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GATE 2022 CY – Question 56

Physical Chemistry · Structure: Postulates of quantum mechanics. Operators. Commutation relations and expectations values; the role of the wave function and its absolute squares, time dependent and time independent Schrödinger equations. Born interpretation. Dirac bra-ket notation. Examples of model solvable quantum mechanical systems. · 2 marks · Multiple select

For operators $\hat A,\hat B,\hat C$, $[\hat A,[\hat B,\hat C]]+[\hat B,[\hat C,\hat A]]=$ ______.

  1. $[\hat C,[\hat A,\hat B]]$
  2. $[\hat C,[\hat B,\hat A]]$
  3. $[[\hat B,\hat A],\hat C]$
  4. $[[\hat A,\hat B],\hat C]$

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Correct answer: (B) $[\hat C,[\hat B,\hat A]]$; (D) $[[\hat A,\hat B],\hat C]$

Explanation

Jacobi identity makes the sum equal−[C,[A, B]]. Reversing the inner commutator gives[C,[B, A]], option B. Reversing the outer commutator instead gives[[A, B], C], option D.

Thus **B, D** are the same operator expression; A/C have the opposite sign.