Summarizing Probability

Summary of Probability: The Heisenberg Uncertainty Principle

• Matter is found to have the same interference characteristics as any other wave.
• There is now a probability distribution for the location of a particle rather than a definite position.
• Another consequence of the wave character of all particles is the Heisenberg uncertainty principle, which limits the precision with which certain physical quantities can be known simultaneously. For position and momentum, the uncertainty principle is \(\Delta x\Delta p\ge \cfrac{h}{4\pi }\), where \(\Delta x\) is the uncertainty in position and \(\Delta p\) is the uncertainty in momentum.
• For energy and time, the uncertainty principle is \(\Delta E\Delta t\ge \cfrac{h}{4\pi }\) where \(\Delta E\) is the uncertainty in energy and \(\Delta t\) is the uncertainty in time.
• These small limits are fundamentally important on the quantum-mechanical scale.

Glossary

Heisenberg’s uncertainty principle

a fundamental limit to the precision with which pairs of quantities (momentum and position, and energy and time) can be measured

uncertainty in energy

lack of precision or lack of knowledge of precise results in measurements of energy

uncertainty in time

lack of precision or lack of knowledge of precise results in measurements of time

uncertainty in momentum

lack of precision or lack of knowledge of precise results in measurements of momentum

uncertainty in position

lack of precision or lack of knowledge of precise results in measurements of position

probability distribution

the overall spatial distribution of probabilities to find a particle at a given location