# Quantum Mechanics/Introduction to QM

## Introduction

Before the early 1900s, scientists used Classical Mechanics, or Newtonian Physics to describe the macroscopic world that we see today. The problem was that come the twentieth century, physicists began to notice that when we got down to a smaller scale, the scale dealing with atoms and bonds, classical mechanics no longer applied and particles could no longer be described by the rules and the system that they were familiar with. As a result, there was soon a need for a new system that would be able to accurately predict the outcome of an experiment involving objects on the angstrom scale. The goal of the development of a new system was not to replace classical mechanics, but rather to complement it. In fact, the new system needed to "become" classical mechanics once expanded to a large enough scale. This new system is known as quantum mechanics and this necessity for quantum mechanics to become classical mechanics and vice versa (depending on the scale of the experiment) is formally known as the correspondence principle.

## Quantum Mechanics Model

One of the hard things to accept about quantum mechanics is that when considering a particle such as an electron, you can no longer measure something that seemed trivial before in classical mechanics: position. The position of an electron cannot be known exactly, rather you can only know the probability that an electron might be somewhere. The uncertainty that is associated with quantum mechanical measurement is illustrated in the system used to describe objects. In quantum mechanics, the state of a given system is described mathematically by a wave. Everything that you would want to extract about the particle can be extracted from the wave function. An important feature of the wave function is the superposition principle, which states that wave functions can be written as a linear combination of multiple wave functions. Conceptually, this means that the state of function is equal to the sum of the probability of all possible states. For example, in the case of the position of an electron, its position can be described by the sum of the possibility of every place it could be.

## Nature of Measuring Observables

In physics, an observable is anything that you can explicitly measure in a system, such as energy, position, momentum, velocity, etc. In classical mechanics these observables can be known simultaneously and the accuracy of the measurement is dependent on the instrument used to take the measurement. Making a measurement on a quantum mechanical system is equivalent to collapsing a superposition wave function system down to a single wave function. In the case of quantum mechanics, only certain observables can be known at a time, and there are some observables that can never be known simultaneously. For example, the position and momentum of a particle in a quantum mechanical system cannot be known at the same time. This is because if you were to measure position, for example, then measure momentum, and then measure position again, you will not get the value for position that you measured before, this is because measuring momentum perturbed the system and caused the wave function describing position to return to its superposition state.

### Experimental Observables that Contradicted Classical Mechanics

Classically, it was observed that energy depends on intensity (amplitude). This resulted in the ultraviolet catastrophe, which described a blackbody at low wavelengths, predicting that the emitted radiation would approach infinity. This was not observed; instead it was discovered experimentally that energy can only be emitted in discrete packets of energy proportional to frequency.

#### Photoelectric Effect

The photoelectric effect is the phenomenon that is observed when energy in the form of light is propelled at a sheet of metal, namely, the ejection of an electron. Once again, the classical belief that energy depends on intensity was not observed experimentally. It was predicted based on classical mechanics that the energy of the ejected electron would depend on the intensity of the photons shot at the sheet of metal. However, it was observed that the kinetic energy of ejected electron is independent of intensity, but dependent on the frequency of the light.

#### Double Slit Experiment

Classically, waves have been observed to diffract when they encounter a small opening or an obstacle. In one particular wave-related experiment a wave is allowed to propagate through space until it encounters two small slits, the result is a readily observable diffraction pattern. A double-slit diffraction pattern is experimentally observed because of the deconstructive and constructive interference of waves. If this experiment were attempted with a particle, the logical assumption would be that the particle would go through one of the slits. Interestingly enough, a double slit diffraction pattern was observed with electrons (a particle), a pattern that looked just like the one observed with waves. The result is the realization that the particle must have simultaneously gone through both slits just as a wave does. As it turns out, it is actually the components of the superposition wave function that were interacting, causing a diffraction pattern.

#### Atomic Spectra

In the case of the observed atomic spectra, it was believed classically that the electron inside an atom would continuously lose its energy, while the frequency of the light would continuously increase. This is not the case, instead, light emitted from atoms can only be emitted as discrete packets each with its own narrow frequency distribution.