By Werner O. Amrein, etc.
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Additional resources for Scattering Theory in Quantum Mechanics
Sample text
In these cases, one says that a 'linear variational' calculation is being performed. The set of functions {ΦJ} are usually constructed to obey all of the boundary conditions that the exact state Ψ obeys, to be functions of the the same coordinates as Ψ, and to be of the same spatial and spin symmetry as Ψ. , convenient to evaluate Hamiltonian matrix elements <ΦI|H|ΦJ>) and that can, in principle, be made complete if more and more such functions are included. For such a trial wavefunction, the energy depends quadratically on the 'linear variational' CJ coefficients: <Φ|H|Φ> = Σ IJ CICJ <ΦΙ |H|ΦJ>.
Only subsequent measurement of another property whose operator does not commute with F, G, or J will destroy precise knowledge of the values of the properties measured earlier. Example: Assume that an experiment has been carried out on an atom to measure its total angular momentum L2. According to quantum mechanics, only values equal to L(L+1) h2 will be observed. Further assume, for the particular experimental sample subjected to observation, that values of L2 equal to 2 h2 and 0 h2 were detected in relative amounts of 64 % and 36 % , respectively.
The probability of finding a particle at some position can not be multivalued nor can it be 'jerky' or discontinuous). , at the wall of an infinitely high and steep potential barrier). This condition relates to the fact that the momentum must be continuous except at infinitely 'steep' potential barriers where the momentum undergoes a 'sudden' reversal. 2. An experimental measurement of any quantity (whose corresponding operator is F) must result in one of the eigenvalues fj of the operator F.