Commutative Property - Non-commuting Operators in Quantum Mechanics

Non-commuting Operators in Quantum Mechanics

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and d/dx. These two operators do not commute as may be seen by considering the effect of their products x(d/dx) and (d/dx)x on a one-dimensional wave function ψ(x):

x{d\over dx}\psi = x\psi' \neq {d\over dx}x\psi = \psi + x\psi'

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum of a particle are represented respectively (in the x-direction) by the operators x and (ħ/i)d/dx (where ħ is the reduced Planck constant). This is the same example except for the constant (ħ/i), so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.

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