Final answer:
The system involves transforming a time-domain signal into a frequency-domain representation, passing it through a frequency response, and obtaining the output. The wave functions provided do not satisfy Schrödinger's time-dependent equation, which requires complex numbers. Superpositions of solutions to Schrödinger's equation are also solutions.
Step-by-step explanation:
The system described in the question involves a sequence in which a time-domain signal, s(t), undergoes a Fourier Transform to become a frequency-domain signal, S(f). This signal is then passed through a system characterized by a frequency response, H(f), resulting in an output signal, Y(f). The mathematical expression for the output signal in the time domain, y(t), can be found using the inverse Fourier Transform of Y(f).
The given wave functions, such as Y(x, t) = A sin (kx - wt) and Y(x, t) = A cos (kx - wt), represent sinusoidal waves in space and time. However, these expressions do not satisfy Schrödinger's time-dependent equation because they do not account for the complex nature of quantum mechanical wave functions, which require the use of complex numbers due to the presence of the imaginary unit i in the equation. The correct form of a quantum mechanical wave function that satisfies Schrödinger's equation uses complex exponentials, for instance, Y(x, t) = Aei(kx-wt).
For a superposition of two functions Y₁(x, t) and Y₂(x, t) that are solutions to the time-dependent Schrödinger equation, the superposition Y(x, t) = AY₁(x, t) + BY₂(x, t) is also a solution. Additionally, a sinusoidal wave function like y(x, t) = A sin(kx = wt + p) incorporates an initial phase shift and illustrates a simple harmonic motion.