Virtual terms in the Dyson series (time dependent perturbation theory) - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-06-18T13:13:01Z https://physics.stackexchange.com/feeds/question/471265 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/471265 2 Virtual terms in the Dyson series (time dependent perturbation theory) Pedro Agostini https://physics.stackexchange.com/users/227633 2019-04-08T10:27:09Z 2019-04-08T10:33:31Z <p>Let the interaction evolution operator in the interaction picture be</p> <p><span class="math-container">$$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$</span></p> <p>where <span class="math-container">$T$</span> is the time order operator and <span class="math-container">$H_I=H-H_0$</span> is the interaction hamiltonian which is supposed to be "small". If <span class="math-container">$| i\rangle$</span> is a eigenstate of <span class="math-container">$H_0$</span> at a time <span class="math-container">$t_0$</span> with eigenvalue <span class="math-container">$E_i$</span> we can write the evolution of this state at a time <span class="math-container">$t$</span> as </p> <p><span class="math-container">$$| n \rangle (t) = U_I(t,t_0) | i\rangle= \Big( 1-i \int_{t_0}^t dt_1 H_I(t_1) -\int_{t_0}^t dt_1 H_I(t_1) \int_{t_0}^{t_1} dt_2 H_I(t_2) + \cdots\Big)| i\rangle \\ = | i\rangle - \sum_j | j\rangle \frac{\langle j|H_{int}| i\rangle}{E_j-E_i} + \sum_{f,j} | j\rangle \frac{\langle j|H_{int}| f\rangle \langle f|H_{int}| i\rangle}{(E_j-E_i)(E_f-E_i)} + \cdots ,$$</span></p> <p>where <span class="math-container">$H_{int}$</span> a time independent interaction hamiltonian in the Schrodinger picture (see <a href="https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#Time-dependent_perturbation_theory" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#Time-dependent_perturbation_theory</a> for details)</p> <p>My question is: Looking at the second order expansion in the r.h.s. of last equation what happens when the initial state and the final state are the same (<span class="math-container">$j=i$</span>), that is,when we have a virtual contribution?</p> <p>I know that the anwer for this term should be a contribution</p> <p><span class="math-container">$$-\frac{1}{2} \sum_f |i \rangle \frac{\langle i|H_{int}| f\rangle \langle f|H_{int}| i\rangle}{(E_f-E_i)^2}$$</span></p> <p>which is what I get if, when doing the expansion of the evolution operator in the second equation, I suppose that <span class="math-container">$t_1=t_2$</span> and therefore the time order opertor is <span class="math-container">$T=1$</span>. So this implies (talking in terms of particle physics) that, in a virtual term in which the initial and final state are the same, we have to take the time in which the virtual particle is emitted and the time in which it is absorbed to be the same?</p>