Question

Consider a particle in one dimension moving under the influence of some time- independent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known. We now subject the particle to a traveling pulse represented by a time-dependent potential, V(t)= A8(x - ct).

Suppose that at t = -the particle is known to be in the ground state whose energy eigenfunction is (xſi) = ui(x).

Answer 1

The time evolution of the particle's state can be expressed as ψ(x, t) = ∫ui(x') * K(x, t; x') dx', where K(x, t; x') is the propagator for the system.

Step-by-step explanation:

When a particle in a known ground state (x_sub_i) = u_sub_i(x) at t = - encounters a time-dependent potential, its subsequent state evolution can be determined by integrating the initial state u_sub_i(x) with the propagator K(x, t; x'), which describes how the state transitions over time. For a particle subject to the traveling pulse represented by V(t) = A * δ(x - ct), the propagator is typically derived from the time-dependent Schrödinger equation.

This evolution is characterized by the propagator K(x, t; x'), which captures the transition amplitude between positions x and x' at time t. It involves integrating the initial state u_sub_i(x') weighted by the influence of the pulse potential over all possible initial positions x'. Mathematically, this is represented as ψ(x, t) = ∫u_sub_i(x') * K(x, t; x') dx', describing the state of the particle at any subsequent time t.

The specific calculation of the propagator involves solving the time-dependent Schrödinger equation for the given potential V(t) = A * δ(x - ct). This equation allows for determining how the initial state evolves over time under the influence of this time-dependent potential, thereby providing insight into the particle's behavior as it interacts with the traveling pulse throughout its motion in one dimension.