Expectation values theorem

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

The quantity Gy is an estimation of G, and the fundamental theorem of Monte Carlo guarantees that the expected value of Gy is G, if G exists (Ref. 161). The error in the calculation is given by... 

After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Equivalently, expectation values of three-dimensional dynamical quantities may be evaluated for each dimension and then combined, if appropriate, into vector notation. For example, the two Ehrenfest theorems in three dimensions are... 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

Although the expectation value r )ni cannot be obtained from equation (6.70), it can be evaluated by regarding the azimuthal quantum number I as the parameter in the Hellmann-Feynman theorem (equation (3.71)). Thus, we have... 

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

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... 

For a closed system, therefore, the expectation value of the energy is constant in time, which is the energy theorem of quantum mechanics. 

These energy bounds are based on two theorems (10-12) which utilize the fact that in the quantum chemical energy expectation value functional the nuclear charges can be regarded as continuous variables. A series of energy relations can be derived for iso-electronic molecules which contain different nuclei, or the same nuclei in different positions. A corollary of the first theorem states (eq.32 in (10)) that... 

A virial theorem (5-8) applied to the electronic coordinates gives the kinetic energy (T) in terms of (V) and expectation values of the first derivative of V with respect to nuclear coordinates (66-72). For bound electronic states f)... 

By the Hellmann-Feynman theorem, the expectation value < f -dV/dRua I f) is the force on nucleus N in the a direction. The force on each nucleus vanishes for a molecule in its equilibrium nuclear configuration the force also vanishes for an isolated atom. In these cases the virial theorem becomes (T) = -other cases, however, the second term on the right in Eq. (17) is non-vanishing. 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

In Table 4 we compare the most physical momentum expectation values a = 2 and a = - 1 with the T/Fones. For a=2 (related with the kinetic energy) we compare with the value -2E obtained by our method in order to check how close the aproximations utilized verify the virial theorem. 

This result represents the most important advantage of the particle-hole formalism. Many-body perturbation theory (MBPT) consists mainly in the evaluation of expectation values (with respect to the physical vacuum) of products of excitation operators. This is easily done by means of Wick s theorem in the particle-hole formalism. 

The Hohenberg-Kohn theorem can be used to redefine entanglement measures in terms of new physical quantities expectation values of observables, ai, instead of external control parameters, li. Consider an arbitrary entanglement measure M for the ground state of Hamiltonian (85). For a bipartite entanglement, one can prove a central lemma, which very generally connects M and energy derivatives. 

The variation theorem has been an extremely powerful tool in quantum chemistry. One important technique made possible by the variation theorem is the expression of a wave function in terms of variables, the values of which are selected by minimizing the expectation value of the energy. 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

To proceed, first, assume we have some well-behaved candidate density that integrates to the proper number of electrons, N. In that case, the first theorem indicates that this density determines a candidate wave function and Hamiltonian. That being the case, we can evaluate the energy expectation value... 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

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