Hermitian operators examples

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

A comparison with equation (3.8) shows that if A is hermitian, then we have = A and A is said to be self-adjoint. The two terms, hermitian and self-adjoint, are synonymous. To find the adjoint of a non-hermitian operator, we apply equations (3.33). For example, we see from equation (3.10) that the adjoint of the operator d/dx is —d/dx. 

When the integral in brackets in (7.6) vanishes, the electronic transition is forbidden. For example, since and )//, are both eigenfunctions of the Hermitian operator 5 (provided spin-orbit interaction is small), and since S2 commutes with del, we conclude [Equation (1.51)1 that electronic transitions with a change in S are forbidden (just as in atoms) ... 

The matrices in equation (35) for a system of n spins of 1/2 have dimensions of 22n. This means that, for example, a four-spin system must be considered within a space of 256 dimensions. If we deal with the motion of a spin system in a static magnetic field (as in pulse-type experiments), significant simplifications are possible owing to the rules of commutation. Namely, if the Hermitian operators A and 6 commute in Hilbert space, then all the corresponding superoperators AL, AR, AD, BL, BR, and BD in Liouville space also commute. The proof of this is given in reference (12). In Hilbert space, the following commutation takes place ... 

For example, suppose that after a full Cl calculation with a complete N-electron set, we have obtained the exact wavefunction, F. The formal rela-fions regarding energy are given by Eqs. (3-7). Now consider a one-electron Hermitian operator, representing a property, say 0(0, where i are electron... 

To demonstrate results for an open-shell system. Table 20 reports results for 02- ° Intensities can also be obtained by considering the left eigenvector of the (non-Hermitian) operator H, along with R, and such EOM-CC examples are shown for electronic excited states later. 

Examples of Hermitian Operators. Let us show that some of the operators we have been using are indeed Hermitian. For simplicity, we shall work in one dimension. To prove that an operator is Hermitian, it suffices to show that it satisfies (7.6) for all well-behaved functions. However, we shall make things a bit harder by proving that (7.11) is satisfied. 

We now postulate that the set of eigenfunctions of any Hermitian operator that represents a physical quantity forms a complete set. (Completeness of the eigenfunctions can be proved in the one-dimensional case and in certain multidimensional cases, but must be postulated for some multidimensional systems.) Thus, any well-behaved function that satisfies the same boundary conditions as the set of eigenfunctions can be expanded according to (7.39). Equation (7.29) is an example of (7.39). 

Postulate 2 For every physically observable variable in classical mechanics, there exists a corresponding linear, Hermitian operator in quantum mechanics. Examples are shown in Table 3.2, where the symbol indicates a quantum mechanical operator and h = h/2n. A Hermitian operator is one which satisfies Equation (3.28). 

The formal properties of operator L eq 2.18 (known as the symplectic structure ) allow the introduction of a variational principle eq D3, " a scalar product (eq Bl), and ultimately to reduce the original non-Hermitian eigenvalue problem (eq 2.18) to the equivalent Hermitian problem which may be solved using standard numerical algorithms (Appendices B—E). For example, F is a Hermitian operator. Lowdin s symmetric orthogonalization procedure " " leads to the Hermitian eigenvalue problem as well (eq E5), which may be subsequently solved by Davidson s algorithm (Appendix E). The spectral transform Lanczos method developed by Ruhe and Ericsson is another example of such transformation. 

The point of the above example is that all of our proofs about eigenvalues or eigenfunctions of hermitian operators refer to cases where the eigenfunctions satisfy the requirement that /i/t = 0. Square-integrability guarantees this, but some... 

Let us look at Hermitian operators more carefully, considering them in the example of Hamiltonian. It has already been mentioned that such operators have real eigenvalues. Furthermore, the eigenfunctions of the Hermitian operator that correspond to different eigenvalues are orthogonal. In other words, for... 

The use of the orbital concept in the Hartree-Fock (HE) and Kohn-Sham (KS) methods leads to similar variational equations a coupled set of eigenvalue equations with a hermitian operator (See for example [7, 25]). This system of integro-differential equations is transformed into a matrix problem when we use a basis set. In both methods, one has to solve a generalized eigenvalue equation ... 

Hermitian operators are very important in quantum mechanics because their eigenvalues are real. As a result, hermitian operators are used to represent observables since an observation must result in a real number. Examples of hermitian operators include position, momentum, and kinetic and potential energy. An operator is hermitian if it satisfies the following relation ... 

The Fock space as introduced in Chapter I is defined in terms of a set of orthonormal spin orbitals. In many situations - for example, during the optimization of an electronic state or in the calculation of the response of an electronic state to an external perturbation - it becomes necessary to carry out transformations between different sets of orthonormal spin orbitals. In this chapter, we consider the unitary transformations of creation and annihilation operators and of Fock-space states that are generated by such transformations of the underlying spin-orbital basis. In particular, we shall see how, in second quantization, the unitary transformations can be conveniently carried out by the exponential of an anti-Hermitian operator, written as a linear combination of excitation operators. 

For illustration, we consider some examples involving only one variable, namely, the cartesian coordinate x, for which w x) = 1. An operator that results in multiplying by a real function /(x) is hermitian, since in this case fix) = fix) and equation (3.8) is an identity. Likewise, the momentum operator p = (i)/i)(d/dx), which was introduced in Section 2.3, is hermitian since... 

The hermitian character of an operator depends not only on the operator itself, but also on the functions on which it acts and on the range of integration. An operator may be hermitian with respect to one set of functions, but not with respect to another set. It may be hermitian with respect to a set of functions defined over one range of variables, but not with respect to the same set over a different range. For example, the hermiticity of the momentum operator p is dependent on the vanishing of the functions ipi at infinity. 

There is no proper perturbative basis for the mnemonic diagram in Fig. 3.58, because the non-orthogonal unperturbed orbitals cannot correspond to any physical (Hermitian) unperturbed Hamiltonian operator,79 as illustrated in Examples 3.17 and 3.18 below. The PMO interpretation of Fig. 3.58 therefore rests on an nnphysical starting point. Removal of orbital overlap (to restore Hermiticity) eliminates the supposed effect. 80... 

However, this Hamiltonian is manifestly non-Hermitian (unless S = 0) and therefore cannot correspond to a physical unperturbed system. Neither of the operators H/(0) and H"(°) can serve as a proper unperturbed Hamiltonian for the PMO rationalization (unless S = 0, when both are equivalent to a standard H(0) such as that underlying, e.g., Example 1.1 or Fig. 3.13). 

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