Senior Physics Challenge Day 3

I have just finished my third day here at Cambridge, and my second day of actual work. And it is actual work, and I'll be reasonably relieved to have the pace drop a little at the end of the week to be honest - but I'm also really glad to be working with all this stuff and learning things properly rather than just looking up stuff and making sure that it makes sense.

Today we had a pretty similar time-table to yesterday, classical mechanics was up first and involved some potential energy problems. Of note is the following question, which you can try:

A large circle is constructed and a ramp built so that it is a chord of the circle that touches the lowest point. Assuming the ramp to be frictionless, how long will it take for a mass of m to fall down the slope? Solve first using forces and accelerations and secondly using potentials.

Quantum mechanics continued from yesterday, and we solved the infinite square well potential by plugging in the boundary conditions that w(x) = 0 at the endpoints. We did this for wells starting at the origin (i.e. from 0 to a) which gives sines and wells centred on the origin (-a/2 to +a/2) which give cosine solutions. We then had a quick break from square wells by introducing the de Broglie relation and calculating the wavelength of an electron and a tennis ball (which is a standard As Physics problem) and then used de Broglie to find the energies of the different states for the square well system.

The way you do that is as follows: start with the de Broglie relation lp=h. This can be rewritten as p=hbar * k , which I'm going to do just because lambda's are difficult to format. (k here is 2pi/l , and hbar = h / 2pi). Square both sides and use the fact that E=p^2/2m to solve for E, giving you E=k^2 hbar^2/(2m). Substituting in the value of k for the square well (2npi/a) gives you a dependence of E on n^2, where n is the number of the solution e.g. n=5 for the fifth largest wavelength solution.

The next question was normalising our solutions, which required you to evaluate the integral Int[w^2(x)]dx and divide by the answer. In order that the probabilities add up to 1, the sines needed to be multiplied by a factor of sqrt(2/a). Some thought tells you that the same constant is needed for the cosine solutions (or you can just do the integral again for practice!) The quantum mechanics we're doing has started to involve more and longer integrals, which is a pain to be honest, but that just means I need to concentrate a bit more and put some work in tonight.

Labs were centred around single slit diffraction which should be covered on this site already - I'll double check the detail after this is over. Our experimental setup was a laser pointed at a mirror which reflected the light onto a screen to increase the path length. A single slit was placed in front of the laser and the mirror adjusted until it reflected as many maxima onto the screen as possible for that particular slit. We then turned all the lights off, and sketched over the diffraction pattern on a piece of paper, turned the lights back on and recorded the distances between pairs of minima going out from the centre. We plotted a graph of distance from centre against order, and measured the optical path length using a measuring tape. We estimated errors in all of our measurements, combined them and calculated the wavelength of the laser - me and my lab partner measured 609nm compared to the true value of 633nm which isn't terrible, but our estimated error was 17nm compared to our actual error of 24nm, which suggests we underestimated the errors in our measurement or had a systematic errors causing our measurement to be low. Personally, I think the path length was very difficult to measure since you couldn't hold the tape to the exact location of the slit, but we tried to minimise this as much as possible so I'm not 100% sure what caused our result.

I have two more days of lectures, but only one more day of labs tomorrow. Tomorrow's lab is focussed around filament lamps and an exploration of the Stefan-Boltzmann law, and I expect quantum mechanics to focus on confinement energy and we might get up to the ionisation energy of Hydrogen and the intrusion of relativistic effects into quantum theory. I don't know what we're doing in classical mech but it would be a fair guess to expect more potentials and some orbits.

See you tomorrow!