Identiti Komutator Mekanik Kuantum

Sebagai rujukan untuk penyelesaian dalam mekanik kuantum, dilampirkan identiti-identiti komutator;

i) [A , B] = -[B , A]
[A , B] = AB-BA
= -(BA-AB)
= -[B , A]

ii) [A+B , C] = [A , C] + [B , C] ; sama juga untuk operasi TOLAK
[A+B, C] = (A+B)C - C(A+B)
= AC+BC - CA-CB
= (AC - CA) + (BC - CB)
= [A , C] + [B , C]

iii) [AB , C] = A[B,C] + [A, C]B
[AB , C] = (AB)C - C(AB)
= ABC - ACB -CAB + ACB
= A(BC - CB) + (AC - CA)B
= A[B, C] + [A , C]B ; cuba selesaikan [A, BC]

iv) [A , A] = 0

v) [AB, CD] = ?
= (AB)(CD) - (CD)(AB)
= AB(CD) - A(CD)B - (CD)AB +A(CD)B
= A ( B(CD) - (CD)B) + ( A(CD) - (CD)A )B
= A[B, CD] + [A, CD]B

gunakan identiti iii) menunjukkan;
= A( C[B, D] + [B, C]D ) + ( C[A, D] + [A, C]D )B

[AB, CD] = AC[B, D] + A[B, C]D + C[A, D]B + [A, C]DB