As a 1-dimensional problem, you have Schrodinger's equation, given by: a -h? a2 iħ (x, t) =- at 2m Əx? 4(x, t) + V(x) Þ(x,t) Suppose for a specific V(x) and certain boundary conditions, the function w,(x, t) is a solution to the above equation and w2 (x, t) is also a solution. Show that 4(x, t) = a,(x, t) + b2(x, t) also solves the above equation, where a, b are complex numbers.

We have a free particle in one dimension at a time t = 0, the initial wave function is (x, 0) = Ae-r|æ| where A and r are positive real constants. Calculate the expectation value (p).

Problem 3. Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation h² d²v 2m dx² EV, with boundary conditions (0) = 0 and (1) = 0. Second, the Quantum Harmonic Oscillator (QHO) V = EV h² d² 2m da² +ka²) 1 +kx² 2 (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system. (d) Explain why you can observe a particle outside of the "classically allowed region". Hint: you can use any state and compute an integral to determine a probability of a particle being in a given region.

Consider the superposition wave function (x) = c sin (πx/a) + d sin (2πx/a). a. Is (x) an acceptable wave function for the particle in the box? b. Is(x) an eigenfunction of the total energy operator Ĥ? c. Is(x) normalized?