# Solving a time independent Schrodinger equation

• collpitt
In summary, the conversation discusses a problem with a time independent Schrodinger equation involving a hyperbolic potential and asks for help in finding an analytical solution without using numerical techniques. The equation is provided and various attempts at solving it are mentioned, including trying a series solution and defining a new variable. The solution is eventually discovered to be in hypergeometric form.
collpitt

## Homework Statement

Hello!
I am currently stuck with a time independent Schrodinger equation where the potential "V(x)" is hyperbolic in nature. I was wondering if anyone could give me a hint as to how I should approach this problem in order to get an analytical solution (without using numerical techniques).

## Homework Equations

The equation is of the form,
d²/dx²(Ѱ) + 2m/ħ²{E + δp²/x(p-x)}(Ѱ) = 0
where the voltage V(x) = -{δp²/x(p-x)}

## The Attempt at a Solution

Thank you!

Stupid question: is $\delta$ just a number here?

Yes, δ, E, p are all constants.
I am trying to solve it by defining z = y(x) and changing the differential equation. But it is getting quite complicated. :(

Any help would be gladly appreciated.

Did u try series solution for diffrntial eqn

Yes. But I am unable to reduce the equation to any of the forms I know (bessel, legendre, laguerre, hermite, chebyshev).

collpitt said:
Yes. But I am unable to reduce the equation to any of the forms I know (bessel, legendre, laguerre, hermite, chebyshev).

I believe the solution is a series, I'll give you a clue; hyper

Yesss! Thank you! I believe it can be reduced to the hypergeometric form. Thank you very much [bold]genericusrnme[/bold] !
I will post the solution as soon as I finish.

## 1. What is the time independent Schrodinger equation?

The time independent Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in terms of its energy and wave function. It is used to find the allowed energy levels and corresponding wave functions of a quantum system.

## 2. How do you solve the time independent Schrodinger equation?

The time independent Schrodinger equation can be solved by using mathematical techniques such as separation of variables, perturbation theory, or numerical methods. The exact method used depends on the specific form of the equation and the properties of the system being studied.

## 3. What are the applications of solving the time independent Schrodinger equation?

The time independent Schrodinger equation has a wide range of applications in quantum mechanics, including calculating the energy levels and wave functions of atoms, molecules, and other quantum systems. It is also used in the study of quantum mechanical phenomena such as tunneling and superposition.

## 4. What are some challenges in solving the time independent Schrodinger equation?

Solving the time independent Schrodinger equation can be challenging due to the complex mathematical techniques and calculations involved. It also requires a good understanding of quantum mechanics and the properties of the system being studied. Additionally, for certain systems, the equation may not have an exact solution and approximations must be made.

## 5. How does the time independent Schrodinger equation relate to the time dependent Schrodinger equation?

The time independent Schrodinger equation is a special case of the more general time dependent Schrodinger equation, where the system's properties do not vary with time. The time independent equation can be derived from the time dependent equation by assuming that the wave function is a product of a time-dependent and a time-independent component.

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