- #1
Erwin Kreyszig
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Homework Statement
(the actual question is now as an attachment)
Assuming that the perturbation V(x,t)=[tex]\beta[/tex]x exp(-[tex]\gamma[/tex] 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[tex]\geq[/tex]1 (here beta and gamma are constants). HO eigenstates are expressed in terms of the Hermite polynomials, H[tex]_{n}[/tex](y), as [tex]\left\lfloor[/tex]n[tex]\right\rangle[/tex]=[tex]\left([/tex][tex]\alpha[/tex]/(2[tex]^{2}[/tex] n! [tex]\sqrt{\pi}[/tex][tex]\right)[/tex][tex]^{1/2}[/tex] e[tex]^{-\alpha^{2} x^{2}}[/tex] H[tex]_{n}[/tex]([tex]\alpha[/tex]x) with [tex]\alpha[/tex]=[tex]\sqrt{M\varpi / \hbar}[/tex] , M is the mass of the oscillator, and [tex]\varpi[/tex] is the frequency. Use the recursion relation 2yH[tex]_{n}[/tex] = H[tex]_{n+1}[/tex] (y) = 2nH[tex]_{n-1}[/tex], 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
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