Kara386 said:How would ##[p_x, r]## be expanded? Where ##r=(x,y,z)##, the position operators. Do you do the commutators of ##p_x## with ##x, y,z## individually? So ##[p_x,x]+[p_x,y+p_x,z]## for example?
Ah, thank you. :)PeroK said:More generally, a vector operator such as ##\mathbf{\hat{r}}## represents three operators ##(\hat{x}, \hat{y}, \hat{z})##, related in the same way as the components of a vector.
In this case, essentially by definition:
##[\hat{p}_x, \mathbf{\hat{r}}] = ([\hat{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])##
The commutator of position and momentum is a mathematical operator that describes the relationship between the position and momentum of a particle in quantum mechanics. It is denoted by [x,p] and is defined as [x,p] = xp - px, where x is the position operator and p is the momentum operator.
The commutator of position and momentum tells us about the uncertainty in simultaneously measuring the position and momentum of a particle. It is related to the Heisenberg uncertainty principle, which states that the more precisely we know the position of a particle, the less precisely we know its momentum, and vice versa.
The commutator of position and momentum is related to classical mechanics through the correspondence principle, which states that in the classical limit (when h, the Planck's constant, approaches 0), the commutator of two operators is equal to the Poisson bracket of the corresponding classical quantities. In the case of position and momentum, the Poisson bracket is equal to the classical expression for the product of position and momentum.
The commutator of position and momentum has physical significance as it is related to the uncertainty principle and plays a crucial role in the quantum description of particles. It also helps us understand the wave-like nature of particles, where their position and momentum are not definite but rather described by a probability distribution. It also appears in the formulation of important equations such as the Schrödinger equation and the Heisenberg equation of motion.
The commutator of position and momentum is used in quantum mechanics to describe the relationship between these two fundamental quantities and to derive important equations and principles, such as the uncertainty principle. It is also used in the calculation of commutation relations between other operators and in the formulation of the Heisenberg uncertainty principle. It is an essential tool for understanding the behavior of particles at the quantum level.