Download Orbital Angular Momentum - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! Physical origin for orbital angular momentum p r is the position vector and p is the linear momentum vector. zyx ppp zyx kji rrr r =LprL rr r ×= docsity.com Since the operators, do not commute with ,we can only measure their averages. This leads to these sort of pictures: z,̂ˆ ll 2 yx ,̂ˆ ll We can express the eigenfunctions in terms of x, y, and z, although not all are eigenfunctions of Lz Despite this, it is useful to think about the quantum number, m, as defining the orientation of the orbital. docsity.com It is customary to write the wave functions of the electron which are eigenfunctions of S2 and Sz as α and β so that: βββ ααα hh hh 2 1ˆ 2 1ˆ −== +== sz sz mS mS Common thought is that the spin of the electron can be understood by think of the electron as a spinning sphere. This is wrong. Spin does not depend on spatial coordinates. It is an intrinsic quantum property which has no classical analogue. docsity.com Electron spin is a bit weird! Let α = |↑> and β = |↓> = basis set functions of “spin space”. Can construct the matrix operator for Sz as: ↓><↓↑><↓ ↓><↑↑><↑ |ˆ||ˆ| |ˆ||ˆ| zz zz SS SS ↓><↓−↑><↓ ↓><↑−↑><↑ = | 2 | 2 | 2 | 2 hh hh − = 10 01 2 h 3ˆ2 σh= σ3 = one of three Pauli matrices = σz Eigenfunctions are |+> and |-> docsity.com In the basis set of |↑> and |↓> matrices for Sx and Sy can also be derived; − = = 0 0 2 ˆ and 01 10 2 ˆ i i SS yx hh This leads to the two other Pauli matrices: − == == 0 0 ˆˆ and 01 10 ˆˆ 21 i i yx σσσσ Note: There is nothing special about the z-direction. The eigenvalues for electron spin along the x- or y-directions are also ±½; that is, the electron has spin one-half no matter which way you look at it. Let the eigenfunctions of Sx = |→> and |←> Let the eigenfunctions of Sy = >•>⊗ |and| docsity.com The expectation value of Sz combines both values (up and down) ( ) ( ) 0 22 ˆ =↓ −+↑ +>=< PPSz hh One will get zero only through doing many measurement. A single measurement won’t work because ms = 0 is not an eigenvalue of Sz. One can only measure ±½. docsity.com Other particles have intrinsic spin: ms =o, ±1s = 1Deuterons ms = ±1*s = 1Photons ms = ±½s = ½Neutrons ms = ±½s = ½Protons *: light is a special case; no ms = 0. ms = ±1 refers to left-hand and right- hand circularly polarized light. We know many atomic nuclei are NMR active which implies an intrinsic nuclear spin. Phenomenon is common. docsity.com Since electron spin is a separate independent variable we can write: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]zyx zyxzz zyx zyx SS SLSL SLSL SHSH ,, 2 ,, 2 ,, 222 ,,2 ˆ,ˆˆ,ˆ ˆ,ˆˆ,ˆ ˆ,ˆˆ,ˆ ˆ,ˆˆ,ˆ Π=Π= == == = Here Π is the parity operator which take r → -r. If we call the spin eigenfunctions Χ(ms) the complete H-atom wave functions for example would be written: ( ) ( ) ( )smnsmmn mrmrs χψψ rr ll ll ,,,,, , = and ( ) ( ) ',',',',',',',,',',',, )'(||)'(| ss mmmmnnssmnmnsmnsmn mmmm δδδδχχψψχψχψ llllll llllll == docsity.com
