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GATE 2026 PH – Question 52

Quantum Mechanics · uncertainty principle · 2 marks · Multiple select

Consider operators $\hat A,\hat B$, and $\hat C$ for three observables of a quantum system satisfying $[\hat A,\hat B]=0$, $[\hat B,\hat C]=0$, and $[\hat A,\hat C]\ne0$, with uncertainties $\Delta A,\Delta B,\Delta C$, respectively. From the options given below, which is/are implied by the commutation relations among $\hat A,\hat B$, and $\hat C$?

  1. $\Delta A\Delta B>0$
  2. $\Delta A\Delta C>0$
  3. $\hat A,\hat B$ can be simultaneously diagonalized.
  4. $\hat A,\hat B,\hat C$ can be simultaneously diagonalized.

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Show answer and explanation

Correct answer: (B) $\Delta A\Delta C>0$; (C) $\hat A,\hat B$ can be simultaneously diagonalized.

Explanation

The official answer is B and C. C follows from simultaneous diagonalization of commuting Hermitian observables. B is the uncertainty statement intended for the noncommuting pair. Strictly, Robertson gives $\Delta A\Delta C\geq\tfrac12|\langle[\hat A,\hat C]\rangle|$; operator noncommutation alone does not force a strictly positive product in every possible state. Use B,C for this paper’s scoring, while retaining this mathematical qualification.

Official GATE 2026 answer key: https://gate2026.iitg.ac.in/doc/download/2026/Keys/PH_Keys.pdf