Uncertainty Restrictions on Orbital Angular Momentum

The general uncertainty relation of quantum mechanics forbids the simultaneous and completely accurate measurements of two observables whose operators do not commute. As will be shown here, the one-dimensional orbital angular momentum operators do not mutually commute. It follows then that, for example, one cannot suggest a system in which there is non-zero orbital angular momentum in only one dimension of orientation, as the knowledge of the orbital angular momentum in said dimension would imply that it is impossible to know that there is zero orbital angular momentum in the other two dimensions. Certain restrictions can thus be derived from the general uncertainty relation for orbital angular momentum.

Zero-Point Energy

The zero-point energy of a quantum mechanical system is the energy of the system in its ground state (when the system has the lowest possible energy). Take for example, a harmonically oscillating particle in a potential well. While classically, it may be possible to suppose that the particle could have zero energy if its kinetic energy and the energy potential at its position are both zero, quantum mechanically speaking, such a system is impossible. The Heisenberg uncertainty relation forbids such a scenario where both the particle's position and momentum are known. Here, the Heisenberg uncertainty relation and the Schrodinger equation are used to derive the zero-point energy of a quantum harmonic oscillator.

Uncertainty Principle

The uncertainty principle, simply put, states that certain measurements cannot be simultaneously taken of a system such that there is absolute certainty in every measurement. Perhaps the most well known of all uncertainty relations is Heisenberg's between momentum and position measurements. To derive the relations that dictate the general uncertainty relation and the Heisenberg uncertainty relation, we'll need to do some playing around using the matrix interpretation of quantum mechanics.