Alchemy to Astrophysics
Quantum Mechanics Homework
1.Principles of Quantum Mechanics
a.Particles have multiple virtual motions and each motion is accompanied by a wave. The strength of the total particle wave at each point corresponds to the probability that the particle may be found there. Applying this principle we can explain all kinds of phenomena, from the properties of atoms and radioactivity to light reflection. 2.Electron Double Slit Experiment
a.Electrons are fired (possibly one at a time) toward a screen with two slits and each electron that passes through leaves a dot on a film plate. The dots accumulate on specific places, as shown in the diagram below, separated by b lank zones. Therefore, there is a certain probability that an electron may land on C, E, G… and zero probability that it may go to D, F…
b.The probability that an electron may land on some point on the film arises from the interference of two electron virtual waves meeting at that point, one coming from the top slit and the other from the bottom slit. The interference of the infinite electron waves fanning out of the slits produces a succession of high and low intensities of the electron wave function on the film that corresponds to highs and lows of the probability of where the electron may end up.
a.This is when a particle can pass through a barrier and come out the other side instead of bouncing back. b.Consider a particle of kinetic energy E hitting a higher energy V barrier. Most of the incoming particle wave bounces back becoming a reflected wave, but, at the same time, a small part penetrates the barrier and a transmitting wave emerges on the other side. Accordingly, the particle most probably will be reflected, but there is also a small probability that it may transmit over, as if passing through a hidden tunnel, hence the term tunneling.
a.Electron nucleus scatter is when incoming electrons hit a nucleus and then scatter out in various directions. We can measure the strength of the wave function in each direction by looking at the percentage of electrons that scatter in a given direction. If 15% of the electrons scatter in the forward direction and 8% in the up direction, then we can conclude the wave function in the forward direction is almost twice as strong as the wave function in the up direction.
b.To calculate the wave function for a given angle, we add all of the scattered waves (easily done by integration) and we obtain the strengths of the outgoing waves in the various directions to be, say, | Ψ45|2 = 0.15. This calculated result predicts that every time an electron with this energy collides with such a nucleus, there is a 15% probability that it may deflect at a 45 degree angle, etc. The theoretical predictions are in agreement with the experimental measurements.
5.Electron in a Box
a.We consider an electron inside a one-dimensional , L wide box. Since the electron is confined, its average momentum is zero having opposite virtual motions. It is easy to detect the electron momentum and find that the electron moves 50% of the time to the right with momentum +p1 and 50% of the time to the left with momentum –p1. The lowest energy wave function is the sum of two oppositely moving waves having wavelength twice the size of the box. The wavelength of the ground state electron would be twice the length of the box. The interference of these two waves produces the wave function that is strongest at the center of the box and zero at the edges, as indicated by the width of the shaded are in the bottom right diagram below:
b.The electron is bound around the nucleus somehow like being in a box and we have seen that the smaller the box the greater is the electron energy. Thus, when two Hydrogen atoms join together, the electrons spread over a...