In quantum mechanics, there is a fundamental limit to the precision with which certain pairs of physical properties can be simultaneously measured. This limit is known as the Heisenberg uncertainty principle, which states that there is inherent uncertainty in the measurement of certain pairs of complementary variables.
The Heisenberg uncertainty principle sets a lower bound on the product of the uncertainties in the measurements of position and momentum, as well as other pairs of conjugate variables, such as energy and time. Mathematically, the uncertainty principle is expressed as:
Δx · Δp ≥ ħ/2
where Δx represents the uncertainty in position measurement, Δp represents the uncertainty in momentum measurement, and ħ (pronounced "h-bar") is the reduced Planck's constant (approximately 1.054 × 10^-34 joule-seconds).
The Heisenberg uncertainty principle implies that there is a fundamental limit to the precision with which certain pairs of properties can be simultaneously known. In other words, it is impossible to measure both the position and momentum of a particle with arbitrary precision simultaneously.
However, it's important to note that the Heisenberg uncertainty principle does not impose a limit on the precision of individual measurements. It is about the trade-off between the uncertainties of two complementary variables when they are measured simultaneously.
The specific limit on the smallest measurable quantity in quantum mechanics depends on the particular physical property being measured and the experimental setup. For example, the precision with which the position of a particle can be measured is limited by factors such as the wavelength of the probing particle (e.g., light or electrons) and the measurement apparatus used.
Therefore, while quantum mechanics sets fundamental limits on the precision of certain measurements, the smallest measurable quantity can vary depending on the context and experimental conditions.