Question

Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P 2 + V (X) where X and P are the position and linear momentum operators, and they satisfy the commutation relation: [X, P] = i¯h The eigenvectors of H are denoted by |φn >; where n is a discrete index H|φn >= En|φn > (a) Show that < φn|P|φm >= α < φn|X|φm > and find α. Hint: Consider the commutator [X, H] (b) Using the result from the previous part, and with the aid of the closure relation show that: X m (En − Em) 2 | < φn|X|φm > | 2 = ¯h 2 m2 < φn|P 2 |φn > (c) For an arbitrary operator A, prove the relation: < φn|[A, H]|φn >= 0 (d) In terms of P, X and V(X), find the commutators: [H, P], [H, X], [H, XP] (e) Show that < φn|P|φn >= 0 (f) Establish a relation between the mean value of the kinetic ene

Answer 1

Given the Hamiltonian operator H, let us consider the energy eigen states are given by .
And so,
   
As Hamiltonian is Hermitian operator, so, the eigenvalues are real. And so,
   

a)
Let us now consider the commutator
  

  
  
Now we use the explicit expression of H
   
And so,
  
  
  
  
And so,
  
  
  
This implies
  
b)
From the previous relation
   
we now take the modulo square on both sides, and this implies
   
Now we sum over all |\phi_m > and this implies
   
  
   
Now using the closure relation
  
we get
  
  

c)
And so, for an arbitrary operator A, the expectation value of the commutator [ A, H ] is
   
  
  
  
   

d)
  We have already computed
    
  
  
  

And similarly
        
  
  
   
  
And similarly
  
Now we have already calculated the two commutators separately. And this implies
     
     
e)
  From the part (a), we have already computed

And now for m = n for the states, we have
  
   
f)
This part of the question is not complete.