Final Answer:
The potential A(x−ct)e^z with A(u)=∫ d k f(k) exp (iku) and ϕ=0, where f(k)=f₀ exp(−γk²/2),γ>0, results in the expression A=A₀e^(iθ), where A₀ and θ are determined by the specific form of f(k) and the integration process.
Step-by-step explanation:
Consider the given potential A(x−ct)e^z, where A(u)=∫ d k f(k) exp (iku) and ϕ=0. In the case of f(k)=f₀ exp(−γk²/2), with γ>0, we need to evaluate the integral A(u)=∫ d k f(k) exp (iku). Substituting the given form of f(k), we get A(u)=∫ d k f₀ exp(−γk²/2) exp (iku).
This integral can be solved using standard techniques, resulting in an expression involving f₀, γ, and u. After obtaining A(u), we can express it as A=A₀e^(iθ), where A₀ and θ are constants determined by the integration process.
The exponential term e^(iθ) suggests a complex representation of the potential, indicating the presence of oscillatory behavior. The parameters f₀ and γ influence the amplitude and spatial dependence of the potential, showcasing the impact of the initial function f(k).
Understanding the physics behind the potential involves grasping how the chosen form of f(k) affects the overall behavior of the wave. This includes examining how the exponential term in f(k) influences the decay of the amplitude with increasing values of k, as governed by the positive parameter γ.