The commutator of raising and lowering operators for angular momentum to the free particle’s hamiltonian

Abstract

Abstract In general some operators in quantum mechanics are not commutable, which means the measurement of two or more operators can’t be done simultaneously. Angular momentum operator and Hamiltonian are examples of operators that can only be measured using a mathematical approach in the form of a commutation relationship. Based on the fifth postulate of quantum mechanics about a constant of motion, if an operator commute with Hamiltonian operator then the operator is a constant of motion. This study was conducted to examine the commutation of the angular momentum operators to the Hamiltonian operator. The commutation of the angular momentum operators L ^ x , L ^ y , L ^ z , L ^ + , and L ^ − to the Hamiltonian operator shows that the operators are commute because the values are zero. Otherwise, the results of the component quadratic angular momentum operators commutator L ^ x 2 , L ^ y 2 , L ^ z 2 , L ^ 2 , L + 2 and L − 2 against Hamiltonian operator show that the operators aren’t commute with the Hamiltonian operator because the values are not zero. Based on the results of the commutation relation indicate that the angular momentum operators are the constant of motion otherwise the quadratic angular operator are not the constant of motion.