Physics
Quantum Angular Momentum
Quantum angular momentum refers to the intrinsic angular momentum of a particle in quantum mechanics. It is a fundamental property of particles, and its quantization leads to the concept of spin. In quantum physics, angular momentum is quantized in discrete units, and it plays a crucial role in describing the behavior of particles at the atomic and subatomic levels.
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11 Key excerpts on "Quantum Angular Momentum"
eBook - PDF- A. R. Edmonds(Author)
- 2016(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R 2 The Quantization of Angular Momentum 2.1. Definition of Angular Momentum in Quantum Mechanics A n g u l a r M o m e n t u m i n C l a ss ic a l M e c h a n ic s . In the classical theory the angular momentum of a system of n massive particles is defined as a vector, given by n L = Z) X P. *-1 where r,, pt are the position vector and linear momentum respectively of the ith particle. We may write down a similar integral expression for a continuous distribution of matter. Provided that there are no external torques operating on the system, all three components of L are constants of the motion, and may take any finite values whatever. T h e I n t r o d u c t io n o f Q u a n t iz a t io n . The historic paper of Bohr (1913) on the spectrum of the hydrogen atom introduced for the first time the postulate that the angular momentum of a system was quantized, i.e. that it could only take values which were integer multiples of the quantum of action h times l / 2w. Sommerfeld (1916) suggested that the direction as well as the magnitude of the angular momentum of an electron in a closed orbit was quantized; that is, that only certain directions of orientation of the angular momentum vector with respect to a fixed axis were possible. From that time onwards spectroscopists studying the structure of atoms made use of empirical rules for dealing with the coupling of the angular momenta involved (cf. Lande (1923)). Difficulties in inter- pretation of these rules continued until the discovery of wave and matrix mechanics, and the establishment of a definite procedure for making the step from the classical to the quantum theory. D e r iv a t io n o f t h e C o m m u t a t io n R u l e s . In classical mechanics the angular momentum of a particle about a point 0 is defined as (2 . 1 . 1 ) L = r X p where r is the position vector of the particle with respect to 0 and p is its linear momentum.
eBook - PDF- Henrik Smith(Author)
- 1991(Publication Date)
- WSPC(Publisher)
152 Introduction to Quantum Mechanics 7 ANGULAR MOMENTUM According to classical mechanics the angular momentum of a particle with respect to a given point in space is conserved, when the potential energy of the particle only depends on the distance from the particle to the given point. Such a potential is called a central field. Since the force on the particle moving in a central field is directed towards or away from the given point, it follows that the moment of the force with respect to this point must be zero, and the angular momentum with respect to the same point is therefore independent of time. The conservation of angular momentum is a consequence of the symmetry of the system, the potential energy being invariant under rotations about the given point. As we shall see, the existence of this symmetry means that the operators for each component of the angular momentum commute with the Hamiltonian. However, the individual components of the angular momentum operator do not commute with each other. This may be seen from the classical expression for the angular momentum L, L = r x p , (7.1) where r and p in quantum mechanics are represented by operators which do not commute. The commutation relations for the components of the angular momentum will be derived in Section 7.1.2 below, starting from the classical expression (7.1). These commutation relations imply that the three components cannot take on definite values at the same time. It is however possible to find states which are simultaneous eigenstates for one of the components of the angular momentum and the square of its length. The eigenvalues for one of the com-ponents of the angular momentum turn out to be an integer or a half-integer times the Planck constant ft. The half-integer values cannot be derived from the orbital angular momentum alone. They are a consequence of the Lorentz invariance that must apply to the wave equation describing the motion of the particle.
eBook - PDF- Ajit Kumar(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Chapter 8 Quantum Mechanical Theory of the Spin Angular Momentum 8.1 Spin Spin angular momentum or simply spin is a fundamental property of all particles, irrespective of whether they are elementary or composite. It belongs to an internal degree of freedom (completely independent of the spatial degrees of freedom) and manifests itself as some intrinsic angular momentum of the particle. It was introduced in quantum mechanics as an attempt to explain the experimentally observed fine structures of the spectral lines in the emission spectra of alkali metals and the peculiarities involved in the anomalous (complex) Zeeman effect that showed the unusual splitting pattern of atomic energy levels in the presence of a weak external magnetic field. Note that all efforts, prior to the conjecture about spin, to explain the aforementioned experimental results on the basis of the Schr¨ odinger equation without spin had miserably failed. An atom of any of the alkali metals has an almost inert core, consisting of the nucleus and (Z - 1) inner electrons, together with a single outer electron. The transitions of the outer electron between energy levels are responsible for the aforementioned spectral lines. Therefore, any additional property required to be postulated for the explanation of the fine structures of the spectral lines or anomalous Zeeman effect, had to be attributed to the valence electron. It is because of this reason that Uhlenbeck and Goudsmit put forward their conjecture about electron’s spin. They assumed that, similar to Earth’s spinning motion about its axis, an electron, in addition to its orbital motion about the nucleus, also possessed a spinning motion about its axis of symmetry. The angular momentum related to this spinning motion was given the name ‘spin’.
- David A. B. Miller(Author)
- 2008(Publication Date)
- Cambridge University Press(Publisher)
Chapter 9 Angular momentum Prerequisites: Chapters 2–5. Thus far, we have dealt primarily with energy, position, and linear momentum and have proposed operators for each of these. One other quantity that is important in classical mechanics, angular momentum , is particularly important also in quantum mechanics. Here, we introduce angular momentum, its operators, eigenvalues, and eigenfunctions. If this discussion seems somewhat abstract, the reader can be assured that the concepts of angular momentum will become very concrete in the discussion of the hydrogen atom. One aspect of angular momentum that is different from the quantities and operators discussed previously is that its operators always have discrete eigenvalues. Whereas linear momentum is associated with eigenfunctions that are functions of position along a specific spatial direction, angular momentum is associated with eigenfunctions that are functions of angle or angles about a specific axis. The fact that the eigenvalues are discrete is associated with the fact that for a single-valued spatial function, once we have gone an angle 2 π about a particular axis, we are back to where we started. The wavefunction is presumably continuous and single-valued 1 and, hence, we must therefore have integral numbers of periods of oscillation with angle within this angular range; this requirement of integer numbers of periods leads to the discrete quantization of angular momentum. Another surprising aspect of angular momentum operators is that the operators corresponding to angular momentum about different orthogonal axes (e.g., ˆ x L , ˆ y L , and ˆ z L ) do not commute with one another (in contrast, e.g., to the linear momentum operators for the different orthogonal coordinate directions). We do, however, find that there is another useful angular momentum operator, 2 ˆ L , which does commute with each of ˆ x L , ˆ y L , and ˆ z L separately.
eBook - PDFFrom Atoms to Galaxies
A Conceptual Physics Approach to Scientific Awareness
- Sadri Hassani(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Therefore, the quantization of projections applies to any direction. We summarize our findings in the following Box. However, as we shall see shortly, our conclusions apply only to orbital angular momentum, i.e., the angular momentum resulting from the orbital motion of a particle around a center. Box 23.3.2. ( Quantization of orbital angular momentum ) In quantum theory the (magnitude of the) orbital angular momentum can take on only values given by | J | = p j ( j + 1) ~ , where j is a nonnegative integer . The projection of the orbital angular momentum along any line can take on values between -j ~ and j ~ incremented by ~ from the lowest to highest values. The number of these projections is 2 j + 1 . Sometimes the quantization of the projection of angular momentum is interpreted as “quan-tization of direction.” This is not entirely correct, because there is no restriction in quantum Quantization of direction? theory on our choice of direction for the axes of our coordinate system. We can choose any direction we want along which to project the angular momentum vector. However, once this (arbitrary) projection axis is chosen, the angular momentum vector can have “quantized” direction (angle) with respect to it. What do you know? 23.8. What is | J | for j = 3? What are the values for its projection along an arbitrary line? How many directions does angular momentum have? Section 23.3 Angular Momentum and Spin 337 23.3.2 Spin The pace at which the quantum weirdness popped at the early twentieth century physi-cists was phenomenal. No sooner had one weirdness been “resolved” than another jumped at them. People were just beginning to get used to the weird idea of quantized angular momentum when they were hit with the weirder notion of spin. Angular momentum is closely related to a quantity called magnetic moment .
eBook - PDF- James D Louck(Author)
- 2008(Publication Date)
- World Scientific(Publisher)
There are, fortunately, general invariance properties of the Schr¨ odinger equation and its solution state functions for a complex composite phys-ical systems that can be used to classify the quantum states of physical systems into substates available to the system. Our focus here is on the properties of the total angular momentum of a physical system, which is a quantity L that has a vector expression L = L 1 e 1 + L 2 e 2 + L 3 e 3 in the right-handed frame ( e 1 , e 2 , e 3 ) and the expression L = L 1 e 1 + L 2 e 2 + L 3 e 3 in a second rotated right-handed frame ( e 1 , e 2 , e 3 ) . At a given instant of time, necessarily L = L , since these quantities are just redescriptions of the total angular momentum of the system at a given time. The total angular momentum is a conserved quantity; that is, d L /dt = 0 , for all time t, and this property makes the total angular momentum an important quantity for the study of the behavior of complex physical systems. For a system of n point particles, the total angular momentum relative to the origin of the reference frame ( e 1 , e 2 , e 3 ) is obtained by vector addition of that of the individual parti-cles by L = ∑ n i =1 L i , where L i is expressed by the vector cross product L i = x i × p i in terms of the vector position x i = x 1 i e 1 + x 2 i e 2 + x 3 i e 3 and the vector linear momentum p i = p 1 i e 1 + p 2 i e 2 + p 3 i e 3 of the particle labeled i. While angular momentum can be exchanged between interact-ing particles, the total angular momentum remains constant in time for an isolated physical system of n particles. The quantum-mechanical op-erator interpretation of such classical physical quantities is obtained by Schr¨ odinger’s rule p i → − i ∇ i , = h/ 2 π, where h is Planck’s constant. The reference frame vectors ( e 1 , e 2 , e 3 ) remain intact. 1.1. BACKGROUND AND VIEWPOINT 13 The viewpoints of Newtonian physics and quantum physics may be contrasted in many ways.
eBook - PDFAtomic Structure and Lifetimes
A Conceptual Approach
- Lorenzo J. Curtis(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
For example, the language by which atomic levels are described is based on hydrogenic quantum numbers, and the characterization of spin and space properties is made as if they were the independent quantities that are obtained for single-electron systems of low nuclear charge. These concepts and notational assumptions must be continually re-examined in complex atomic systems. 4.2 The intrinsic angular momentum and magnetic moment of the electron The ratio of the magnetic moment to the angular momentum of an electron is approximately twice the so-called “classical” value e / 2 m that occurs when mass and charge are assumed 74 4 The vector model of angular momentum to have identical distributions. There is really nothing “nonclassical” about such a value, since the same gyromagnetic ratio of two occurs for any uniform solid cylinder of mass that spins about its axis, and has a uniform charge confined to its outer cylindrical surface. However, attempts to apply this macroscopic model to the electron invariably lead to self-contradictory results. The minimum radius for a mechanically spinning electron model that will yield a value for the electromagnetic inertia that does not exceed its observed mass (the so-called “classical electron radius”) leads to a tangential velocity much greater than the speed of light. Thus, any attempt to gain conceptual insights by considering the electron as anything other than a point particle are ill-conceived and counter-pedagogic. However, a very attractive model does exist that provides a clear conceptual picture of a mechanism by which a point particle can possess both an angular momentum and a magnetic moment. This lies in the formulation of the Foldy–Wouthuysen transformation. In 1950, L. L. Foldy and S. A. Wouthuysen reported [98] on a phenomenon similar to that encountered in Schr¨ odinger’s first and second equations, in that it involved differing choices in evaluating a nonrelativistic limit.
- Michael A. Parker(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
However, the classical angular momentum is not well de fi ned for point particles such as electrons. Further, measuring the spin of the electron along a given direction produces only one of two possible values — the spin is quantized. As a result, the quantum mechanical model for electron spin uses a 2-D Hilbert space with a complex number fi eld. The complete set of operators to describe the fundamental spin states consists of the magnitude and z -component of the spin. We cannot simultaneous specify the spin angular momentum along more than one of the spatial axes. 5.6.1 B ASIC I DEA OF S PIN Classically, we picture the electron as a small particle spinning at fi xed speed about an axis. The rotating mass has angular momentum ~ S with a direction given by the right-hand rule. Moving charge produces a magnetic fi eld. Because the electron has negative charge, the magnetic fi eld (of the electron) ~ B e and hence its dipole moment ~ m must point in a direction opposite to the angular momentum at the position of the electron. Figure 5.28 illustrates the relationship. Keep in mind that r ~ B ¼ 0 holds so that the magnetic fi eld forms continuous loops and therefore does not every-where point along ~ B e shown. Any change in angular momentum must be related to an applied torque ~ t ¼ ~ m ~ B . The magnitude of the electron spin cannot change, but its direction can change. The magnetic moment ~ m of the electron comes from the spinning electron and the references relate it to the spin angular momentum ~ m ¼ 2 m B h ~ S where m B ¼ j e j h 2 m represents the Bohr magneton. As usual, we are most interested in the dynamics of a spin system. We therefore need the Hamiltonian that describes the interaction of the spin with an applied fi eld. The word ‘‘ interaction ’’ S B e FIGURE 5.28 Classical picture of the rotating electron. Quantum Mechanics 309
- Robert Kolenkow(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
One of his most important works was calculation of the anomalous magnetic moment of the electron (1948). g S 2 1 C ˛ 2 : : : The first correction term is engraved on his tombstone in Mt. Auburn cemetery in Cambridge, Massachusetts. He taught primarily at Harvard University and UCLA. Disclosure: As a graduate student at Harvard, the author took a course on quantum electrodynamics with Prof. Schwinger. 222 8 Quantum Angular Momentum Summary of Chapter 8 Chapter 8 discusses representations of the rotation group and angular momentum operators, central to applications in quantum mechanics. a) The Stern–Gerlach experiment showed that the magnetic moment of atoms, hence the angular momentum, is spatially quantized – directed only in discrete directions. b) The Stern–Gerlach experiment stimulated the development of quantum num- bers: principal quantum number n, orbital angular momentum quantum number L and its projection m L , spin quantum number S and its projection m S . Quantum numbers are the labels for wave functions. c) Angular momentum in quantum mechanics is represented by operators analo- gous to classical angular momentum but with de Broglie momentum p D i „r. A finite angle rotation can be generated by repeated application of an angular momentum operator. The rotation is represented by a complex exponential with angular momentum operator in the argument. d) Angular momentum operators obey nonzero commutation relations, showing that different components of angular momentum do not commute, hence are not all simultaneously measurable. Matrices for the angular momentum components can be derived from first-order rotation matrices, from which the commutation relations follow. e) Rotations about a fixed axis form an Abelian group, which has only 1- dimensional representations e ˙im where m is an integer. f) Ladder operators are complex linear combinations of angular momentum com- ponents I ˙ D I x ˙iI y .
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
The orbital angular momentum is a function of only angular variables, and the commuting operators hatwide L z and hatwide L 2 take the form hatwide L z = – i planckover2pi1 ∂ ∂ϕ , hatwide L 2 = – planckover2pi1 2 hatwide Λ , where hatwide Λ = 1 sin θ ∂ ∂θ parenleftBig sin θ ∂ ∂θ parenrightBig + 1 sin 2 θ ∂ 2 ∂ϕ 2 . (P6. 2 . 4 . 3 ) In what follows, it is important that the operator hatwide Λ coincides with the angular part of the Laplace operator written in spherical coordinates: Δ = 1 r 2 ∂ ∂r parenleftBig r 2 ∂ ∂r parenrightBig + 1 r 2 hatwide Λ . (P6. 2 . 4 . 4 ) The eigenfunctions can be represented as products of functions of different angular variables: ψ l , m = Θ l , m ( θ ) Φ m ( ϕ ). The equation for Φ – i planckover2pi1 d Φ m dϕ = m planckover2pi1 Φ m 582 Q UANTUM M ECHANICS . A TOMIC P HYSICS has the solution Φ m = A exp( imϕ ), which is a unique function of the angle ϕ only for integer values of m . Hence, in the case of orbital angular momentum, only integer values of the numbers l and m are realized. The number l is called the orbital quantum number and the number m is called the magnetic quantum number (denoted by m l ). Half-integer values are realized in the case of intrinsic angular momentum — spin ; the corresponding spin quantum numbers are denoted by s and m s . The equation for the eigenfunctions of the operator hatwide Λ has solution in the form of spherical functions Y lm ( θ , ϕ ) = Φ m ( ϕ ) Θ ml ( θ ) with eigenvalues – l ( l + 1 ), which are well known in mathematical physics. (The functions Θ ( θ ) are expressed in terms of associated Legendre functions ; see Subsection M13.10.1). For example, Φ m ( φ ) = 1 √ 2 π e imφ , Θ 1 , 0 ( θ ) = radicalbigg 3 2 cos θ , Θ 1 , ±1 = ± radicalbigg 3 4 sin θ , Θ 2 , 0 = radicalbigg 5 8 ( 3 cos 2 θ – 1 ). Each of the functions is normalized as follows: integraltext 2 π 0 Φ 2 dϕ = 1 and integraltext π 0 Θ 2 sin θdθ = 1 .
eBook - PDF- J. J. Sakurai, Jim Napolitano(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
3.8.1 Simple Examples of Angular-Momentum Addition Before studying a formal theory of angular-momentum addition, it is worth looking at two simple examples with which the reader may be familiar: (1) how to add orbital angular momentum and spin-angular momentum and (2) how to add the spin-angular momenta of two spin 1 2 particles. 206 Theory of Angular Momentum Previously we studied both spin 1 2 systems with all quantum-mechanical degrees of freedom other than spin, such as position and momentum, ignored and quantum- mechanical particles with the space degrees of freedom (such as position or momentum) taken into account but the internal degrees of freedom (such as spin) ignored. A realistic description of a particle with spin must of course take into account both the space degree of freedom and the internal degrees of freedom. The base ket for a spin 1 2 particle may be visualized to be in the direct-product space of the infinite-dimensional ket space spanned by the position eigenkets {|x } and the two-dimensional spin space spanned by |+ and |−. Explicitly, we have for the base ket |x , ± = |x ⊗ |±, (3.321) where any operator in the space spanned by {|x } commutes with any operator in the two-dimensional space spanned by |±. The rotation operator still takes the form exp(−iJ· ˆ nφ/ ¯ h) but J, the generator of rotations, is now made up of two parts, namely, J = L + S. (3.322) It is actually more obvious to write (3.322) as J = L ⊗ 1 + 1 ⊗ S, (3.323) where the 1 in L ⊗ 1 stands for the identity operator in the spin space, while the 1 in 1 ⊗ S stands for the identity operator in the infinite-dimensional ket space spanned by the position eigenkets. Because L and S commute, we can write D (R) = D (orb) (R) ⊗ D (spin) (R) = exp −iL· ˆ nφ ¯ h ⊗ exp −iS· ˆ nφ ¯ h . (3.324) The wave function for a particle with spin is written as x , ±| α = ψ ± (x ).
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