Eigenvalue of an operator

In quantum mechanics, the eigenvalues of an operator represent the only numerical values that can be observed if the physical property corresponding to that operator is measured. Operators for which the eigenvalue spectrum (i.e., the list of eigenvalues) is discrete thus possess discrete spectra when probed experimentally. 

In the vector space L defined over the field of real numbers, every operator acting on L does not necessarily have eigenvalues and eigenvectors. Thus for the operation of 7t/2 rotation on a two-dimensional vector space of (real) position vectors, the operator has no eigenvectors since there is no non-zero vector in this space which transforms into a real multiple of itself. However, if L is a vector space over the field of complex numbers, every operator on L has eigenvectors. The number of eigenvalues is equal to the dimension of the space L. The set of eigenvalues of an operator is called its spectrum. 

The wave functions have the form (5.54), but since Pc does not commute with H, we cannot separate out a chi factor the Schrodinger equation is not separable, and we will try another method of dealing with the problem. We saw in Section 2.3 that the eigenvalues of an operator H can be found by expanding the unknown eigenfunctions in terms of some known complete orthonormal set [Pg.361]

Absolute values of A are smaller or equal to one by construction (they are eigenvalues of an operator consisting of unitary transformations and projections), and, therefore, the logarithm operator has eigenvalues with real part smaller than or equal to zero. 

Even systems as seemingly simple as diatomic molecules often act as complex, many-body systems. Mechanistic understanding and insight, as opposed to mere empirical description, are based on the existence and discovery of patterns that owe their existence to approximate constants of motion. An approximate constant of motion is the eigenvalue of an operator that commutes with most, but not all, terms in the exact molecular Hamiltonian. Nonconservation of this quantity results in subtle rather than catastrophic corruption of the simple patterns on which spectroscopic assignments and mechanistic interpretations are based. 

How do we interpret (7.69) We postulated in Section 3.3 that the eigenvalues of an operator are the only possible numbers we can get when we measure the property that the operator represents. In any measurement of B, we get one of the values b, (assuming there is no experimental error). Now recall Eq. (3.81) ... 

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

Upon computing the eigenvalues of the operator H(q), the equations (3)-(5) can be solved exactly. However, this is, in general, an expensive undertaken. Therefore we proceed with the following multiple-time-stepping approach The first step is to consider the identity... 

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... 

Let us dwell on the properties of eigenvalues and eigenvectors of a linear self-adjoint operator A. A number A such that there exists a vector 0 with = A is called an eigenvalue of the operator A. This vector... 

Prikazchikov, V. (1965) A difference problem on eigenvalues of an elliptic operator. Zh. Vychisl. Mat. i Mat. Fiz., 5, 648-657 (in Russian) English transl. in USSR Comput. Vlathem. and Mathem. Physics. 

These N equations have the appearance of eigenvalue equations, where the Lagrangian multipliers are the eigenvalues of the operator f. The have the physical interpretation of orbital energies. The Fock operator f is an effective one-electron operator defined as... 

Li is an atomic configuration of the site i, with probability p Li) in the GWF and po Li) in the HWF respectively, whereas L is a configuration of the remaining sites of the lattice. Note that this prescription does not change the phase of the wave function as the eigenvalues of the operators Ti are real. The correlations are local, and the configuration probabilities for different sites are independent. 

We now assume that an external magnetic field is directed in the z-direction. Each eigenstate 0 (r,A) has a spin that is an eigenvalue of the operator 5. We seek the driving potential Vpp and the vector potential App that generates 0 (r,A )exp -i E m ))dt /n... 

Exercise 1.17 Suppose Ai, 3-2,. .. are distinct eigenvalues of an energy operator. Suppose that fi, (/)2y the associated eigenvectors. Consider the state corresponding to the wave function... 

According to (13.12), the eigenvalues of this operator will also be zero and unity, however, unity is now an eigenvalue of the one-determinant wave funotion in which this one-electron state a is vacant and zero is an eigenvalue of a function for which this state is filled. Thus, quantity /,na is the occupation number operator for the hole state a. 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

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