By Sven Larsson
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Additional resources for Chemical physics. Electrons and excitations
3 Expectation Value If electronic wave packets are going to behave as particles, their wave packets must have properties like velocity and acceleration. A wave packet is not necessarily an eigenfunction to the operator related to a certain property or observable, for example, angular momentum or kinetic energy. A distribution of values is observed. The expectation value is defined as a kind of average value for the region of space where the electron can be found: 20 Chemical Physics: Electrons and Excitations ˆ ˆ = ψΩψ = Ω ψ ψ ∫ ψ Ωˆ ψdx .
In the first case, they are close (r12 = r1 − r2 small) and in the second case, distant (r12 large). The probability density is expected to be higher at a large distance than at a small distance since the electrons repel each other. Despite the fact that they are at the same distance from the nucleus, the probability density of the positions in the right figure is higher than that of the positions in the left figure. In the wave function, the two orbitals may be identical 1s orbitals whose value depends only on the distance from the nucleus.
17) ψ n = sin ( nπx L ) . 18 define stationary states and standing waves. The ground state corresponds to n = 1 and the excited states to n > 1. 17 are one-electron states. The time-independent SE can also be an equation for a wave function that depends on the position of several electron coordinates or nuclear coordinates. If the particles are electrons (or fermions in general), they obey the Pauli principle, according to which at most two electrons, with different spins, are allowed to have the same spatial wave function.
Chemical physics. Electrons and excitations by Sven Larsson