Energy eigenvalue

In the high-field approxunation with B z, the energy eigenvalues classified by the magnetic spin quantum numbers, and Mj, are given by... 

Since shallow-level impurities have energy eigenvalues very near Arose of tire perfect crystal, tliey can be described using a perturbative approach first developed in tire 1950s and known as effective mass theoiy (EMT). The idea is to approximate tire band nearest to tire shallow level by a parabola, tire curvature of which is characterized by an effective mass parameter m. ... 

In order to remove tlie unwanted electrical activity associated witli deep-level impurities or defects, one can eitlier physically displace tlie defect away from tlie active region of tlie device (gettering) or force it to react witli anotlier impurity to remove (or at least change) its energy eigenvalues and tlierefore its electrical activity passivation). 

We ai e free to pick a tefei ence poitit of energy once, but otily otice, for each system, l,et us choose the reference point t.. We have obtained the energy eigenvalues of the x bond in ethylene as one [f greater than y. 011)11 bunding) and one p lower than y ( bunding) (Fig, 6-3),... 

The excess energies can be measured for a known by essentially a stopping potential method, giving a spechum. This spectrum is then matched with calculated orbital energies (eigenvalues) derived from molecular orbital calculations. 

Clearly the energy (eigenvalue) expression must be independent of x and the two terms eontaining x2 terms must eaneel upon insertion of a ... 

Note that both the numerator and denominator in the final expression are always positive expressions in the case of the denominator, we know this because is the lowest energy eigenvalue of the unperturbed system. (The denominator reduces to a difference in orbital energies.)... 

Lowdin, P-O., Studies in Perturbation Theory. X. Bounds to Energy Eigenvalues in Perturbation Theory Ground State, Physical Review, 1965 139A 357-364. 

For a degenerate energy eigenvalue, the several corresponding eigenfunctions of H may not initially have a definite parity. However, each eigenfunction may be written as the sum of an even part V e(q) and an odd part V o(q)... 

Since the Hamiltonian operator is hermitian, the energy eigenvalues E are real. 

In the determination of the energy eigenvalues, we first show that the eigenvalues X of N are positive (X 0). Since the expectation value of the operator N for an oscillator in state Xi) is X, we have... 

Bond strengths are essentially controlled by valence ionization potentials. In the well established extended Hiickel theory (EHT) products of atomic orbital overlap integrals and valence ionization potentials are used to construct the non-diagonal matrix elements which then appear in the energy eigenvalues. The data in Table 1 fit our second basic rule perfectly. 

The eigenfunctions of the spin Hamiltonian [eqn (1.7)] are expressed in terms of an electron- and nuclear-spin basis set ms, mr), corresponding to the electron and nuclear spin quantum numbers ms and mr, respectively. The energy eigenvalues of eqn (1.7) are ... 

These coefficients (Equation 7.30) are required to calculate the transition probability or spectral amplitude (cf. Chapter 8). Note that for systems with more than two spin wavefunctions (S > 1/2) the energy eigenvalue problem is usually not solvable analytically (unless the matrix can be reduced to one of lower dimensionality because it has sufficient off-diagonal elements equal to zero) and numerical diagonalization is the only option. 

This can only hold for a = E/h. Since u(t) differs from v only by the phase factor exp (—iEt/h) it is physically the same at all times and therefore represents a stationary state or energy eigenstate. The frequency of oscillation of the phase factor is v = E/2-kH = E/h, which confirms that E is an energy eigenvalue. 

The energy eigenvalues of the hydrogen electronic bound states are inversely proportional to the square of the principal quantum number, in SI units,... 

From (27) and (29) it follows that every component of the total angular momentum operator J = L + S and J2 commute with the Dirac Hamiltonian. The eigenvalues of J2 and Jz are j(j + 1 )h2 and rrijh respectively and they can be defined simultaneously with the energy eigenvalues E. 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

This equation is the same as the angular momentum equation (5.8) of the hydrogen atom, with 2IE/H2 instead of l(l + 1). For molecular rotation it is conventional to symbolize the quantum number as J, rather than l. The energy eigenvalues are therefore determined by setting... 

Herman and Skillman [79] used an HFS algorithm to calculate radial atomic wave functions and energy eigenvalues for all atoms, tabulating all results and the computer software at the same time. They treated all single electronic... 

It follows that the wave function for nuclear motion is calculated in an effective potential E (Q) obtained from the energy eigenvalues of the stationary electronic state. 

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