Variational method in a finite square well

[jcsimon89]

I am trying to prove that there is always one bound state for a finite square well using variational method, and I am stuck. I've tried using e^(-bx^2) as my trial wave function, but I am left with E(b)=(hbar^2)b/2m - V, where V is the depth of the well. In this equation, taking the derivative in terms of b and setting it to zero doesn't lead to a critical value of b, am I doing something wrong?

Thanks,
Jacob

 

[kanato]

I think you evaluated your potential wrong. If your well goes from 0 to a, then you need to evaluate
$$-V \int_0^a \psi^*(x) \psi(x) dx$$
which should give you error functions.

 

[Entropia]

The variational method is a powerful tool in quantum mechanics for finding approximate solutions to problems that do not have exact analytical solutions. In the case of a finite square well, the variational method can be used to find an approximate solution for the bound state energy of the system.

The trial wave function you have chosen, e^(-bx^2), is a common choice for the variational method in problems involving a quadratic potential. However, it is important to note that this is just one possible trial wave function and may not always lead to a critical value of b.

It is possible that in this case, the derivative of E(b) with respect to b does not have a critical value, indicating that this trial wave function may not be the best choice for this particular problem. It is important to explore other trial wave functions and see if they lead to a critical value of b.

Additionally, it is important to consider the limitations of the variational method and the fact that it provides an approximate solution, rather than an exact one. It is possible that there may not be a unique bound state for this particular problem, and the variational method may not be able to accurately capture this. Further analysis and exploration of the problem may be needed to fully understand the behavior of the system.

In conclusion, it is important to carefully choose a trial wave function and consider the limitations of the variational method when using it to find solutions to problems in quantum mechanics. Keep exploring different trial wave functions and analyzing the behavior of the system to gain a better understanding of the problem at hand.