commutator operator Interview Questions and Answers

1. What is the commutator operator?

   • Answer: The commutator operator, denoted as [A, B], is a binary operation on two operators A and B, defined as [A, B] = AB - BA. It measures the extent to which the application of the operators A and B in different orders affects the outcome.

2. What does it mean if [A, B] = 0?

   • Answer: If [A, B] = 0, the operators A and B commute. This means that the order of applying A and B does not matter; ABψ = BAψ for any state vector ψ. They possess simultaneous eigenstates.

3. What does it mean if [A, B] ≠ 0?

   • Answer: If [A, B] ≠ 0, the operators A and B do not commute. The order of applying A and B matters, and they generally do not possess simultaneous eigenstates. The outcome depends on the order of operations.

4. Calculate the commutator of position and momentum operators.

   • Answer: [x, p] = xp - px = iħ, where x is the position operator, p is the momentum operator, and ħ is the reduced Planck constant. This is a fundamental result in quantum mechanics.

5. What is the significance of the non-commutation of position and momentum?

   • Answer: The non-commutation of position and momentum is a manifestation of the Heisenberg uncertainty principle. It implies that we cannot simultaneously know both the position and momentum of a particle with perfect accuracy.

6. What is the commutator of two angular momentum operators, Lx and Ly?

   • Answer: [Lx, Ly] = iħLz. Similarly, [Ly, Lz] = iħLx and [Lz, Lx] = iħLy. These commutation relations are fundamental to the understanding of angular momentum in quantum mechanics.

7. How does the commutator relate to the uncertainty principle?

   • Answer: The commutator quantifies the extent to which two operators do not commute. A non-zero commutator implies that there's a fundamental limit to the precision with which the corresponding observables can be simultaneously measured, as described by the uncertainty principle.

8. What is the commutator of the Hamiltonian and the momentum operator in a free particle system?

   • Answer: For a free particle, the Hamiltonian is H = p²/2m. [H, p] = 0. The momentum is conserved.

9. What is the commutator of the Hamiltonian and the position operator in a free particle system?

   • Answer: For a free particle, [H, x] ≠ 0. The position is not conserved; it changes with time.

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