How do you find the matrix element using Dirac notation?

[physics2004]

I've been trying to solve some questions using dirac notation, and most seem to be pretty straight forward (once you set everything up) but i always seem to get stuck when i try to find the matrix element and i can't seem to find the proper way to express the eigenstates given...

so for example we got some practice problems where we have to find the matrix element <a|U(t,tb)|b>. A and B are just position eigenstates and U(t,tb) is just the time evolution operator just:exp((-i*H/h)*(t-tb))

H = Hamiltonian
h = H bar (i.e 1.05e-34)
tb= intial time
t = final timeI get that the time operator just shifts the parameter t from tb to t, but all the other ones I've done i was given all the eigenstates. For example, griffiths 3.23 where you atleast have a orthonormal basis to start with and then you just need to find the eigenvalues and vectors. But for this question we are expected to find the matrix element given abstract position eigenstates? not quite sure how to get started. Maybe iam just struggling with dirac notation in general as i just learned it not to long ago, but iam not quite sure how to approach the question.

Thanks any help would be appreciated, also there's a similar question but its with the displacement operator but i figure if i can get this one i should be able to try the other one too.

 

[fzero]

If the Hamiltonian is that of a free particle, you can compute the matrix element of U in the momentum basis and then transform to the position basis.

 

[physics2004]

The particle is confined to move in the x-direction and has a potential of zero
, V(x) = 0
heres how i did it but not quite sure if its right

just expand
e^{x}=\\sum_n x^n/n!

and use the hamiltonian operator to get the energy and use the delta function to normalize.

I think taking the matrix element of the time oeprator on a momentum basis then transforming to the position would be a little redundant and iam not quite sure how you would transform afterwards

 

[fzero]

If you found a Gaussian exponential of the distance $$x_a-x_b$$ then you probably did it correctly. If not, there's a reason why it's easier to compute the matrix element involving the Hamiltonian in the momentum basis.

 

[physics2004]

You mean from tb to t ? and how would you compute the matrix element involving the momentum basis, as the time evolution operator contains no x terms.

 

[fzero]

You mean from tb to t ? and how would you compute the matrix element involving the momentum basis, as the time evolution operator contains no x terms.

The matrix element that you want to compute has the functional form (not quite a Gaussian)

$$\langle x'' | U(t,t_0) | x''\rangle \sim A \exp\left( i \alpha \frac{(x'-x'')^2}{t-t_0}\right).$$

The position basis is the Fourier transform of the momentum basis. In the momentum basis, the matrix element is proportional to a delta function of the momenta. The Fourier transform involves two integrals, one of which is of a Gaussian function of a momentum.