4. Prove that if there are two operators A, ˆ Bˆ, such that their commutator [A,...

4. Prove that if there are two operators A, ˆ Bˆ, such that their commutator [A, ˆ Bˆ] = λ where
λ is a constant. Show that (6 points)

exp[μ(Aˆ + Bˆ)] = exp(μAˆ) exp(μBˆ) exp(−μ
2λ/2)

where given an operator Xˆ; exp(tXˆ) = 1 + tXˆ +
1
2
t
2Xˆ 2 + ... +
1
n!
t
nXˆ n + ... (the usual

exponential series implemented for operators).

Expert Solution

genius_generous answered 1 year ago

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