Transformation function from ground state -> nth energy state, force applied, HO

[Imanbk]

Hello Everyone!

I have a question regarding a Quantum problem I am trying to solve in L. Brown's Quantum Field Theory book, Chapter 1, Problem 4.f.

Homework Statement

I have a question which asks me to compute [p][/n], i.e. the probability that the ground state (n=0) is brought to the n-th energy state with the effect of an applied external force f using the transformation:

[<z*, [t][/2]|0,[t][/1]>][/f] ,

Where z* is a coherent state eigenvalue. Here is the problem statement:

Use [<z*, [t][/2]|0,[t][/1]>][/f] as a generating function to compute:

[<n, [t][/2]|0,[t][/1]>][/f] , and hence the probability that the force changes the ground state into the n-th energy eigenstate,

[p][/n]= [|[<n, [t][/2]|0,[t][/1]>][/f]|][/2]

Homework Equations

The relevant equations are the equations given above, and the result I got for [<z*, [t][/2]|0,[t][/1]>][/f]. The book states that the answer for <z*, [t][/2]|0,[t][/1]> should have the form:

<z*, [t][/2]|0,[t][/1]> = <z*|Z>f(Z).

To arrive at the solution for the transformation of z* -> 0 with f=0 (why? Well, it wasn't stated otherwise so I assume f=0 in <z*, [t][/2]|0,[t][/1]>), I take out the time dependence by writing the time-evolution operator, U([t][/2], [t][/1]) = exp{-i[H][/n]([t][/2]-[t][/1])}, since there is no force the Hamiltonian is time-independent.

Than I act [H][/n] (where H is an operator) on the |0> energy state and this rids the equation of any time dependence. So I get:

<z*, [t][/2]|0,[t][/1]> = <z*|0> .

Now to get the form required, I write the |0> state as a coherent state using

exp{Za}|0>= |Z> hence |0> = exp{-Za}|Z>, than I expand the exponential, and than act [a][/+] on <z*|, and I get:


<z*, [t][/2]|0,[t][/1]> = exp{-z*Z} <z*|Z> = 1.

The Attempt at a Solution

The first thing I do to solve the current problem is use the resolution of the identity for coherent states to decompose [<n, [t][/2]|0,[t][/1]>][/f] into two parts, one of which is [<z*, [t][/2]|0,[t][/1]>][/f] and the other is [<n, [t][/2]|z,[t][/2]>][/f].


We have right away that [<n, [t][/2]|z,[t][/2]>][/f] = [<n|z>][/f] since the bra and ket are evaluated at the same time (i.e. the time evolution operator is integrated from time t2 to time t2, hence equals the exponential of zero. the second term is slightly more complicated. It now involves a time-dependent hamiltonian because there is an applied force, and hence the time-evolution operator will be written as the time integral of the hamiltonian.


I write the time-evolution operator in terms of H([a][/+], a) (i.e. in terms of the annihiliation and creation operators. I use H=/omega*([a][/+]*a+1/2) - f/sqrt(2)*([a][/+]+a) as was asked in part b) of the same problem, and I act the with the annihilation and creation operators on the <z*| and |0> states. I can show the steps if requested (because my post is getting quite long), but the rest is just acting on the states, putting both transformation functions together under the dzdz*/(2*pi*i) exp{-z*z} integral, simplifying the expression, and making a change of variables to compute each integral. I get the product of two integrals at the end. One looks like a gamma function, except that the integral bounds are from -\infty to \infty instead of 0 to \infty and the integrand is not an even function. And the other integral converges but I don't recognize the form of the integral.


Can someone please let me know if I'm on the right track and/or let me know if the integrals sound like they are in the correct form? Or let me know of a reference which might help me with my problem? If something isn't clear in the question or my approach kindly let me know.


Thank you very much,

imanbk

 

[Imanbk]

Hello,

I found one way to do it which requires changing the Schrodinger representation into the Heisenberg representation and writing

p_n(t) = <n|0>^(f) *<0|n>^(f) = <0| P_n(t)|0>^(f),

where p_n(t) is the probability that the force changes the ground state into the n-th energy state.

After making a change to the Heisenberg picture I use the Hamilton equations of motion for the annihliation and creation operators resp. and solve for a(t), a^+(t). I than use normal ordering to order P_n(t) and computer p_n(t) with the time-dependent a and a^+ I computed.

Can someone please let me know why my posts never get answered? Is it because of the form I write the equations in? If this is the problem, can someone explain to me what I can use to write the equations using something the forum provides? I tried using the subscript and superscript bottons but those didn't print out the way they should.

Thank you for replying about the possible problem of why I don't get replies to my post.

 

[vela]

Thank you for replying about the possible problem of why I don't get replies to my post.

It's probably because your notation is unfamiliar and thus confusing. You might take a look at the FAQs pointed to at the top of the forum to learn how to use LaTeX and the forum typesetting features to make your post easier to understand.

 

[Imanbk]

Thank you vela for your reply! I'll try it out :)

 

[paranormal]

Hello imanbk,

It seems like you are on the right track with your approach to solving the problem. The transformation function you are using, [<z*, [t][/2]|0,[t][/1]>][/f], is a coherent state eigenvalue which represents the probability amplitude for a coherent state to evolve from time t1 to t2 with an applied force f.

In order to calculate the probability of the ground state being transformed to the nth energy state, you are decomposing the transformation function into two parts, one involving the coherent state eigenvalue [<z*, [t][/2]|0,[t][/1]>][/f] and the other involving the energy eigenstate [<n, [t][/2]|z,[t][/2]>][/f].

Your approach to solving for the coherent state eigenvalue <z*, [t][/2]|0,[t][/1]> is correct. It is important to note that f=0 in this case because the problem does not specify an applied force, so it is assumed to be 0.

For the energy eigenstate [<n, [t][/2]|z,[t][/2]>][/f], you are correct in using the time-evolution operator to integrate the Hamiltonian over the time interval t1 to t2. This will take the form of a product of two integrals, one of which will have the form of a gamma function. The other integral may not have a recognizable form, but as long as it converges and can be solved, your approach is correct.

Overall, your approach to solving this problem is correct. It may be helpful to refer to other resources or textbooks for guidance on solving similar problems. I hope this helps!