Math Help: Commutator & Relation [f(\hat{A}),\hat{B}]

Thread starter noospace
Start date Feb 19, 2008
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The discussion centers on the commutation relation [f(â, A), B] = [A, B] f'(A) and its validity under specific conditions. It is established that this relation holds true when operator A commutes with the commutator [A, B], not solely when [A, B] equals 1. The derivation involves utilizing a Taylor expansion of the function f(A), which is expressed as f(A) = f(0) + f'(0) A + (1/2) f''(0) A² + ... . This mathematical framework is crucial for understanding operator algebra in quantum mechanics.

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Feb 19, 2008

noospace

Does the relation [f(\hat{A}),\hat{B}] = df(\hat{A})/d\hat{A} follow when A commutes with [A,B]? or is this only valid when [A,B]=1?

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Feb 19, 2008

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Homework Helper

If A commutes with [A,B] then:

[f(A),B] = [A,B] f'(A)

You should try to derive this formula. Use a taylor expansion of f(A), ie:

f(A) = f(0) + f'(0) A + \frac{1}{2} f''(0) A^2 + ...