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42 ELECTROMAGNETIC RADIATION AND MATTER

or (reinserting ε∓)

(m − m ∓ 1) , m ε∓x± ] , m = 0 (1.162)

Thus for the matrix element , m ε∓ x± , m to be non-zero we requirem − m ∓ 1 = 0 or m = m − m = ± 1. The polarization ε is associatedwith the absorption of a photon and ε∗ is associated with the creation of a photon. We see from Equation (1.155) that ε∗± = ε∓ . Substitution of thisrelation in Equation (1.162) gives

( m ∓ 1) , m ε∗± x± , m = 0 (1.163)

For the upper signs in Equation (1.163) we nd that for the matrix element, m ε∗± x± ] , m to be non-zero m must be one unit larger than m.Conservation of the component of angular momentum along the axis of quantization k implies that the created photon has Sz = − 1, that is, thecreated photon is a left-handed (LH) one. Thus ε∗+ = εx − iεy is associatedwith the creation of a LH photon. Similarly we nd using the lower signin Equation (1.163) that ε∗− = εx + iεy is associated with the creation of aright-handed (RH) photon. Likewise, ε+ = εx + iεy absorbs a RH photonwhile ε− = εx − iεy absorbs a LH photon. It is left to the reader to prove

these statements, see Section 1.2.4 on Polarization and Spin.We now turn to M1 transitions. From Equation (1.128) and Equation(1.129) we see that the relevant quantity is ( k × ε ) · , m L , m . The com-ponent of L parallel to k does not contribute to the triple product, so we onlyneed to consider the components of L that are perpendicular to k , the axisof quantization along which we choose the z-axis to be. Take for exampleL = (Lx ,0,0) ∝ (L+ + L− , 0, 0) with L± the raising and lowering operators.If we sandwich this between , m and , m we nd that the matrixelement , m L , m can be non-zero only if m = m ± 1. We can also

sandwich the commutator relation [ L 2 , L ] = 0 between that bra and ket tond that the matrix element , m L , m can be non-zero only if = . Wesee from the examples given thus far that selection rules can be obtained byusing magic commutation relations sandwiched between the bra , m andthe ket , m . Although a more systematic approach can be taken, we willnot do that here because it is outside the scope of this book. Instead we willgive a few more examples using the methods employed earlier in this section.

Consider for example E2 transitions. Among the nine quantitiesε ik j B xix j A to be considered, see Equation (1.133), we chose B xy A .

One can easily show that xy = (x2+ − x2− )/ (4i) and that

[Lz, x2± ] = ± 2x2+ (1.164)