This video is actually a very good general description of how modern physics, and also differential equations works (I also really like Futurama). You see, after Einstein had to be a jerk and show how hundreds and even thousands of years worth of scientific knowledge was either completely wrong or at best incomplete, we started to look at things differently. Parameters, such as mass, that were thought to be constant with regards to other parameters such as speed, became known to change, which led to the need for increasingly complicated differential equations to describe phenomena and to accurately predict their behavior. Differential equations refers to an equation that involves one or more possibly unknown functions and it’s derivative (a derivative being the change in something with respect to something else. An example of a simple differential equation from classical mechanics is velocity. Velocity is the change in speed over the change in time, represented by the differential equation V = dx/dt. So, if the velocity of an object can be found using the eqation V = 5t + 10, the position can be found by separating the differential equation into dx = (5t + 10)dt (since V also = dx/dt) and integrating both sides to get x = (5/2)t^2 + 10t + C. C is an arbitrary constant that arises from the definition of an integral (see the Fundamental Theorem of Calculus for more info on Riemann Sums and integration). The value of this constant must be figured out through experimental observation and more advanced mathematical techniques. In modern physics, many phenomena involve so many potentially changing parameters that have to be identified that we just have standard equations, like Schrodinger’s Equation or Euler-Lagrange equations and we must figure out the different parameters and find out what is changing with respect to what parameter in order to find a solution to the equation and thus describe and predict the behavior of the phenomenon.
Since many parameters are unknown without direct analysis and observation of a phenomenon, we are unable to solve the differential equations for these phenomena until we can observe and analyze these parameters and thus input them into the appropriate equation. Thus, until the parameters of a phenomenon are observable and measurable, the phenomenon can be thought of as a superposition of all possible states until the actual parameters are known and measured.
One of the best examples of this is Schrodinger’s Cat, which is explained humorously in the video above. Basically, you have a perfectly healthy cat in a box that is completely cut off from the outside world, but enough air for the cat to breath for over an hour, with a Geiger counter and a number of radioactive elements that would put the probability of one atom decaying in one hour at 50%. The Geiger counter is also somehow attached through a lever to a vial of poisonous gas that will break if the Geiger counter detects radiation. At the end of one hour there is a 50% chance that the cat will be dead and a 50% chance that the cat will still be alive but really pissed off. Without some way of knowing what the real conditions are inside the box (i.e. looking in the box, shaking the box to see if the cat reacts, etc.) we do not know and thus have no way of knowing what parameters to input into the differential equation, and thus the cat can be thought of as a superposition of both states, alive and dead, which is a fancy way of saying that we do not know. Once we do know, we can input those parameters into the differential equation and thus find the solution. This is what is meant by the phrase “collapse the wave function.”
