Another problem - time dependent perturbation and transition probabilities

[Erwin Kreyszig]

Homework Statement

(the actual question is now as an attachment)

Assuming that the perturbation V(x,t)=$$\beta$$x exp(-$$\gamma$$ t) is applied at t = 0 to a harmonic oscillator (HO) in the ground state, calculate to the first approximation the transition probabilty to any excited state n$$\geq$$1 (here beta and gamma are constants). HO eigenstates are expressed in terms of the Hermite polynomials, H$$_{n}$$(y), as $$\left\lfloor$$n$$\right\rangle$$=$$\left($$$$\alpha$$/(2$$^{2}$$ n! $$\sqrt{\pi}$$$$\right)$$$$^{1/2}$$ e$$^{-\alpha^{2} x^{2}}$$ H$$_{n}$$($$\alpha$$x) with $$\alpha$$=$$\sqrt{M\varpi / \hbar}$$ , M is the mass of the oscillator, and $$\varpi$$ is the frequency. Use the recursion relation 2yH$$_{n}$$ = H$$_{n+1}$$ (y) = 2nH$$_{n-1}$$, and orthogonality of different eigenstates.

Homework Equations

That is the question, but what i am struggling on this question is, where to start, lol. I am completely lost as to what the recursion relation is, and how to go about finding the transition probability. Please, any help would be great, maybe a little more advise on what to read or what steps i need to take to find this probability.

The Attempt at a Solution

As i said, i not had much/any luck with this. i am truly struggling. I am aware the question is part of a time dependent perturbation and i am looking to find a probability of an electron making a jump from one energy level to another, but there i find myself scratching my head.

ANy help or advise on how to approach the problem would be greatfully recieved.

Thanks for your help...again EK

 

[Avodyne]

Do you know a formula for the transition probability? That would be a good place to start.

 

[Erwin Kreyszig]

Do you know a formula for the transition probability? That would be a good place to start.

I have the equation for the probability, "Fermi goldern rule" But the matrix element through me a little. is that the expectation value for the event occurance?

Thanks for your help EK

 

[Avodyne]

This is not a "Golden Rule" problem, you need a more basic formula of time-dependent perturbation theory.

But it will involve a matrix element of V(x,t) between the initial state (the ground state) and the final state (an energy eigenstate labeled by n). Since V(x,t) is proportional to x, you need to compute $\langle n|x|0\rangle$. Any idea how to use the hints you were given to do that?

 

[Erwin Kreyszig]

This is not a "Golden Rule" problem, you need a more basic formula of time-dependent perturbation theory.

But it will involve a matrix element of V(x,t) between the initial state (the ground state) and the final state (an energy eigenstate labeled by n). Since V(x,t) is proportional to x, you need to compute $\langle n|x|0\rangle$. Any idea how to use the hints you were given to do that?

Not really, i have not encountered the recursion relation, and am fairly confused about that. Is the eqauation for the hamiltonian just the [ hamiltonian for the stationary state + a the pertibation ] ? I am truly at a loss atm. Thanks

EK

 

[Avodyne]

You may be confused by the Dirac bra-ket notation. It means
$$\langle n|x|0\rangle = \int_{-\infty}^{+\infty}dx\,\psi^*_n(x)x\psi_0(x)$$
Of course, you know that, without the factor of x in the middle, the energy eigenfunctions are orthogonal:
$$\langle n|n'\rangle = \int_{-\infty}^{+\infty}dx\,\psi^*_n(x)\psi_{n'}(x)=\delta_{nn'}$$
Perhaps now you can see how the recursion relations helps ...

 

[Erwin Kreyszig]

I understand the Kronecker and the dirac notation, but the recursion relation still has me. so far on this problem, i am thinking about the Probability being
$$\frac{\left|H''\right|^2}{2 \hbar^2} g(w)t$$
where g(w) being the density of states.
And in this case, we only deal with the ground state and the 1st state, right? As from that relation that would only give me one part of that equation. I.e. $$H_{0}$$ gives only the $$H_{n+1}$$ term?

Thanks for your continued help.

EK

 

[Dakes]

I'm doing the same problem - Erwin, are you at Loughborough?

The problem is, using the orthogonality conditions and the recursion relations...

Basically, you get an integral (I think) of this form:

INT(-inf..inf) exp(-2a^2 * x^2) Hn Ho dx

Even if this is the correct method, I don't know what to do next.

PS - Apologies for the ASCII maths, I'm re-downloading LaTeX presently.

 

[Erwin Kreyszig]

I'm doing the same problem - Erwin, are you at Loughborough?

The problem is, using the orthogonality conditions and the recursion relations...

Basically, you get an integral (I think) of this form:

INT(-inf..inf) exp(-2a^2 * x^2) Hn Ho dx

Even if this is the correct method, I don't know what to do next.

PS - Apologies for the ASCII maths, I'm re-downloading LaTeX presently.

Yeah Loughborough, i am pretty sure you take the matrix element of perturbation, at t=0, then find the constant $$C_{n}$$ with that. Then from that you can find the probability. Maybe...

That or you go straight from $$V_{nk}$$ to probability

EK

[

 

[Avodyne]

Can you use the recursion relation to express $x\psi_0(x)$ in terms of other energy eigenstates?

And $$\frac{\left|H''\right|^2}{2 \hbar^2} g(w)t$$ is not correct. This is a "golden rule" formula that does not apply to this problem.

 

[Erwin Kreyszig]

Can you use the recursion relation to express $x\psi_0(x)$ in terms of other energy eigenstates?

And $$\frac{\left|H''\right|^2}{2 \hbar^2} g(w)t$$ is not correct. This is a "golden rule" formula that does not apply to this problem.

I am lost as to how to use the recursion relation to be honest. I am thinking you use this to find the wave function of the problem. Is there any good internet sites or books in which the recursion relation and hermite polynomials are described fully? I am struggling to find out how to use them.

Thank you EK

 

[Dakes]

Using Hermite, xH1 = (1/2)H2 + nHo

But that doesn't really help.

 

[Erwin Kreyszig]

Thanks for your help Avodyne, i think i was being a bit retarded. now i know what i am looking for using the recursion relation, i should be fine. I think i got an answer for the solution, and its less that 1 so i am happy that its sensible. Thank you again.

EK

 

[Avodyne]

Dakes: it helps a lot! See my post #6.

EK: you're welcome!