Schrodinger equation for a weird potential

In summary, the Schrodinger equation looks like this:-\frac{\hbar^2}{2m}\frac{d^2\Psi(x)}{dx^2} + \left(\frac{U_1}{ \left( 1+e^{x/a}\right)^2 } - \frac{U_2}{ \left( 1+e^{x/a}\right)}\right)\Psi(x)=E\Psi(x),for the bound states we have E=-|E|. Using these following notations: \begin{equation}w=1+e^{x/a},\end{equation}\
  • #1
4
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Hello everyone,

I have this weirdo potential for homework

\begin{equation}
U(x) = \frac{U_1}{ \left( 1+e^{x/a}\right)^2 } - \frac{U_2}{ \left( 1+e^{x/a}\right)}
\end{equation}

where U1,U2 and "a" are positive

and I need to find the energies for the bound states and also the wave functions for them. I got stuck on one part for like 2 or 3 hours, tried tons of ways, but still nothing. So I'll post the most relevant way by far and I'm hoping to get some help from you. By the way, If anyone knows the name of this potential or some kind of a book (I prefer English, Russian or German, but any other language would do :D ) where I can get some help, please tell me. I looked up in Landau, Flugge and Greiner but couldn't find anything that could help...

So here it goes:

The Schrodinger equation looks like this:

\begin{equation}
-\frac{\hbar^2}{2m}\frac{d^2\Psi(x)}{dx^2} + \left(\frac{U_1}{ \left( 1+e^{x/a}\right)^2 } - \frac{U_2}{ \left( 1+e^{x/a}\right)}\right)\Psi(x)=E\Psi(x),
\end{equation}

For the Bound states we have E=-|E|. Using these following notations:
\begin{equation}w=1+e^{x/a},\end{equation}
\begin{equation}\frac{2ma^2U_1}{\hbar^2} = \alpha^2, \end{equation}
\begin{equation} \frac{2ma^2U_2}{\hbar^2} = \beta^2, \end{equation}
and
\begin{equation} \frac{2ma^2|E|}{\hbar^2} = k^2\end{equation}
I get

\begin{equation}
(w-1)^2 \Psi''(w) + (w-1) \Psi'(w) -\left(\frac{\alpha^2}{w^2} - \frac{\beta^2}{w} + k^2 \right) \Psi(w)
\end{equation}

and \begin{equation} 1<w< \infty \end{equation}.

When \begin{equation} w \to \infty \end{equation}, then the equation is

\begin{equation}
(w-1)^2 \Psi''(w) + (w-1) \Psi'(w) - k^2 \Psi(w)=0
\end{equation}
This is Eulers differential equation and the solution is \begin{equation} \Psi(w) = (w-1)^{-k} \end{equation}.

So here is the problem... When
\begin{equation} w \to 1 \end{equation}
then I can clearly say that

\begin{equation}
\frac{\alpha^2}{w^2} - \frac{\beta^2}{w} = \alpha^2 - \beta^2
\end{equation},
but I have no idea what should I do about the first and the second derivative. Any ideas?








 
Last edited:
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  • #2
If w is going to 1, then you don't have to worry about the derivatives because the coefficients both go to zero.
 
  • #3
Yes that would be true if we where guaranteed that both derivatives are not infinitely large.. or are we :D I have no idea what should i do at this point
 
  • #4
I don't know how you'd get an infinitely large derivative at a point without the wavefunction being asymptotically large. Ψ*Ψ has to be a positive number less than 1. Don't know if you could normalize a wavefunction with an infinite derivative at a point.
 

What is the Schrodinger equation for a weird potential?

The Schrodinger equation for a weird potential is a mathematical equation that describes the behavior of a quantum system with a non-standard potential energy function. It is a key equation in quantum mechanics and is used to determine the wave function of a particle in a given potential.

How is the Schrodinger equation for a weird potential different from the standard equation?

The Schrodinger equation for a weird potential is different from the standard equation in that it takes into account non-standard potential energy functions that do not follow the traditional laws of physics. This can include potentials that are infinitely high, discontinuous, or have complex shapes.

What are some applications of the Schrodinger equation for a weird potential?

The Schrodinger equation for a weird potential has various applications in quantum mechanics, including in the study of quantum tunneling, quantum chaos, and the behavior of particles in non-standard environments. It is also used in the design of quantum devices and simulations of quantum systems.

How is the Schrodinger equation for a weird potential solved?

The Schrodinger equation for a weird potential is typically solved using numerical methods, such as the finite difference method or the spectral method. These methods involve discretizing the wave function and potential energy function and solving the resulting equations numerically.

What are some limitations of the Schrodinger equation for a weird potential?

One limitation of the Schrodinger equation for a weird potential is that it is a non-relativistic equation and does not take into account the effects of special relativity. It also assumes a single particle system and does not account for interactions between multiple particles. Additionally, the applicability of the equation may be limited in extreme environments, such as near black holes or at very high energies.

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