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lesya [120]
3 years ago
12

Show that any three linear operators A, B, and Ĉ satisfy the following (Ja- cobi) identity (10 pt) [[A, B] Ĉ] + [[B,C), A] + [[C

, A, B) = 0
Physics
1 answer:
DerKrebs [107]3 years ago
7 0

Answer:

Three linear operators A,B, and C will satisfy the condition [[A, B],C] + [[B,C), A] + [[C, A], B] = 0.

Explanation:

According to the question we have to prove.

[[A, B],C] + [[B,C), A] + [[C, A], B] = 0

Now taking Left hand side of the equation and solve.

[[A, B],C] + [[B,C), A] + [[C, A], B]

Now use commutator property on it as,

=[A,B] C-C[A,B]+[B,C]A-A[B,C]+[C,A]B-B[C,A]\\=(AB-BA)C-C(AB-BA)+(BC-CA)A-A(BC-CB)+(CA-AC)B-B(CA-AC)\\=ABC-BAC-CAB+CBA+BCA-CAB-ABC+ACB+CAB-ACB-BCA+BAC\\=0

Therefore, it is proved that [[A, B],C] + [[B,C), A] + [[C, A], B] = 0.

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