Bound-State Solutions of the Schrödinger Equation

Abstract

We present a new method for solving the Schrödinger equation for attractive potentials. The basic idea is to use in conjuction both the differential and integral forms of the equation. In one dimension, for example, a formal solution for $\psi \mathrm{}\left(x\right)\mathrm{}$ can be found from the differential equation; since this is of second order, the solution is fixed, in principle, by $\psi \mathrm{}\left(0\right)\mathrm{}$ and ${\psi }^{\prime }\mathrm{}\left(0\right)\mathrm{}\equiv \frac{d\psi }{\mathrm{dx}}|x_{=0\mathrm{}}$. These quantities are in fact unknown, but from the integral equation we can calculate integral expressions for them. We insert the formal solution into the integrands of these expressions and eliminate $\psi \mathrm{}\left(0\right)\mathrm{}$ and ${\psi }^{\prime }\mathrm{}\left(0\right)\mathrm{}$ from the resulting equations to get an equation for the energy eigenvalues. In three dimensions, the method works much the same way. In terms of $Q_l\mathrm{}\left(r\right)\mathrm{}$, where the radial wave function $R_l\mathrm{}\left(r\right)\mathrm{}$ for the $l\mathrm{th}$ angular momentum state is $R_l\mathrm{}\left(r\right)=r^l{Q}_l\mathrm{}\left(r\right)\mathrm{}$, it leads to the basic equation for the eigenvalues, $1=-\frac{K{v}_0{\mathrm{}\left(\mathrm{ik}\right)\mathrm{}}^l}{1\times 3\times 5\times \cdots \times \mathrm{}(2l+1)\mathrm{}}\int 0^{\infty }{h}_l\left(iK{r}^{\prime }\right)u\left(r^{\prime }\right)Q_l\left(r^{\prime }\right)r^{\prime l+2\mathrm{}}d{r}^{\prime },$ where $K^2$ is essentially the magnitude of the energy, and the potential is written as $-v_{0}u\left(r\right)\mathrm{}$. For nonsingular potentials, the formal solution for $Q_l\mathrm{}\left(r\right)\mathrm{}$ can be put in the form $Q_l\mathrm{}\left(r\right)=S_l\mathrm{}\left(r\right)+\Sigma \stackrel{}{s}{S}_{l,s}\mathrm{}\left(r\right)\mathrm{}{U_0}^{\mathrm{}\left(s\right)\mathrm{}}+\Sigma \stackrel{}{\mathrm{st}}{S}_{l,st}\mathrm{}\left(r\right)\mathrm{}{U_0}^{\mathrm{}\left(s\right)\mathrm{}}{U_0}^{\mathrm{}\left(t\right)\mathrm{}}+\cdots ,$ where $U\left(r\right)=v_{0}u\left(r\right)-K^2$, and where ${U_0}^{\mathrm{}\left(s\right)\mathrm{}}$ is the $s\mathrm{th}$ derivative of $U\left(r\right)\mathrm{}$ at the origin. The functions $S_l\mathrm{}\left(r\right)\mathrm{}$ and $S_{l,s}\mathrm{}\left(r\right)\mathrm{}$ are calculated, and then the results are applied to the case of a Gaussian potential. A modification of these ideas for singular potentials is also discussed, and is applied to the Yukawa potential.