Final answer:
The function y(x) = A cos kx is not a solution to the stationary Schrödinger equation because it does not satisfy the boundary conditions of the infinite square well, and the functions Y(x, t) = A sin (kx - wt) and Y(x, t) = A cos (kx - wt) fail to comply with the time-dependent Schrödinger equation due to missing complex exponential time-dependence. Additionally, logistic curve solutions are unrelated to quantum systems and cannot be applied to the Schrödinger equation.
Step-by-step explanation:
The function y(x) = A cos kx is not a solution to the stationary Schrödinger equation for a particle in a box which represents an infinite square well with wall boundaries at x = 0 and x = L, because it does not satisfy the boundary conditions of the well. The stationary Schrödinger equation requires that the wave function must be equal to zero at the walls of the well (x = 0 and x = L). However, the cosine function can have non-zero values at these points depending on the value of k. The correct solutions must be of the form sin(nxπ/L) which is zero at x = 0 and x = L for integer values of n, thereby satisfying the boundary conditions.
The functions Y(x, t) = A sin (kx - wt) and Y(x, t) = A cos (kx - wt) do not obey Schrödinger's time-dependent equation because they represent classical wave functions. Schrödinger's time-dependent equation involves both the spatial part and the time part of the wave function (Ψ(x, t)) being separable and the time part having a form that involves the exponential of the imaginary number 'i' times the energy eigenvalue 'E' and the time 't' divided by Planck's constant. These functions are missing the complex exponential time dependence required for quantum mechanics.
The logistic curve is a solution to a modified differential equation that is different from the linear Schrödinger equation used to describe quantum systems. It cannot be applied to solve the Schrödinger equation because their underlying mathematical structures and physical interpretations differ significantly.