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The relative amplitudes Ck are determined by knowledge of the state at the initial time; this depends on how the system has been prepared in an earlier experiment. , the coordinates and momenta must be known at some initial time), the Schrödinger equation must be accompanied by initial conditions to fully determine Ψ(qj,t). 09769Å)2 π  that was created by the fast ionization of N2. Subsequent to ionization, this N 2 function is not an eigenfunction of the new vibrational Schrödinger equation appropriate to N2+.

0 Continuum State r The solutions of this one-electron problem form the qualitative basis for much of atomic and molecular orbital theory. For this reason, the reader is encouraged to use Appendix B to gain a firmer understanding of the nature of the radial and angular parts of these wavefunctions. The orbitals that result are labeled by n, l, and m quantum numbers for the bound states and by l and m quantum numbers and the energy E for the continuum states. Much as the particle-in-a-box orbitals are used to qualitatively describe π- electrons in conjugated polyenes, these so-called hydrogen-like orbitals provide qualitative descriptions of orbitals of atoms with more than a single electron.

Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions {ψj}. Any function Φ that depends on the same spatial and spin variables on which H operates and obeys the same boundary conditions that the {ψj} obey can be expanded in this complete set Φ = Σ j Cj ψj. The expectation value of the Hamiltonian for any such function can be expressed in terms of its Cj coefficients and the exact energy levels Ej of H as follows: <Φ|H|Φ> = Σij CiCj <ψi|H|ψj> = Σ j|Cj|2 Ej. If the function Φ is normalized, the sum Σ j |Cj|2 is equal to unity.

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