In general ##[A, B] = - [B, A]##. The proof is as easy as they come.Dennmac said:So we know [Sz, Sx] = ihbar Sy (S with hats on) so what happens if you get [Sx, Sz]? Is it the same result? Just trying to work out if I've gone wrong somewhere
Sx and Sz are components of the spin angular momentum operator for quantum particles, such as electrons. They represent the spin along the x-axis and z-axis, respectively, in a given quantum state. These operators are fundamental in quantum mechanics, particularly in describing the behavior of spin-1/2 systems.
To commute two operators means that their order of application does not affect the outcome. Mathematically, two operators A and B commute if AB = BA. For spin operators Sx and Sz, commuting means that applying Sx followed by Sz gives the same result as applying Sz followed by Sx.
No, Sx and Sz do not commute. The commutation relation for the spin operators follows the algebra of angular momentum, specifically: [Sx, Sz] = iħSy, where [A, B] = AB - BA is the commutator. This non-commutation reflects the fundamental uncertainty in measuring different components of spin simultaneously.
The non-commutation of Sx and Sz implies that it is impossible to simultaneously know the exact values of both spin components. This is a manifestation of the Heisenberg uncertainty principle, which states that certain pairs of physical properties cannot be precisely measured at the same time. Therefore, measuring the spin along one axis will disturb the measurement along the other axis.
The non-commutation affects how quantum states are represented and transformed. For example, if a system is in an eigenstate of Sz, measuring Sx will yield a probabilistic outcome, collapsing the state into a superposition of Sx eigenstates. This behavior is crucial in quantum mechanics, influencing phenomena like quantum entanglement and spin dynamics in magnetic fields.