Inverse square root potential

The inverse square root potential is a three-parametric quantum-mechanical potential for which the one-dimensional Schrödinger equation is exactly solvable in terms of the confluent hypergeometric functions.^[1][2] The potential is defined as:

$V(x) = V_c+\frac{V}{\sqrt{x-x_0}}$.

Comments

Omitting the non-essential constants $V_c , x_0$ the general solution of the Schrödinger equation

$\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(E-V(x))\psi=0$

for the potential $V(x) = V_0/{\sqrt x}$ for arbitrary $V_0$ is written as

$\psi(x)= e^{-\delta x/2}\frac{du}{dy}$,

where

$u=e^{-\sqrt{2a}y}\left(c_1 \cdot H_a(y)+c_2 \cdot {}_1F_1(-\frac{a}{2};\frac{1}{2};y^2)\right)$.

Here $c_{1,2}$ are arbitrary constants, $H_a$ is the Hermite function (for a non-negative integer $a$ it becomes the Hermite polynomial; however, in general $a$ is arbitrary). ${}_1F_1$ is the Kummer confluent hypergeometric function, the auxiliary dimensionless argument $y$ defines a scaling of the coordinate followed by deformation and shift:

$y=\sgn(V_0)\sqrt{\delta x}+\sqrt{2a}$,

and the involved parameters $\delta$ and $a$ are given as

$\delta=\sqrt{-8mE/\hbar^2}$,

$a=\frac{m^2 V_0^2}{\hbar (-2 m E)^{3/2}}$.

Bound states and Energy spectrum

A set of bounded quasi-polynomial solutions for an attractive potential with $V_0<0$ is achieved by putting $a=n , n\in N$. Then, the Hermite function in the solution becomes the Hermite polynomial and one should put $c_2=0$ to ensure vanishing of the solution at infinity. The energy eigenvalues for these polynomial solutions are

$E_n=\frac{V_0}{2}\left(\frac{-m V_0}{\hbar}\right)^{1/3}n^{-2/3}, n=1,2,3... ,$

and the corresponding solutions are written as

$\psi_n=e^{-\sqrt{2 n}y-\delta x/2}(H_n (y) - \sqrt{2 n} H_{n-1}(y)), y=\sqrt{2 n}-\sqrt{ \delta x}.$

A peculiarity of this set of quasi-polynomial functions is that the solutions do not vanish at the origin. Depending on the particular problem (for instance, if one considers the one-dimensional Schrödinger equation as the s-wave radial equation for the three-dimensional Schrödinger equation with the potential $V=V_0/\sqrt{r}$ ), it is useful to have a set of bounded wave functions that vanish at the origin ( $\psi(0)=0$ ). The exact spectrum in this case is determined through the roots of the transcendental equation

$\sqrt{2 a} H_{a-1}(-\sqrt{2 a})+H_a (-\sqrt{2 a})=0.$

A highly accurate approximation for the resultant energy spectrum is given as

$E_n=\frac{V_0}{2}\left(\frac{-m V_0}{\hbar^2}\right)^{1/3} \left(n-\frac{1}{2 \pi}\right)^{-2/3}, n=1,2,3,... .$

Since the roots $a_n$ of the spectrum equation are not integers the wave functions of the bound states for this spectrum are not quasi-polynomials in contrast to the spectrum provided by above polynomial reductions.

See also

a/ Confluent hypergeometric potentials

• Quantum harmonic oscillator
• Hydrogen atom
• Morse potential
• Kratzer potential
• Lambert-W step-potential

b/ Hypergeometric potentials

• Pöschl–Teller potential
• Eckart potential
• woods-Saxon potential

c/ Other potentials

• Rectangular potential barrier
• Finite potential well
• Infinite potential well
• Delta potential barrier (QM)
• Finite potential barrier (QM)

References

1. "Exact solution of the Schrödinger equation for the potential V0/√x"
2. "Discretization of Natanzon potentials"