Question

If the perturbation is a constant potential, how many terms are needed for an exact solution?

Question 2 options:

	a) Infinite
	b) One
	c) It depends on the nature of the system.
	d) Zero
	e) Two

Answer 1

If the perturbation potential is constant, then, we need only one term, i.e.,  the first order correction term in the perturbation theory for the exact solution. This can be seen quantitatively by looking at the generic correction terms in the perturbation theory:

Let the full Hamiltonian is
   
where, is the unperturbed Hamiltonian and is the constant perturbation. If we denote the orthonormal eigenstates of the unperturbed Hamiltonian as ,and the corresponding eigenvalues as ,  then, we have
  
And so,
The 1st order correction term :

The 2nd order correction term :
  

The above thing is zero because for
   

And similarly higher order terms are also zero.

There is a qualitative explanation for the above result :

For a constant potential perturbation, the Hamiltonian gets a constant shift and so, all the eigenvalues of the full Hamiltonian is shifted by the same constant.
Now, the 1st order perturbation term itself produces the constant shift. So, all the higher order perturbation terms should vanish.
So, the eigenvalues of the full Hamiltonian is
  
  
  
So, we see that only one term is needed for the exact solution. i.e., the 1st order term .
So, option (b) is correct.