Abstract Instead of (or in addition to) the common vector diagram with cones, we propose to visualize the peculiarities of quantum mechanical angular momentum by a completely quantized 3D model. It spotlights the discrete eigenvalues and noncommutativity of components of angular momentum and corresponds to outcomes of measurements—real or computer-simulated. The latter can be easily realized by an interactive worksheet of a suitable program package of algebraic calculations. The proposed complementary method of visualization helps undergraduate students to better understand the counterintuitive properties of this quantum mechanical observable.
Abstract Instead of (or in addition to) the common vector diagram with cones, we propose to visualize the peculiarities of quantum mechanical angular momentum by a completely quantized 3D model. It spotlights the discrete eigenvalues and noncommutativity of components of angular momentum and corresponds to outcomes of measurements—real or computer-simulated. The latter can be easily realized by an interactive worksheet of a suitable program package of algebraic calculations. The proposed complementary method of visualization helps undergraduate students to better understand the counterintuitive properties of this quantum mechanical observable.
The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.
Angular momentum is a difficult knowledge point for undergraduates to understand in quantum mechanics. In this paper, a number of numerical experiments are designed to help students build visual images of angular momentum. The design concept is to break the boundaries between elementary and advanced quantum mechanics, basic theory and cutting-edge research, guide students to calculate and draw by themselves, discuss with each other, and build their own physical pictures. Through this efficient way of learning by doing, students can get an intuitive sense of abstract concepts from some simple calculations. The discussion starts from rotation operation, gradually increases the difficulty to squeezing operation, and introduces the basic concepts of spin squeezed state and Husimi Q representation. This design skillfully links classroom teaching with frontier research, bringing deeper thinking and inspiration to undergraduates.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
The magnitude of the angular momentum (J2) in quantum mechanics is larger than expected from a classical model. We explain this deviation in terms of quantum fluctuations. A standard quantum mechanical calculation gives the correct interpretation of the components of the angular momentum in the vector model in terms of projections and fluctuations. We show that the addition of angular momentum in quantum mechanics gives results consistent with the classical intuition in this vector model.
The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
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.