##### Quantum Probability

Can you determine exactly where everything is? Of course not. One cannot determine the exact position of a particle due to measurement errors. If one tried to measure the position of a pen relative to a piece of tape in centimeters, there would always be errors when doing so, and they would never be sure of the precise measurement. As an example, they could get measurements of 141 cm, 142 cm, 141.8 cm, 141.5 cm, and 142.7 cm when performing this experiment five times. Given this set of distances, we could be reasonably sure that the actual distance is between 140 and 144 centimeters, but we are still uncertain of the actual position. We can guess that the distance is most likely to be very close to the average, which is 141.8 cm, so we could approximate the distance to have the highest probability of being between 141.5 cm and 142 cm and that it is very unlikely for the distance to be less than 140 cm or greater than 144 cm.

Thus, we know that there is a probability that a particle, which the pen is acting as in this experiment, has position X for every X, since the position of that particle is obviously not fixed. What is that probability? Zero. Why? Say that we make a graph that shows where the particle is most likely to be and least likely to be, with an x-axis of position and a y-axis of probability. If the area of that graph is, say, one, then in order to find the probability that the particle is at an exact point, we need to find the area of that point and divide it by 1. This area is obviously zero since the length of a particle is zero, thus making the probability that a particle is at a certain point zero. This should intuitively make sense. In our experiment, while we are sure that the distance is very close to 141.8 cm, there is an infinitely low probability that the particle is exactly 141.8 cm due to measurement errors. It could be at 141.79 cm, 141.81 cm, 141.8000001 cm, which are all very close to 141.8 cm, but not equal to it. However, with such a graph, we can determine the probability that a particle is between two locations. For example, we could determine that the probability that the pen is between 141.5 and 142 cm is 60% from such a graph.

Where do we even get this graph? The idea of particle-wave duality suggests that every particle(not just light) also has a wave function that goes along with it. The amplitude of the wave function of a particle at a certain point is squared to determine the probability that it will be at that point. When measuring a particle, we are more sure of its location, so the wave function collapses. Or does it? The measurement problem questions how and whether a wave function collapses when we measure an object.

The measurement problem is yet to be solved, but there are several theories. One is that we live in many worlds and that the collapse of the wave function occurs in a parallel universe. The observer effect, another explanation of the measurement problem, says that systems are changed when one observes it. Yet another explanation, Schrödinger's cat, involves a hypothetical cat being alive or dead based on a random subatomic event that may or may not happen. Despite all of these possible explanations, nothing has been definitive, and the measurement problem still has not been solved, ever since it was first investigated in the 1930s by several physicists such as Niels Bohr and Werner Heisenberg, making the measurement problem yet another interesting but unsolved question of life.