## Non-determinism & Superposition in Quantum Mechanics

A system is deterministic if the outcome is easily predictable. Ever since the beginning of Quantum Mechanics, precisely when Einstein had proposed the existence of photons, scientists were concerned that the quantum mechanical model of physics was not deterministic.

Einstein had proposed that light was made of quanta called photons. Previously, light had always been considered a wave (as explained by Maxwell’s equations) and now it was being considered as particles.

Einstein’s formula for the energy of a photon: E = h*ν, where ν is the frequency (ν = c/λ).

One issue arose in the case of an experiment with a polarizer. A polarizer is a system that abosrbs energy of light that is not focused in the direction of the polarizer. The question arose then, how does one determine whether a photon in a beam that forms an angle with the polarizer will go through the polarizer? In vector mechanics, only the fraction of the magnitude that is in the direction would enter. In Quantum Mechanics, presenting an idea of a fraction of a photon went against the theory. In the end, it was decided that it could not predictably be determined whether or not a particular photon would be allowed through the system and it was a system represented by probability.

The method of writing a photon moving in a particular direction is written in the following way in Quantum Mechanics:

Photon in the x direction:

- |photon; x>

Photon in the y direction:

- |photon; y>

**Superposition in Quantum Mechanics**

The nature of superposition in Quantum Mechanics is different than in classical mechanics.

The optical experiment using the Mach-Zehnder Interferometer bacome of interest to physicists dealing with Quantum Mechanics. Questions arose, such as how to understand the interference between photons in the interferometer. If two photons were able to interfere with each other in a cancelling manner, this would result in a violation of the conservation of energy. Likewise, a constructive interference would result in the creationof photons, which is also problematic. The current understanding that resulted from this debate was that photons interfered with themselves and that photons are unable to interfere with other photons. Further it was proposed that after moving through the beamsplitter, a single photon will exist in either path devised by the beamsplitter. As it comes to the second beamsplitter, the path of the photon towards either detector is a probabilistic expression. The existence of a single photon in both paths is the understanding of superposition in Quantum Mechanics.

In the superposition of states in the quantum mechanical model, the result of the two states is the outcome of either of the two added states with a probability between either condition. The addition of both states with a scaling factor will effect the probability of the outcome state, however there is no intermediate, average or ‘total’ outcome state. States in a quantum mechanical model are notated as |A> and |B> for states A and B.

There is an assumption in quantum mechanics that the superposition of a state on itself does not change the outcome.

Using the above assumption, we are able to alter a system of photon states in two directions to simplify the expression to one complex parameter.

Consider you would like to design a new quantum state. It is measured that two quatum states are for one system, a positive particle spin in the z direction and a negative (downward) spin in the z direction. A superposition of these two states together is referred to as a new quatum state. An experiment could be done many times to determine the probability of outcome. The spin of the particle will however only be either in the positive z direction or negative z direction.

Barton Zwiebach. *8.04 Quantum Physics I. *Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

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