For a static (time-independent) perturbation V suddenly switched on at time t=0:
show that the transition probability for a transition of the system from a state
up to first order in perturbation theory is just:
How does the transition probability change if the time of application is doubled in the limit t→0. (6+4)
To solve this problem numerically, we need to compute the transition probability using first-order time-dependent perturbation theory. Let's break it down step by step: --- ### Step 1: Expression for Transition Probability Using first-order time-dependent perturbation theory, the probability amplitude for a transition from an initial state \( |i\rangle \) to a final state \( |n\rangle \) is given by: \[ c_n(t) = -\frac{i}{\hbar} \int_0^t V_{ni} e^{i\omega_{ni}t'} dt' \] where: - \( V_{ni} = \langle n | V | i \rangle \) is the matrix element of the perturbation. - \( \omega_{ni} = \frac{E_n - E_i}{\hbar} \) is the transition frequency. Evaluating the integral: ______ ________ _________ ____ _________ ____ _________ __________.
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