Time_Dep_PT

Time_Dep_PT - Time-Dependent Perturbation Theory Michael...

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Time-Dependent Perturbation Theory Michael Fowler 7/6/07 Introduction: General Formalism We look at a Hamiltonian ( ) 0 HH V t =+ , with ( ) Vt some time-dependent perturbation, so now the wave function will have perturbation-induced time dependence. Our starting point is the set of eigenstates n of the unperturbed Hamiltonian 0 n Hn En = , notice we are not labeling with a zero, no 0 n E , because with a time-dependent Hamiltonian, energy will not be conserved, so it is pointless to look for energy corrections. What happens instead, provided the perturbation is not too large, is that the system makes transitions between the eigenstates n of . 0 H Of course, even for V = 0, the wave functions have the usual time dependence, ( ) / n iE t n n tc e ψ = n = with the c n ’s constant. What happens on introducing ( ) is that the c n ’s themselves acquire time dependence, ( ) ( ) / n iE t n n t e = n = and this time dependence is determined by Schrödinger’s equation with : () 0 t // 0 nn iE t iE t ic t e n H V t c t e t −− ∑∑ == = n so ( ) ( ) iE t iE t i c te n Vt n = ± = Taking the inner product with the bra / m iE t me = , and introducing mn mn E E ω = = , ( ) mn mn it mVt nce V e c ωω ± = m n n This is a matrix differential equation for the c n ’s :
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2 12 21 11 11 12 22 21 22 33 33 .. . . . . . . . . . . it cc VV e Ve V i V ω ⎛⎞ ⎜⎟ = ⎝⎠ ± ± ± = and solving this set of coupled equations will give us the c n ( t )’s, and hence the probability of finding the system in any particular state at any later time. If the system is in initial state i at t = 0, the probability amplitude for it being in state f at time t is to leading order in the perturbation () 0 . fi t ff if i i ct Vte d t δ =− = The probability that the system is in fact in state f at time t is therefore 2 2 2 0 1 . fi t i t = = Obviously, this is only going to be a good approximation if it predicts that the probability of transition is small—otherwise we need to go to higher order, using the Interaction Representation (or an exact solution like that in the next section). Example : kicking an oscillator . Suppose a simple harmonic oscillator is in its ground state 0 at t = . It is perturbed by a small time-dependent potential / . t V t eExe τ What is the probability of finding it in the first excited state 1 at t = + ? Here / 10 t fi Vt e E x e ′ =− , and ( ) /2 x ma a = = + . , from which the probability can be evaluated. It is 22 2 2 / 2 // 2 eE m e ωτ ωπτ == It’s worth thinking through the physical interpretations for very long and for very short times, and explaining the significance of the time for which the probability is a maximum.
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3 The Two-State System: an Exact Solution For the particular case of a two-state system perturbed by a periodic external field, the matrix equation above can be solved exactly. Of course, real physical systems have more than two states, but in fact for some important cases two of the states may be only weakly coupled to other degrees of freedom and the analysis then becomes relevant. A famous example, the ammonia maser, is discussed at the end of the section.
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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Time_Dep_PT - Time-Dependent Perturbation Theory Michael...

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