Physics
Quantum Orbital Angular Momentum
Quantum orbital angular momentum refers to the intrinsic angular momentum of a particle as it orbits around a central point, as described by quantum mechanics. It is quantized, meaning it can only take on certain discrete values. This concept is fundamental in understanding the behavior of electrons in atoms and plays a crucial role in the interpretation of atomic spectra.
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10 Key excerpts on "Quantum Orbital 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 - PDFQuantum Mechanics I
A Problem Text
- David DeBruyne, Larry Sorensen(Authors)
- 2018(Publication Date)
- Sciendo(Publisher)
Chapter 12 Orbital Angular Momentum Angular momentum comes in three flavors, orbital angular momentum, spin, and total angular momentum, which is the vector sum of orbital angular momentum and spin. This chapter is primarily about orbital angular momentum. Classical orbital angular momentum is a vector quantity denoted vector L = vector r X vector p . A common mnemonic to calculate the components is vector L = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle hatwide i hatwide j hatwide k x y z p x p y p z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ( yp z -zp y ) hatwide i + ( zp x -xp z ) hatwide j + ( xp y -yp x ) hatwide k = L x hatwide i + L y hatwide j + L z hatwide k. Let’s focus on one component of angular momentum, say L x = yp z -zp y . On the right side of the equation are two components of position and two components of linear momentum. Quantum mechanically, all four quantities are dynamic variables which are operators. The components of quantum mechanical orbital angular momentum do not commute. Thus, we are forced to seek another operator, like L 2 that does commute with the component operators, to obtain a complete set of commuting observables. As with the SHO, raising and lowering operators will be useful. Unlike all other systems encountered where one quantum number uniquely identifies the state, two quantum numbers are required to uniquely identify the quantum state of orbital angular momentum because two operators are required for the complete set of commuting observables. The first part of this chapter is dominated by arguments from linear algebra and commutator algebra. The middle portion is calculus-based arguments in position space. Problems concerning angular momentum often use, some practically require, spherical polar coordinates.
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.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
Since the orbital angular momentum operators hatwide L z and hatwide L 2 commute with hatwide r 2 and hatwide p 2 , they also commute with the Hamiltonian. Hence, just as in classical mechanics, the value of the angular momentum and its projection onto the z -axis are preserved in the motion of a particle in a central field. The wave function of any state ψ ( r , θ , ϕ ) can be expanded in the common eigenfunctions ψ E , l , m l of the three commuting operators hatwide H , hatwide L z and hatwide L 2 corresponding to the three simultaneously measurable physical quantities: the energy, the orbital angular momentum projection, and its value. We use the separation of variables; that is, we seek the eigenfunctions in the form: ψ E , l , m l = R E , l ( r ) Φ m l ( ϕ ) Θ l , m l ( θ ). Writing the Hamiltonian in spherical coordinates and taking (P6.2.4.3) and (P6.2.4.4) into account, we obtain hatwide H = – planckover2pi1 2 2 mr 2 ∂ ∂r parenleftbigg r 2 ∂ ∂r parenrightbigg + hatwide L 2 2 mr 2 + U ( r ). (P6. 2 . 4 . 5 ) The Schr¨ odinger equation in form (P6.2.3.5) after the action of the Hamiltonian (P6.2.4.5) on the function ψ E , l , m l becomes the equation for the radial part of the wave function R E , l ( r ): – planckover2pi1 2 2 m 1 r 2 ∂ ∂r parenleftbigg r 2 ∂ ∂r parenrightbigg R E , l + parenleftbigg planckover2pi1 2 l ( l + 1 ) 2 mr 2 + U ( r ) parenrightbigg R E , l = ER E , l . (P6. 2 . 4 . 6 ) Just as in classical mechanics (see Subsection P1.6.2), the potential energy is supplemented with the centrifugal energy planckover2pi1 2 l ( l + 1 ) / 2 mr 2 . One can see that the magnetic quantum number is not contained in the equation for allowed values of the energy. To each energy level there correspond two quantum numbers: the orbital number l and the radial number n r = 0 , 1 , 2 , ... numbering the allowed discrete values of the energy. The level E n r , l is ( 2 l + 1 )-fold degenerate with respect to the values of the magnetic quantum number m l .
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.
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.
- 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 .
eBook - PDF- J. J. Sakurai, Jim Napolitano(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
(3.226) 191 3.6 Orbital Angular Momentum Before taking the preceding expression between x | and | α, first note that x |x·p| α = x ·(−i ¯ h∇ x | α) = −i ¯ hr ∂ ∂ r x | α. (3.227) Likewise, x |(x·p) 2 | α = −¯ h 2 r ∂ ∂ r r ∂ ∂ r x | α = −¯ h 2 r 2 ∂ 2 ∂ r 2 x | α + r ∂ ∂ r x | α . (3.228) Thus x |L 2 | α = r 2 x |p 2 | α + ¯ h 2 r 2 ∂ 2 ∂ r 2 x | α + 2r ∂ ∂ r x | α . (3.229) In terms of the kinetic energy p 2 /2m, we have 1 2m x |p 2 | α = − ¯ h 2 2m ∇ 2 x | α = − ¯ h 2 2m ∂ 2 ∂ r 2 x | α + 2 r ∂ ∂ r x | α− 1 ¯ h 2 r 2 x |L 2 | α . (3.230) The first two terms in the last line are just the radial part of the Laplacian acting on x | α. The last term must then be the angular part of the Laplacian acting on x | α, in complete agreement with (3.224). 3.6.2 Spherical Harmonics Consider a spinless particle subjected to a spherical symmetrical potential. The wave equation is known to be separable in spherical coordinates and the energy eigenfunctions can be written as x |n, l, m = R nl (r)Y m l (θ , φ), (3.231) where the position vector x is specified by the spherical coordinates r, θ , and φ, and n stands for some quantum number other than l and m, for example, the radial quantum number for bound-state problems or the energy for a free-particle spherical wave. As will be made clearer in Section 3.11, this form can be regarded as a direct consequence of the rotational invariance of the problem. When the Hamiltonian is spherically symmetrical, H commutes with L z and L 2 , and the energy eigenkets are expected to be eigenkets of L 2 and L z also. Because L k with k = 1,2,3 satisfy the angular-momentum commutation relations, the eigenvalues of L 2 and L z are expected to be l(l + 1) ¯ h 2 , and m¯ h = [−l ¯ h, (−l + 1) ¯ h, ... , (l − 1) ¯ h, l ¯ h].
eBook - PDF- Mark Fox(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Their energies will differ due to the residual electrostatic interaction. The atomic states defined by the values of L and S are called terms . For each atomic term, we can find the total angular momentum of the whole atom from: J = L + S . (5.33) The values of J , the quantum number corresponding to J , are found from L and S , according to Eq. (5.22) . The states of different J for each LS-term have different energies due to the spin-orbit interaction. In analogy with Eq. (5.23) , the spin-orbit interaction of the whole atom is written: E so ∝ − μ atom spin · B atom orbital ∝ L · S , (5.34) where the superscript atom indicates that we take the resultant values for the whole atom. The details of the spin-orbit interaction in the LS-coupling limit are considered in Section 7.6 . At this stage, all we need to know is that the spin-orbit interaction splits the LS terms into levels labeled by J , and that the splitting is much smaller than the energy difference between different LS terms. It is convenient to introduce a shorthand notation to label the energy levels that occur in the LS coupling regime. Each level is labeled by the quantum numbers J , L , and S , and is represented in the form: ( 2 S + 1 ) L J . The superscript ( 2 S + 1 ) and subscript J appear as numbers, whereas L is a capital letter that follows the usual rule: S, P, D, F correspond, respectively, to L = 0, 1, 2, 3. Thus, for example, a 2 P 1 / 2 level has quantum numbers S = 1 / 2, L = 1, and J = 1 / 2, while a 3 D 3 level has S = 1, L = 2, and J = 3. For L > 3, the letters increment alphabetically, with the exception that the letter J is omitted in order to avoid confusion with the angular momentum quantum number J . Hence L = 6 is designated by I, but L = 7 is designated by K. The factor of ( 2 S + 1 ) in the top left is called the multiplicity . It indicates the degeneracy of the level due to the spin – i.e., the number of M S states available.
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