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Journal of Physics Conference Series

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

A F Sugihartin, Bambang Supriadi, Subiki Subiki, V Rizqiyah, Novarika Aliya Rizky, Febrianti Utami

Published May 1, 2020• 0 citations
Publisher

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 <?CDATA ${\widehat{L}}_{x}$?> L ^ x , <?CDATA ${\widehat{L}}_{y}$?> L ^ y , <?CDATA ${\widehat{L}}_{z}$?> L ^ z , <?CDATA ${\widehat{L}}_{+}$?> L ^ + , and <?CDATA ${\widehat{L}}_{-}$?> 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 <?CDATA ${\widehat{L}}_{x}{}^{2}$?> L ^ x 2 , <?CDATA ${\widehat{L}}_{y}{}^{2}$?> L ^ y 2 , <?CDATA ${\widehat{L}}_{z}{}^{2}$?> L ^ z 2 , <?CDATA ${\widehat{L}}^{2}$?> L ^ 2 , <?CDATA ${L}_{+}{}^{2}$?> L + 2 and <?CDATA ${L}_{-}{}^{2}$?> L

DOI
10.1088/1742-6596/1538/1/012037

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Observables in Quantum mechanicsare represented by operators. Its measurements cannot be determined simultaneously and the results are only a possibility. This study uses the rules of the commutator in determining the relation between angular momentum to the position and free particle Hamiltonian. The commutator will be zero if the operator that is connected can be determined simultaneously, and it will produce value if cannot be determined simultaneously. This study focused on the commutator of the angular momentum operator on the position and Hamiltonian of free particles in Cartesian coordinates. The result of the commutator of angular momentum operator to the position was zero (commut) if there wasn't a component of the angular momentum that is equal to the position made by the commutation pair. While the results of the commutator angular momentum operator towards the free particle Hamiltonian indicated that angular momentum is the constant of motion.
When a classical particle moves in a central field the energy and the three components of angular momentum are all constants of the motion. The motion is an orbit whose orientation in space is fixed. To discuss the corresponding situation in quantum mechanics we must examine the commutation properties of the associated operators: If they all commute, then the quantities which they represent may all simultaneously be constants of the motion; if not, we must find as many mutually commuting operators as we can and obtain a maximal set of constants of the motion.
This chapter shows that the three operators are Hermitian and (in a rotationally invariant system) commute with the Hamiltonian; therefore they are observable constants of the motion. It finds out what their physical significance is. By knowing from classical mechanics, the constant of the motion arising from rotational invariance is the angular momentum. The angular momentum may split up into two parts: orbital and intrinsic. The orbital part can always be described by the dependence of a wave function, whereas the intrinsic part is by definition contained in the transformation property of the wave function. In the case of a single component (scalar), no intrinsic angular momentum exists. There are—apart from the vector addition and the scalar product—two other ways of combining two vectors: the direct sum and the direct product. Both occur in the theory of angular momentum, in fact whenever symmetry is involved.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.