Probability of Transition in Timedependent Perturbation Theory

In time-dependent perturbation theory, the first-order transition amplitude from an initial state $|m\rangle$ to a final state $|n\rangle$ is given by

$c_n^{(1)}(t) = -\frac{i}{\hbar} \int_0^t \langle n | V(t') | m \rangle , e^{i \omega_{nm} t'} , dt',$

where $\omega_{nm} = \frac{E_n - E_m}{\hbar}$ and $V(t)$ is the time-dependent perturbation.

a) Explain the physical meaning of the transition amplitude $c_n^{(1)}(t)$.

b) Write down the expression for the transition probability from state $|m\rangle$ to $|n\rangle$, and explain how it relates to the amplitude.

In this problem, we are exploring the concept of time-dependent perturbation theory, a powerful tool in quantum mechanics used to study systems that are subject to a time-varying perturbation. The primary focus is on the probability of a quantum system transitioning from one eigenstate to another due to the influence of an external potential. The initial state of the system is an eigenstate of the Hamiltonian, characterized by a specific energy, and the perturbation begins at time zero. To predict the system's state after a given time interval, we utilize the framework of time-dependent perturbation theory, which incorporates the interaction picture of quantum mechanics, emphasizing how states evolve over time under a dynamic potential.

Key to solving such problems is the understanding of how perturbations modify system states according to first-order and sometimes second-order approximations. Typically, the probability of transition is calculated using matrix elements of the perturbing Hamiltonian in the basis of the eigenstates of the unperturbed system. The transition amplitude, which is integrated over the time interval during which the perturbation is active, plays a crucial role in these calculations. Familiarity with Dirac notation and complex exponentials can be particularly helpful in managing the mathematical aspects of the solution. Comprehending these principles enables us to understand how external influences can induce quantum jumps, thereby altering the state of a quantum system in a predictable manner.

This problem also introduces students to the concept of interaction between an external time-varying force and a quantum system, expanding upon the static conditions usually considered in time-independent scenarios. By distinguishing between the unperturbed and perturbed Hamiltonians, students gain insights into how quantum states are transformed through interaction, an essential aspect of predicting behaviors in quantum mechanics.