Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 10

The matrix elements of Jˆz,Jˆ± in the {∣jm〉;j=0,12,1,32,…;m=+j,+j−1,…,−j} basis are (cf. Volume 1, chapter 11):

〈j′m′∣Jˆz∣jm〉=mℏδj′jδm′m,(1.9)

〈j′m′∣Jˆ±∣jm〉=(j∓m)(j±m+1)ℏδj′jδm′m±1.(1.10)

Matrix elements of Jˆx and Jˆy follow from:

Jˆx=12(Jˆ++Jˆ−),(1.11)

Jˆy=12i(Jˆ+−Jˆ−),(1.12)

where, recall Jˆ±≔Jˆx±iJˆy. Thus, the matrix representations of Jˆx,Jˆy, and Jˆz in a ∣jm〉 basis are:

Jˆx↔ℏ200000011000002020202000000300302002030030⋱,(1.13)

Jˆy↔ℏ2i0000001−10000020−2020−2000000300−30200−20300−30⋱,(1.14)

Jˆz↔ℏ200000100−100020000000−200003000010000−10000−3⋱.(1.15)

Note the ‘block-diagonal’ form of Jˆx and Jˆy. These blocks correspond to j=0,12,1,32,…. The matrix representation of Jˆz is diagonal with eigenvalues 0;12ℏ,−12ℏ;ℏ,0,−ℏ;32ℏ,12ℏ, −12ℏ,−32ℏ;…. It is normal practice to reduce these (infinite) matrices by breaking apart the blocks to give finite dimensional matrices. Thus, e.g.

 j=12:Jˆx12↔ℏ20110,Jˆy12↔ℏ20−ii0,Jˆz12↔ℏ2100−1;(1.16)

 j = 1:Jˆx(1)↔ℏ2020202020,Jˆy(1)↔ℏ20−2i02i0−2i02i0,Jˆz(1)↔ℏ220000000−2.(1.17)

The terminology:

Jˆx12≔Sˆx,Jˆy12≔Sˆy,Jˆz12≔Sˆz,(1.18)

σx≔0110,σy≔0−ii0,σz≔100−1,(1.19)

(cf. equation (1.16)), where σx,σy,σz are the Pauli spin matrices, is in common use.