# Wave Probability and Randomness

Friday, February 11, 2022

## Consequences of Wave Mechanics and Uncertainty

The principle of uncertainty displays itself when conducting experiments with wave particles like electrons. Preparing an experiment with an electron in the same way multiple times, for instance, will lead to different outcomes. This violates the classical notion that systems prepared in identical ways show identical behavior.

### Probability

Instead of trying to predict specific outcomes of experiments, for instance, scientists predict the overall *distribution* of outcomes. This is more representative of a phenomenon based on probability like a coin flip. The same is true in quantum mechanics: it is not possible to predict the outcome of any one specific experiment but the distribution can be predicted with math.

## Quantum Theory

Quantum theory provides the mathematical tools to predict statistical distributions of quantum particles rather than single observations. Although it may seem like a major problem, scientists often do not observe a single atom, for example, at a time; it is actually quite helpful to see the distribution of possible results.

*actually* random?

Is quantum physics It is tempting to believe that there are simply hidden variables that exist which explain our inability to predict exact outcomes, like knowing the exact velocity and momentum of a coin to predict its outcome when flipped. However, experimental evidence suggests this is not the case; it is a fundamental aspect of nature that quantum physics is truly random.

## The Probability Amplitude

The probability of finding a particle at a particular location depends on the amplitude of its de Broglie wave at that point. Particles' wave packets, for instance, show that the particle is likely to be found where the packet is and very unlikely to be found elsewhere. In fact, the probability to observe particles is proportional to the square of the magnitude of the de Broglie wave amplitude (much like how the probability to observe a photon is proportional to the square of the magnitude of the electric field!).