Solving the Schrodinger Equation for V(x)=A sech^2(αx)

[Mahasweta]

1. How can I solve the Schrödinger equation for a potential V(x)= A sech^2(αx) ? How do I come to know that whether sech(αx) is a non-node bound state of the particular or not?




2. p^2/2m + V(x) = E



3. exp(kx)[A tanh(αx) + C]

 

[Simon Bridge]

Welcome to PF;

How can I solve the Schrödinger equation for a potential V(x)= A sech^2(αx) ?

You put the potential into the Schrödinger equation with appropriate boundary conditions - just like any DE.
Note: $$\text{sech}(x)=\frac{2e^{-x}}{1+e^{-2x}}$$

How do I come to know that whether sech(αx) is a non-node bound state of the particular or not?

... "non bound state of a particular" what? That sentence is incomplete.

i.e. are you saying that you are given $\psi=\text{sech}(ax)$ and you want to know if it is the wavefuction of a bound energy eigenstate of the potential you've been given, if it is a bound state of any potential or what?

You can figure out a lot about a potential by plotting it and using your experience of solving for different wells - like what sorts of potentials have bound states etc.

 

[Mahasweta]

I meant that for a particular potential how do I come to know that among a set of wave functions for that potential which one is non-node bound state?

 

[Simon Bridge]

Well you have two conditions to be satisfied here.
1. the state is bound
2. the state has no nodes

Do you know how to test for these conditions separately?
Do you know what these conditions mean?

Perhaps this will help?
http://arxiv.org/pdf/quant-ph/0702260.pdf