Angular momentum quantum-mechanical operators

L, S, J AAA L, S, J L, S, J A J j =jl +j2 orbital, spin, and total angular momenta quantum mechanical operators corresponding to L, S, and J quantum numbers that quantize L2, S2, and J2 operator that obeys the angular momentum commutation relations total (j) and individual (ji, j2, ) angular momenta, when angular momenta are coupled... 

The quantum-mechanical operators for the components of the orbital angular momentum are obtained by replacing px, Py, Pz in the classical expressions (5.2) by their corresponding quantum operators. 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

Thus, the quantum-mechanical treatment of generalized angular momentum presented in Section 5.2 may be applied to spin angular momentum. The spin operator S is identified with the operator J and its components Sx, Sy, Sz with Jx, Jy, Jz- Equations (5.26) when applied to spin angular momentum are... 

Formula (58) shows that the angular momentum operator for the photon consists of two terms. The first term is identical with the usual quantum-mechanical operator L for the orbital angular momentum in the momentum... 

In conventional quantum mechanics, a wavefunction d ribing the ground or excited states of a many-particle system must be a simultaneous eigenfunction of the set of operators that commute with the Hamiltonian. Thus, for example, for an adequate description of an atom, one must introduce the angular momentum and spin operators L, S, L, and the parity operator H, in addition to the Hamiltonian operator. 

The relationship of these quantum mechanical operators to experimental measurement will be made clear later in this chapter. For now, suffice it to say that these operators define equations whose solutions determine the values of the corresponding physical property that can be observed when a measurement is carried out only the values so determined can be observed. This should suggest the origins of quantum mechanics prediction that some measurements will produce discrete or quantized values of certain variables (e.g., energy, angular momentum, etc.). 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

The angular momentum variables all enter as quantum-mechanical operators, whose eigenvalues may be measurable. 

Here I is interpreted as a quantum mechanical operator. From the general properties of spin angular momentum in quantum mechanics, it is known that the solution of this Hamiltonian gives energy levels in which... 

The detailed structure of the Mossbauer and ESR spectra for the iron transport compounds can be described in terms of a spin Hamiltonian with effective spin S= 5/2 for the high spin ferric ion. These parameters can give information about the site symmetry of the iron. Since there is no orbital angular momentum in the 6S state, the effective spin is the same as the real spin of the iron ion. There is spherical symmetry in the 6S state to first order, but spin-orbit coupling to excited (non-spherical) orbital states gives rise to asymmetries about the iron site which are reflected in the spin Hamiltonian. The general form of the spin Hamiltonian which we will use here is a quantum mechanical operator which acts on the electronic states ms> and nuclear states mf>. 

In quantum mechanics operators represent observable quantities, such an energy, angular momentum and magnetization. For a single spin-half, the x- y-... 

Show that the x and z components of the angular momentum have quantum mechanical operators that do not commute and find their commutator ... 

Angular momentiun is an important example of quantum mechanical operators in terms of electron field operators. The components of the angular momentiun vector j = I + s are the generators of the 3-dimensional rotation group. In our units, the operator of orbital angular momentum is... 

We get the quantum-mechanical operators for the components of orbital angular momentum of a particle by replacing the coordinates and momenta in the classical equations (5.39) by their corresponding operators [Eqs. (3.21)-(3.23)]. We find... 

In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing p Py,Pz by the appropriate operators. Hie inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them. 

Two wave-functions with the same energy are obtained, contradicting the assumption of a non-degenerate state. Thus, the wave-function can be taken real (cp2 = 0). The orbital angular momentum is also a real quantity. On the other hand, the corresponding quantum mechanical operator of e.g. the x-component. Lx = iHy i - z- ) is imaginary. Thus, the expectation value is zero, Lx l ) = 0. 

The orbital angular momentum quantum number 1 can take the values 0,1,2,3,... (also know as azimuthal quantum number) and the magnetic quantum number m must be in —/, — / + 1,..., / (also known as orientational quantum number). The eigenfunctions can be efficiently constructed through the definition of ladder operators, which is standard in nonrelativistic quantum mechanics and therefore omitted here. The general expression for the spherical harmonics reads [70]... 

Quantum mechanical operators corresponding to the components of an angular momentum vector can be found directly from the familiar rectilinear position and momentum operators, for example. 

Another useful operator can be found from the angular momentum component operators. The square of the angular momentum of a classical system, or of a quantum mechanical system, is a scalar quantity, the dot product of L with itself. The dot product of any vector with itself equals the square of the length of the vector. Thus, L Lis the square of the magnitude of the angular momentum vector. The quantum mechanical operator that corresponds to this dot product is designated U, and its explicit form can be derived from the component expression for a dot product. Thus,... 

The components of the quantum mechanical angular momentum operators along the three principal axes are ... 

ANGULAR MOMENTUM OPERATORS AND ROTATIONS IN SPACE AND TRANSFORMATION THEORY OF QUANTUM MECHANICS ... 

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