Energy eigenfunction

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

As seen from (3.7), the closed paths with x(to) = 3c(0) fully determine the energy spectrum of the system. Propagator K has, in terms of the energy eigenfunctions m>, the form... 

The potential V r) in (2) being identical for all atoms in the extended cell, the problem of finding the energy eigenfunctions and eigenvalues in (8) may be reduced to their calculation for a one-atom cell. 

The appropriate expression for the operator H in the above equations is that appearing in Eq. (8-160). Eor a first example, consider an ideal gas without interactions. Assuming that the one-particle wave functions used in the population density operators are the energy eigenfunctions, then the matrix H0llA is diagonal, and we can write... 

The same result can be found by using energy eigenfunctions for the representation the trace of is independent of time. 

According to the postulates of QM, any if/ representing a physical state of the system can be expressed as a linear combination of energy eigenfunctions forming an infinite orthonormal set ... 

In the actual situation, with a finite barrier, the wave functions are not confined to a single side of the barrier and inversion can happen. The two wave functions now have the shapes shown schematically on the diagram at the right. Since the wells are now linked the functions ip+(z) and ip (z) are not eigenfunctions and not orthogonal to each other. The true energy eigenfunctions are the linear combinations ... 

The proof is based on the fact that the diagonal elements of the commutator [A,H] vanish in the basis of energy eigenfunctions. Because of the Hermitian properties of A... 

On account of (45) the density matrix may be diagonalized in the basis of energy eigenfunctions, such that... 

In the basis of energy eigenfunctions the grand partition function is... 

If, on the other hand, the electron had been prepared in a state /(x,y) that is not a pure eigenstate (i.e., cannot be expressed as a single energy eigenfunction), then the time evolution is more complicated. For example, if at t=0 / were of the form... 

The state function (3.20) is a superposition of the energy eigenfunctions t/30) of H°, the expansion coefficients being... 

Since the symmetric-top energy eigenfunctions are eigenfunctions of Pc and of Pz, they have the form... 

Certain semiclassical properties involving the eigenfunctions can also be calculated with periodic-orbit theory. Considering the Wigner functions corresponding to die energy eigenfunctions H = Enn [28],... 

The energy eigenfunctions may be classified according to the irreducible representa-... 

Derivation of the Wigner-Eckart theorem (WET) will be based on the transformation properties of the energy eigenfunctions expressed by... 

The dominant error term is third order in At. The initial wavefunction (Qx,Qy,t) at t = 0 is normally the lowest energy eigenfunction of the initial state of the spectroscopic transition. The value of the wavefunction at incremental time intervals At is calculated by using Eq. (7) for each point on the (Qx,Qy) grid. The autocorrelation function is then calculated at each time interval and the resulting < (t> is Fourier transformed according to Eq. (2) to give the emission spectrum. 

The total wave function can, accordingly, be written as a product of wave functions corresponding to each mode. The energy eigenfunctions corresponding to each mode are, in particular, just the well-known eigenfunctions for a harmonic oscillator. 

The DMC method achieves the lowest-energy eigenfunction by employing the quantum mechanical evolution operator in imaginary time [25], For an initial function expanded in eigenstates, one finds that contributions of the excited states decay exponentially fast with respect to the ground state. 

Fig. 4.2 Electronic energy of H2 as a function of the internuclear distance, for the lower energy eigenfunctions. (Adapted with permission from ref. 26.)...

Here, we begin by analysing some general properties of the energy eigenfunctions of a confined hydrogenic system, the cusp and inflexion properties, and virial relation. 

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