Momentum operator quantum mechanical

This result is interpreted to mean that angular momentum is described by the operator L(y ) = ihd/dp, which is equivalent to the postulate that linear momentum is quantum-mechanically represented by the operator —ihV. The Bohr operator for angular momentum is converted into cartesian coordinates by writing... 

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... 

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

One can consider in this way that the imaginary factor in front of the gradient operator, which represents momentum in Quantum Mechanics, could... 

The incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

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

Using the fact that the quantum mechanical coordinate operators q = x, y, z as well as the conjugate momentum operators (pj = px, Py, Pz are Hermitian, it is possible to show that Lx, Ly, and L are also Hermitian, as they must be if they are to correspond to experimentally measurable quantities. 

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

Any operator J, which satisfies the commutation rule Eq. (7-18), represents quantum mechanical angular momentum. Orbital angular momentum, L, with components explicitly given by Eq. (7-1), is a special example5 of J. 

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

Fortunately, in quantum mechanics, the corresponding spatial operations for the individual nucleons (4.15) can be replaced by convenient angular momentum operators that act on the total spin I of the nucleus [4]. The corresponding... 

Equation (2.23) is the quantum-mechanical analog of the classical definition of momentum, p = mv = m(Ax/At). This derivation also shows that the association in quantum mechanics of the operator (h/i)(d/dx) with the momentum is consistent with the correspondence principle. 

The second postulate states that a physical quantity or observable is represented in quantum mechanics by a hermitian operator. To every classically defined function A(r, p) of position and momentum there corresponds a quantum-mechanical linear hermitian operator A(r, (h/i)V). Thus, to obtain the quantum-mechanical operator, the momentum p in the classical function is replaced by the operator p... 

The position, momentum, and energy are all dynamical quantities and consequently possess quantum-mechanical operators from which expectation values at any given time may be determined. Time, on the other hand, has a unique role in non-relativistic quantum theory as an independent variable dynamical quantities are functions of time. Thus, the uncertainty in time cannot be related to a range of expectation values. 

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. 

We now apply the results of the quantum-mechanical treatment of generalized angular momentum to the case of orbital angular momentum. The orbital angular momentum operator L, defined in Section 5.1, is identified with the operator J of Section 5.2. Likewise, the operators I , L, Ly, and are identified with J, Jx, Jy, and Jz, respectively. The parameter j of Section 5.2 is denoted by I when applied to orbital angular momentum. The simultaneous eigenfunctions of P and are denoted by Im), so that we have... 

Accordingly, the quantum-mechanical Hamiltonian operator H for this system is proportional to the square of the angular momentum operator U-... 

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... 

As an example, consider the quantum mechanical operator for the linear momentum in one dimension,... 

This expression has the structure of quantum-mechanical expectation values, defining the operator for photon angular momentum as... 

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... 

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

Statistical mechanics deals explicitly with the motion of particles. The common quantization procedure that provides a quantum description of classical particles by the introduction of operators, such as the momentum operator, p —> however, replaces the classical particle description by a wave... 

This vector field can be inserted directly into the quantum mechanical Hamiltonian by a variable change in the momentum operator ... 

---