Inverse square root potential

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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 {}_{1}F_{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). ${}_{1}F_{1}$ is the Kummer confluent hypergeometric function, the auxiliary dimensionless argument $y$ defines a scaling of the coordinate followed by deformation and shift:

y=\operatorname {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 (-2mE)^{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 {-mV_{0}}{\hbar }}\right)^{1/3}n^{-2/3},n=1,2,3...,

and the corresponding solutions are written as

\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 $V=V_{0}/{\sqrt {r}}$ ), 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 {2a}}H_{a-1}(-{\sqrt {2a}})+H_{a}(-{\sqrt {2a}})=0.

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

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.

References

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Inverse square root potential

Comments

Bound states and Energy spectrum

See also

References