Lecture 21 Highlights Up to this point we have only considered static solutions to the Schrödinger equation. It is now time to consider what happens to a quantum system when it is given a time-dependent perturbation. The philosophy of this calculation is as follows. Consider a quantum system governed by a time-independent ‘baseline’ or unperturbed Hamiltonian 0Hthat has solutions to the time-dependent Schrödinger equation ),(),(000trdtditrHnnrhrΨ=Ψ of the formhrr/000)(),(tiEnnnertr−=Ψφ, where is the un-perturbed eigen-energy. Suppose that this system is prepared in a particular eigenstate, say the n0nEth state. Next consider turning on a “small” time-dependent perturbing potential such that the new Hamiltonian is given by),('0trHHrλ+, where 1<<λand the perturbation is in general a function of both position and time. Let this perturbation act for some time ‘t’, and then have it stop. Now the system is governed once again by the unperturbed time-independent Hamiltonian 0H. The question is this: what is the probability that the quantum system is now in some other state “j”? This is equivalent to asking for the probability that the system has made a quantum jump from state ‘n’ to state ‘j’. To address this question we employ a time-dependent version of perturbation theory. While the perturbation is on, the wavefunction becomes ),( trrΨand satisfies the new time-dependent Schrödinger equation: ),(),()],('[0trdtditrtrHHrhrrΨ=Ψ+λ We employ the trick of expanding the new wavefunction around the unperturbed solution plus a series of ever smaller corrections, , and substitute this into the time-dependent Schrödinger equation. Collecting like-powers of ...2210+Ψ+Ψ+Ψ=Ψnnnnλλλyields 0000:nndtdiH Ψ=Ψ hλ, which is the original unperturbed problem, 10101':nnndtdiHH Ψ=Ψ+Ψ hλ. We use the completeness postulate of quantum mechanics to express the first order correction to the wavefunction as an infinite sum over all the unperturbed eigenfunctions: ),()(01trtanlnrllΨ=Ψ∑ with unknown time-dependent coefficients . Substituting this into the equation and projecting out the j)(tanl1λth eigenstate yields the amplitude transition rate from state ‘n’ to state ‘j’: xdxtxxeianjtEEinjnj30*0/)()(),(')(00rrrh&hφφΗ−=∫− (1) Hence if we know the perturbing Hamiltonian, this matrix element can be computed and the result integrated over time to find the transition amplitude from state ‘n’ to state ‘j’, . The probability of the transition is proportional to)(tanj2)(tanj. 1We then considered two-level systems, as discussed by Griffiths in the first few pages of Chapter 9. A 2-level system with states ‘a’ and ‘b’ subject to a time-dependent perturbation will have a wavefunction of the form: hh //)()()(tiEbbtiEaabaetcetct−−+=ΨψψAssuming that the system started in state “a” at time t=0, just before the time-dependent perturbation began, gives the initial conditions: , . 1)0( =ac 0)0( =bcDemanding that satisfies the time-dependent Schrödinger equation we can solve for the rate at which amplitude builds up in state ‘b’: )(tΨ ()()xdxtxHxeitcabtEEibab3*/),(')()(rrrh&hψψ∫−−= This result is a special case of Eq. (1) above. EEbEaab
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