation for the potential


such that $E<0$ (bound states).

Obviously, the solutions have the form

\[\psi= \begin{cases} A_{\left\lfloor\frac{\left|x\right|}{a}\right\rfloor}\mathrm e^{\kappa\left|x\right|} +B_{\left\lfloor\frac{\left|x\right|}{a}\right\rfloor}\mathrm e^{-\kappa\left|x\right|}, &\left|x\right|<na,\\ A_n\mathrm e^{\kappa\left|x\right|}+B_n\mathrm e^{-\kappa\left|x\right|}, &\left|x\right|>na, \end{cases}\]

where $A_j,B_j$ ($j=0,1,\ldots,n$) are constants of integration, and $\kappa:=\frac{\sqrt{-2mE}}{\hbar}$.

Noting that we are finding bound states, we should have $\lim_{x\to\infty}\psi=0$. Therefore,

\[\begin{equation} \label{eq: A_n} A_n=0. \end{equation}\]

Function $\psi$ is naturally continuous at $x=0$. Considering the continuity of $\psi$ at $\left|x\right|=ja$ ($j=1,2,\ldots,n$), we have

\[\begin{equation} \label{eq: continuity} A_{j-1}\mathrm e^{\kappa ja}+B_{j-1}\mathrm e^{-\kappa ja}=A_j\mathrm e^{\kappa ja}+B_j\mathrm e^{-\kappa ja}. \end{equation}\]

For $j=-n,\ldots,n$, integrate both sides of the time-independent Schrödinger equation over interval $\left[ja-\varepsilon,ja+\varepsilon\right]$ and let $\varepsilon\to0$, and we have

\[\left.\frac{\mathrm d\psi}{\mathrm dx}\right|_{ja^-}^{ja^+}=-\beta\left.\psi\right|_{ja},\]

where $\beta:=\frac{2m\alpha}{\hbar^2}$.

For $j=0$, the formula above gives

\[\begin{equation} \label{eq: derivative jump j=0} \left(A_0\kappa-B_0\kappa\right)-\left(-A_0\kappa+B_0\kappa\right)=-\beta\left(A_0+B_0\right). \end{equation}\]

For $j=1,2,\ldots,n$, on the other hand,

\[\begin{equation} \label{eq: derivative jump} \left(A_j\kappa\mathrm e^{\kappa ja}-B_j\kappa\mathrm e^{-\kappa ja}\right) -\left(A_{j-1}\kappa\mathrm e^{\kappa ja}-B_{j-1}\kappa\mathrm e^{-\kappa ja}\right) =-\beta\left(A_j\mathrm e^{\kappa ja}+B_j\mathrm e^{-\kappa ja}\right). \end{equation}\]

Equations \ref{eq: A_n}, \ref{eq: continuity}, \ref{eq: derivative jump j=0}, and \ref{eq: derivative jump} together form a homogeneous linear equation w.r.t. $A_j,B_j$ ($j=0,1,\ldots,n$). To require that the equation has non-zero solutions, the determinant of the coefficient matrix should be zero, and we can find $\kappa$ by this property. However, the equation is transcendental for $n>0$.

If we found the value of $\kappa$, the solution space for the homogeneous linear equation should be one-dimensional, and then we can determine all the constants by normalizing $\psi$.