## Understanding the Role of Uncertainty in Quantum Mechanics

Werner Heisenberg’s uncertainty principle means that on a microscopic scale, we can know where something is or where it is going, but not both at the same time.

### Accurate Measurements

In daily life people might measure a length with a ruler very carefully and say it is exactly 8 inches. Physicists however know that there is no such thing as an exact measurement. There will always be errors in measurements. Even the most careful measurement will have a fundamental limitation to its accuracy.

This limitation does not result from any flaw in the measurement techniques or instruments. Rather it is a fundamental limitation imposed by quantum mechanics, specifically the Heisenberg uncertainty principle. Werner Heisenberg developed the uncertainty principle in terms of momentum and position as well as of time and energy.

### Uncertainty and Action, Preliminary Concepts

Any measurement will have an uncertainty associated with the measurement. If the measurement is very accurate, the uncertainty will be small. The uncertainty of an inaccurate measurement will be large. In the example of the 8 inch length, the uncertainty might be 1/8 of an inch, the smallest division in the ruler. Someone lacking a ruler, might use his hand to estimate the length. In that case the uncertainty might be more than an inch.

Momentum is defined as mass multiplied by velocity. When a momentum is multiplied by a position the units are the same as the units of energy multiplied by time. Physicists call energy multiplied by time action. One of the fundamental constants in physics is Plank’s constant, h=6.63E-34J-s, which is the smallest possible amount of action. Plank’s constant is therefore sometimes called the quantum of action.

### Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states that when measuring the position and momentum of any object, the uncertainty in the position multiplied by the uncertainty in the momentum must be greater than Plank’s constant divided by 2 pi, or

(position uncertainty) X (momentum uncertainty) > h/2pi.

Because Plank’s constant has such a small value this minimum required uncertainty is much much smaller than the other errors associated with measurements of macroscopic objects. For microscopic objects, such as protons and electrons, however the uncertainty principle can have a real effect.

Measuring the position very accurately, will affect the momentum so that we know nothing about the momentum. Conversely, the act of measuring the momentum very accurately will affect the position so we no longer know anything about the particle’s position. The more accurately we know one of these quantities, the less accurately we know the other. We can know were it is or we can know where it is going, but not both at the same time.

The uncertainty principle also applies to energy and time. The uncertainty in the energy multiplied by the uncertainty in the time is greater than plank’s constant divided by 2 pi.

(energy uncertainty) X (time uncertainty) > h/2pi.

### Probability in Quantum Mechanics

The Heisenberg uncertainty principle is related to the idea of probability in quantum mechanics. Prior to quantum mechanics, physicists conceived of the universe as being very deterministic. Newton’s laws allowed accurate predictions. The uncertainty principle however limits the accuracy of predictions. Predictions are limited to probability predictions. For example, the electron has a 40% chance of being in a specific location.

By explaining the photoelectric effect, Einstein sowed the seeds for quantum mechanics, but he never really thought the probability aspect of quantum mechanics was correct. Einstein did not think that God would choose to play dice with the universe.

Heisenberg may have been here!

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