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Extra info for Chemical physics. Electrons and excitations
In a magnetic field, there will be 2S + 1 nondegenerate components. For S = 0, there is a singlet; for S = 1/2, there is a doublet; for S = 1, there is a triplet; in general, thus, a (2S + 1)-tet. 8 it follows that for N = 1, there is one doublet; for N = 2 one singlet and one triplet; for N = 3 one quartet and two doublets; and for N = 4 one pentet, three triplets, and two singlets. The notation for the state is 2S+1X. This can be applied to electrons with spatial angular momentum too. The configuration 1s22s22p2 for the carbon atom with the same radial function consists of 15 different functions on three energy levels: nine in a 3P state (ground state), five in a 1D state, and one in a 1S state.
The expression for the kinetic energy, p2/2m = (hn)2/(2L)2/2m, of the particle in the box may be used to derive the uncertainty in momentum; p may be positive or negative. We find ∆p = 2 p = hn/L. Δx is simply equal to the width of the box, L. Finally, we obtain ∆x ∆p = L hn = hn > L 2. 29 shows that an electron in a molecule satisfies the Heisenberg uncertainty relation if its wave function satisfies time-independent SE with suitable boundary conditions. The product of uncertainties, � x � p , is larger in the excited states.
The first version of the Pauli exclusion principle holds that: any two electrons cannot have the same set of quantum numbers. After the discovery of quantum mechanics, particularly the relativistic quantum theory of Dirac, the Pauli exclusion principle was reformulated as follows: a oneelectron spatial function (orbital) can be occupied by at most two electrons with different spins. An alternative way of saying this is: a spin orbital can be occupied by only one single electron. The two-rule of the chemical bond could now be explained.