SME
Quantum Mechanics · Problem type
Which observables commute?
Use this for “Can the position x and the momentum pᵧ of an electron both be known exactly at once?”
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Can each pair be known exactly at the same time?
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Work out [A, B] = AB − BA: zero means a shared set of eigenstates; anything else means an uncertainty relation.
Assumptions
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The commutator of two operators is [A, B] = AB − BA; when it is not 0, A and B have no full set of shared eigenstates.
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Two that commute can both have sharp values in one state; two that do not obey ΔAΔB ≥ ½|⟨[A, B]⟩|.
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The standard results: [x, pₓ] = iħ, [Lₓ, Lᵧ] = iħLz, [Sz, Sₓ] = iħSᵧ; a position and a momentum along different axes commute, and so do operators of different particles.
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Measuring the energy of a superposition
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