Download Time-Dependent Perturbation Theory: Probability of Quantum Jumps - Prof. Steven Anlage and more Study notes Quantum Physics in PDF only on Docsity! 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 0H that has solutions to the time-dependent Schrödinger equation ),(),( 000 tr dt ditrH nn r h r Ψ=Ψ of the form hrr /00 0 )(),( tiEnn nertr −=Ψ φ , where is the un- perturbed eigen-energy. Suppose that this system is prepared in a particular eigenstate, say the n 0 nE th state. Next consider turning on a “small” time-dependent perturbing potential such that the new Hamiltonian is given by ),('0 trHH rλ+ , where 1<<λ and 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 ),( trrΨ and satisfies the new time-dependent Schrödinger equation: ),(),()],('[ 0 tr dt ditrtrHH rhrr Ψ=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=Ψ nnnn λλ λ yields 0000 : nn dt diH Ψ=Ψ hλ , which is the original unperturbed problem, 10101 ': nnn dt diHH 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: ),()( 01 trtanln r l l Ψ=Ψ ∑ with unknown time- dependent coefficients . Substituting this into the equation and projecting out the j )(tanl 1λ th eigenstate yields the amplitude transition rate from state ‘n’ to state ‘j’: xdxtxxeia nj tEEi nj nj 30*0/)( )(),(')( 00 rrr h & h φφ Η−= ∫− (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)(tanj 2 )(tanj . 1
