Show that when \(\Psi_{1}(x, t)\) and \(\Psi_{2}(x, t)\) are solutions to the time-dependent Schrödinger equation and \(A, B\) are numbers, then a function \(\Psi(x, t)\) that is a superposition of these functions is also a solution: \(\Psi(x, t)=A \Psi_{1}(x, t)+B \Psi_{1}(x, t)\)

The time-dependent Schrödinger equation (TDSE) governs the behavior of quantum mechanical wave functions and is given by: \[i\hbar\frac{\partial\Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t)\] Here, \(i\) is the imaginary unit, \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, \(V(x)\) is the potential energy function, and \(\Psi(x, t)\) is the wave function that we want to find.

We are given the following functions as solutions to the TDSE: \[\Psi_1(x, t)\] \[\Psi_2(x, t)\] We know that if these functions satisfy the TDSE when substituting them for \(\Psi(x,t)\), then they are valid solutions.

The given superposition function is: \[\Psi(x, t) = A \Psi_1(x, t) + B \Psi_2(x, t)\] We will now substitute this function into the TDSE and see if it is a solution. First, let's find the first and second derivatives with respect to \(x\) and the first derivative with respect to \(t\): \[\frac{\partial \Psi(x,t)}{\partial t} = A \frac{\partial \Psi_1(x,t)}{\partial t} + B \frac{\partial \Psi_2(x,t)}{\partial t}\] \[\frac{\partial^2 \Psi(x,t)}{\partial x^2} = A \frac{\partial^2 \Psi_1(x,t)}{\partial x^2} + B \frac{\partial^2 \Psi_2(x,t)}{\partial x^2}\] Now, we substitute these derivatives into the TDSE: \[i\hbar\left(A \frac{\partial \Psi_1(x,t)}{\partial t} + B \frac{\partial \Psi_2(x,t)}{\partial t}\right) = -\frac{\hbar^2}{2m}\left(A \frac{\partial^2 \Psi_1(x,t)}{\partial x^2} + B \frac{\partial^2 \Psi_2(x,t)}{\partial x^2}\right) + V(x)\left(A \Psi_1(x, t) + B \Psi_2(x, t)\right)\] Since we know that both \(\Psi_1(x, t)\) and \(\Psi_2(x, t)\) are solutions to the TDSE, we can see that the given linear superposition, \(\Psi(x, t)\), also satisfies the equation. Therefore, the function \(\Psi(x, t) = A \Psi_1(x, t) + B \Psi_2(x, t)\) is also a valid solution to the time-dependent Schrödinger equation.