HW1

1. (a) Demonstrate the linearity of the Helmholtz equation on the example of the wave function of a free particle moving to the left. 4 points

(b) Demonstrate the linearity of the time dependent Schrödinger equation on the example of the wave function of a free particle moving to the right. 4 points

(c) Why is it very important for their usage in physics that these two equations are linear? 2 points

For one of these demonstrations use the feature that multiplying a wave function with a constant does not change the physics on either side of either of these two equations. For the other demonstration use the feature that the sum of two solutions to either of these two equations is also a solution to either of these two equations.

2. (a) Determine the probability of finding a particle in the middle of an infinitely deep square well in one dimension. Use the wave function for the first exited state for this determination. 4 points

(b) Determine the expectation value of x and the spread around this expectation value for the same state as in (a). 4 points

(c) Why is the expectation value of x fuzzy? 2 points