Commutators

Basic vector operators:

• r_i
• p_i

Vectors defined using r and p:

• L_i = S_j,k e_ijk r_j p_k
  
• a_± = [1/(2m)^1/2] [ p_x ±imwx]

Product rule:

• [A, BC] = [A, B]C + B[A, C]

Fundamental commutators:

1. [r_i, r_j] = 0
2. [p_i, p_j] = 0
3. [r_i, p_j] = i hbar d_ij

Derivable commutators

1. [ r_i, S_j p_j² ] = 2 i hbar p_i
2. [ p_i, S_j r_j² ] = - 2 i hbar r_i
3. [ L_i, L_j ] = i hbar S_k e_ijk L_k
4. [ L_i, S_j L_j² ] = 0
5. [ r_i, L_j ] = i hbar S_k e_ijk r_k
6. [ p_i, L_j ] = i hbar S_k e_ijk p_k
7. [ r_i, S_j L_j² ]
8. [ p_i, S_j L_j² ]
9. [ L_i, S_j r_j² ] = 0
10. [ L_i, S_j p_j² ] = 0
11. [ a₊, a_- ] = - hbar w
12. [ p_x², a_± ] = ± (2m)^1/2 hbar w p_x
13. [ x², a_± ] = (2/m)^1/2 i hbar x

2. Parallel to #1.

3. From Notes 16:

[L_i , L_p] = S_j,k,q,s {e_ijk e_pqs [r_jp_k , r_qp_s]}
= S_j,k,q,s {e_ijk e_pqs {r_j [p_k , r_qp_s] + [r_j , r_qp_s] p_k }} by expanding the commutator
= S_j,k,q,s {e_ijk e_pqs {-i hbar r_j d_kq p_s + i hbar r_q d_js p_k }} by evaluating the two commutators
= S_j,q,s [ i hbar [- e_ijq e_pqs r_j p_s] + S_k,q,s [e_isk e_pqs r_q p_k ] by using the two Kronecker deltas

Now in order that the r and p factors have the same indices in each term, so that they may be factored, we will rename s to q, q to j and k to s in the second term only:

[L_i , L_p] = i hbar S_j,q,s [- e_ijq e_pqs r_j p_s] + i hbar S_j,q,s [e_iqs e_pjq r_j p_s ]
= i hbar S_j,q,s {r_j p_s (-e_ijq e_qsp + e_siq e_qpj)} by factoring r_j p_s
= i hbar S_j,q,s {r_j p_s [-(d_is d_jp -d_ip d_js) +(d_sp d_ij -d_sj d_ip)]} by using eq. [1] of Notes 16 in both terms
= i hbar S_j,q,s {r_j p_s [-(d_is d_jp) +(d_sp d_ij)]} by combining terms

We can now apply eq. [1] of Notes 16 in reverse to get

[L_i , L_p] = i hbar S_j,q,s {e_ipq e_qjs r_j p_s}

and use the definition of L to get

[L_i , L_p] = S_q i hbar e_ipq L_q

4. [L_i, S_j L_j²] = S_j { [L_i , L_j] L_j + L_j [L_i , L_j] = i hbar S_j,k { e_ijk (L_k L_j + L_j L_k )} = 0

5. [r_i , L_j] = S_{k, l} ie_jkl [r_i , r_kp_l] = S_{k, l} ie_jkl r_k [r_i , p_l] = S_{k, l} ie_jkl r_k i hbar d_il = i hbar S_k e_jkir_k = i hbar S_k e_ijkr_k

6. Parallel to # 5.

7. [ r_i, S_j L_j² ] = S_j { [r_i , L_j] L_j + L_i [r_i , L_j]} = S_{j, k} i hbar e_ijk (r_k L_j + L_j r_k) ¹ 0

8. Parallel to # 7

9, 10. Similar to # 4.

11-13. Omitted; no new ideas needed.

Last Revised 03/11/15