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:

${\displaystyle V(x)=V_{c}+{\frac {V}{\sqrt {x-x_{0}}}}}$.

Comments

Omitting the non-essential constants ${\displaystyle V_{c},x_{0}}$ the general solution of the Schrödinger equation

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

for the potential ${\displaystyle V(x)=V_{0}/{\sqrt {x}}}$ for arbitrary $V_0$ is written as

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

where

${\displaystyle u=e^{-{\sqrt {2a}}y}\left(c_{1}\cdot H_{a}(y)+c_{2}\cdot {}_{1}F_{1}(-{\frac {a}{2}};{\frac {1}{2}};y^{2})\right)}$.

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

${\displaystyle y=\operatorname {sgn}(V_{0}){\sqrt {\delta x}}+{\sqrt {2a}}}$,

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

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

${\displaystyle a={\frac {m^{2}V_{0}^{2}}{\hbar (-2mE)^{3/2}}}}$.

Bound states and Energy spectrum

A set of bounded quasi-polynomial solutions for an attractive potential with ${\displaystyle V_{0}<0}$ is achieved by putting ${\displaystyle a=n,n\in N}$. Then, the Hermite function in the solution becomes the Hermite polynomial and one should put ${\displaystyle c_{2}=0}$ to ensure vanishing of the solution at infinity. The energy eigenvalues for these polynomial solutions are

${\displaystyle E_{n}={\frac {V_{0}}{2}}\left({\frac {-mV_{0}}{\hbar }}\right)^{1/3}n^{-2/3},n=1,2,3...,}$

and the corresponding solutions are written as

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

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 ${\displaystyle V=V_{0}/{\sqrt {r}}}$ ), it is useful to have a set of bounded wave functions that vanish at the origin ( ${\displaystyle \psi (0)=0}$ ). The exact spectrum in this case is determined through the roots of the transcendental equation

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

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

${\displaystyle E_{n}={\frac {V_{0}}{2}}\left({\frac {-mV_{0}}{\hbar ^{2}}}\right)^{1/3}\left(n-{\frac {1}{2\pi }}\right)^{-2/3},n=1,2,3,....}$

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"