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. The potential is defined as:
$$V(x) = V_c+\frac{V}{\sqrt{x-x_0}}$$.
Comments
Omitting the non-essential constants Vc, x0 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 V0 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 c1, 2 are arbitrary constants, Ha is the Hermite function (for a non-negative integer a it becomes the Hermite polynomial; however, in general a is arbitrary). 1F1 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 δ 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 V0 < 0 is achieved by putting a = n, n ∈ N. Then, the Hermite function in the solution becomes the Hermite polynomial and one should put c2 = 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 ( ψ(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 an 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)