Theorem momentum operator

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

The B cyclic theorem is a Lorentz invariant construct in the vacuum and is a relation between angular momentum generators [42], As such, it can be used as the starting point for a new type of quantization of electromagnetic radiation, based on quantization of angular momentum operators. This method shares none of the drawbacks of canonical quantization [46], and gives photon creation and annihilation operators self-consistently. It is seen from the B cyclic theorem ... 

It is highly useful to employ symmetry relations and selection rules of angular momentum operators for SOC matrix elements [108, 109], The Wigner-Eckart theorem (WET) allows calculations of just a few matrix elements of manifold S. M. S, M in order to obtain all other matrix elements. The WET states that the dependence of the matrix elements on the M, M quantum numbers can be entirely... 

With the help of the replacement theorem (Section 1.4) these matrix elements can be replaced by those of the total angular momentum operator... 

To conclude, we have seen that for a given wave function and Hamiltonian, the Ehrenfest theorem can be instrumentalized to derive explicit expressions for the density and current-density distributions by rewriting it in such a way that the continuity equation results. We will rely on this option in the relativistic framework in chapters 5, 8, and 12 to define these distributions for relativistic Hamiltonian operators and various approximations of N-particle wave functions. From the derivation, it is obvious that the definition of the current density is determined by the commutator of the Hamiltonian operator with the position operator of a particle. All terms of the Hamiltonian which depend on the momentum operator of the same particle will produce contributions to the current density. In section 5.4.3 we will encounter a case in which the momentum operator is associated with an external vector potential so that an additional term will show up in the commutator. Then, the definition of the current density has to be extended and the additional term can be attributed to an (external-field) induced current density. 

The crystal field potential W cf is developed in Legendre polynomials, and with the help of the Wigner-Eckhart theorem (Edmonds, 1957), can be expressed in terms of operator equivalents O which are functions of angular momentum operators J J+, J- This method of Stevens (1952) is described in detail by Hutchings (1966), whose nomenclature is adopted here. The hamiltonian as a function of angular momentum operators is... 

We could also go through the same exercise above to show that [l, LJ = 0 and [l, Ly] = 0, so all the angular momentum components DO commute with l. While this discussion may seem very abstract, we are leading up to an important result that can be used for both the rigid rotor and the H atom. Recall the proof that operators that commute can have the same set of eigenfunctions (Theorem 3, Chapter 11). Here the situation is that each of the angular momentum operators does... 

As a result of the projection theorem [31], the expectation value of the EDM operator d, which is a vector operator, is proportional to the expectation value of J in the angular momentum eigenstate. This fact, in conjunction with Eq. (9), implies that the electric field modifies the precession frequency of the system because of the additional torque experienced by the system due to the interaction between the electric field and the EDM. It can readily be shown that the modified precession frequency is... 

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,... 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

Utilization of the tensorial properties of the electron creation and annihilation operators allows us to obtain expansions in terms of irreducible tensors of any operators in the second-quantization representation. So, using the Wigner-Eckart theorem (5.15) in (14.11) and (14.12), then coupling ranks of second-quantization operators by (5.12) and utilizing (14.10), we can represent one-shell operators of angular momentum in the irreducible tensor form... 

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

The Wigner-Eckart theorem (Biedenharn and Louck, 1981a, p. 96) can be used to obtain Eq. (40) directly. According to this theorem the matrix elements in an angular momentum basis of a spherical tensor operator V (q = —k, — k + 1,k) have a particularly simple structure given by3... 

The dependence on the m indices of the amplitude for the transition between states JM) and J M ) of total angular momentum due to a tensor operator Tq has a remarkably simple form in which the indices M, M and Q all appear in a single 3-j symbol. It is given by the Wigner—Eckart theorem. 

The simplest example of the Wigner—Eckart theorem is given by the Gaunt integral over three spherical harmonics, which is the matrix element for the transition between eigenstates m) and fm ) of a single orbital angular momentum observable due to a tensor operator Tj. We prefer to use the renormalised tensor operator C, which simplifies the expression. 

In order to obtain the eigenfunctions of H y we have to apply the transformation operator U (Eq. (14)) to according to Eq. (13). This is a nontrivial task. The operator exp(ijSpiPy) acts on a function of the coordinates x and y and the result cannot be derived directly. It is very simple, though, to get the result if the operator acts on a function in momentum space. For the determination of the wave function we, therefore, proceed as follows. First we take the Fourier transform of the eigenfunction of H3. As a next step we apply the operator exp(ij8pjPy) which is in momentum space a simple multiplication. Next we take the inverse Fourier transform of the result in order to obtain the function / , j(x, y) in the coordinate space of H y. In the latter step we use the convolution theorem [13] for Fourier transforms. Subsequently applying the... 

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