Heisenberg Hamiltonian, equation

The functions fk and are the counterparts of the so-called destruction (annihilation) and creation operators in the Heisenberg-Dirac picture. It is noted in anticipation that these operators occur as the solutions a,k(t) = lulkt of the Hamiltonian equation... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

S2 possesses the eigenvalues S(S + 1). For a spin-1/2 dimer (S can be 0 or 1), the same values are obtained from the derivation of the Bleaney-Bowers equation. For spin systems in an external magnetic field, the Zeeman operator Hmag = -g/ BB S accounts for Zeeman splitting. The isotropic Heisenberg Hamiltonian for multiple spin centers can be expanded by adding the individual coupling pairs ... 

Use of the Heisenberg spin Hamiltonian (equation 1) to represent the energy difference of the singlet and triplet spin states is easily demonstrated. Two spins, Sj and Sj, can be added to produce a maximum spin of 5 max = Sj + Sj, and lower values - 1, max - 2 down to a minimum of Si - Sj. When the two spins are both one-half, the two possible values for the total spin are 5 tot = 1 and 5 tot = 0, the spin triplet and singlet, respectively. To evaluate the energies of these states from equation f, it is necessary to know the value of Si Sj for the two states. This can be found by evaluating the vector sum of the spins and employing the basic quantum rule... 

We investigate the spatial evolution of the optical fields propagating in the Raman coupler depicted in Fig. 28. Working in the Heisenberg picture, we solve a set of Heisenberg-Langevin equations for the z-dependent field operators. It is convenient to employ momentum operator G rather than Hamiltonian H to describe the operation of the coupler because this approach allows us to take dispersion into account. 

Finally, we consider in this section the force law in quantum mechanics (Ehrenfest, 1927). From Heisenberg s equation of motion employing the Schrodinger Hamiltonian we have... 

To facilitate the derivation we shall assume that we are in the Heisenberg picture and dealing with a time-independent hamiltonian, i.e., H(t) — 27(0) = 27, in which case Heisenberg operators at different times are related by the equation... 

Assume that there exists a unitary operator U(it) which maps the Heisenberg operator Q(t) at time t into the operator (—<). Assume further that this mapping has the property of leaving the hamiltonian invariant, i.e., that U(it)SU(it)" 1 = H. Consider then the equation satisfied by the transformed operator... 

In this equation, H, the Hamiltonian operator, is defined by H = — (h2/8mir2)V2 + V, where h is Planck s constant (6.6 10 34 Joules), m is the particle s mass, V2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Tint) is in terms of probability I/2 () is the probability of finding the particle between x and x + dx, at time t. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

The Hamiltonian of equation (78) corresponds to so-called Heisenberg exchange. In some systems, particularly chains of atoms, it seems that the coupling is restricted to one direction, z. Then the Hex becomes... 

For organic spin systems, one frequently assumes applicability of Heisenberg spin behavior, in which all interactions can be reasonably modeled by pairwise exchange interactions. A typical Heisenberg spin Hamiltonian for exchange Jy between various spin sites i and j, with spin quantum numbers S, and Sj, is given in the following equation ... 

The ACF of the dipole moment operator of the fast mode may be written in the presence of Fermi resonances by aid of Eq. (10). Besides, the dipole moment operator at time t appearing in this equation is given by a Heisenberg equation involving the full Hamiltonian (225). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the real... 

Generalization of equation 89 to the many-electron system of a crystal is given by the Heisenberg exchange Hamiltonian... 

It was pointed out in Chapter II that the Heisenberg exchange Hamiltonian of equation 90, which can be directly related to the Weiss field parameters at T = 0°K by equation 94, is an excellent formal expression for the interactions between atomic spins (or moments) of neighboring atoms. There remains the problem of establishing the various spin-dependent mechanisms that contribute to the Jij. In general, there are two types of interaction cation- -cation and cation-anion-cation (or even cation-anion-anion-cation) interactions. 

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