Classical mechanics Hermitian operator

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... 

We recall here that, in quantum mechanics, each dynamic variable used in classical mechanics is associated with a linear hermitian operator. As a consequence, energy and momentum of a particle placed in an external potential , = Ep/q, will respectively be associated to the following operators ... 

In quantum mechanics, the independent variables q and p of classical mechanics are represented by the Hermitian operators q and p with the following matrix elements in the Cartesian coordinate basis q), here just written for the coordinate g and the conjugate momentum pp... 

Postulate 2. To every physical observable there corresponds a linear Hermitian operator. To find this operator, write down the classical-mechanical expression for the observable in terms of Cartesian coordinates and corresponding linear-momentum components, and then replace each coordinate x by the operator x and each momentum component p, by the operator — ih didx. 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

Unlike classical quantum mechanics, the spontaneous processes of the damped oscillator are irreversible, so its quantum mechanical description needs changes to some instruments of classical quantum mechanics. To do this, we use the Heisenberg picture of quantum processes. In this picture, the observables are time-dependent linear Hermitian operators, and the state vector of the system is time independent. Using the terminology introduced in the first part, the infinitesimal time transformation of the Hermitian operator could happen in two ways ... 

In classical mechanics the abstract Hermitian operator L, defined as operating on the Hilbert space of distributions, is Lc = -i, H), whereas it is Lq = h l [, H] in quantum mechanics. Here, denotes a Poisson bracket and [, ] denotes a commutator. 

Quantum mechanics postulates that to every observable property of a system there corresponds a linear Hermitian operator. Mechanical operators having a classical analogue can be constructed when the operator forms for coordinates and momenta are known. 

In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing p Py,Pz by the appropriate operators. Hie inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them. 

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). 

In quantum mechanics, dynamical variables are represented by linear Hermitian operators 0 that operate on state vectors in Hilbert space. The spectra of these operators determine possible values of the physical quantities that they represent. Unlike classical systems, specifying the state ) of a quantum system does not necessarily imply exact knowledge of the value of a dynarttical variable. Only for cases in which the system is in an eigenstate of a dynamical variable will the knowledge of that state IV ) provide an exact value. Otherwise, we can only determine the quantum average of the dynamical variable. 

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

The derivation of the Hamiltonian resembles the standard procedure the classical Lagrange function is constructed first, then it is used to express the classical Hamilton function and then quantisation is applied by substituting the canonical variables for corresponding quantum-mechanical operators. There are two additional requirements the Hamiltonian should be symmetric with respect to the interchange of two electrons, and it should be Hermitian. 

The third postulate asserts that there is a hermitian mathematical operator for each mechanical variable. To write the operator for a given variable (1) write the classical expression for the variable in terms of Cartesian coordinates and momentum components, (2) replace each momentum component by the relation... 

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