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(e) [p, f(f)] (f) [â, Ħ] for general potential V (x)

(e) [p, f(f)] (f) [â, Ħ] for general potential V (x)

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Please answer (e) and (f)
4. Calculate the following commutators (Recall that [Â, B] = ÂB – BÂ= -[B, Â) (a) [â, î] (b) [â, î] (c) [â²,p] (d) [â, ƒ(â)] (e) [p, f(â)] (f) [î, H] for general potential V (x)
Transcribed Image Text:4. Calculate the following commutators (Recall that [Â, B] = ÂB – BÂ= -[B, Â) (a) [â, î] (b) [â, î] (c) [â²,p] (d) [â, ƒ(â)] (e) [p, f(â)] (f) [î, H] for general potential V (x)
Expert Solution
Step 1

[?,?(?)]?=(??(?)−?(?)?)?[p,F(x)]ψ=(pF(x)−F(x)p)ψ

knowing that ?=−?ℏ∂∂?p=−iℏ∂∂x you'll get

[?,?(?)]?=−?ℏ∂∂?(?(?)?)+?ℏ?(?)∂∂??=−?ℏ?∂∂??(?)−?ℏ?(?)∂∂??+?ℏ?(?)∂∂??[p,F(x)]ψ=−iℏ∂∂x(F(x)ψ)+iℏF(x)∂∂xψ=−iℏψ∂∂xF(x)−iℏF(x)∂∂xψ+iℏF(x)∂∂xψ

from where you find that [?,?(?)]=−?ℏ∂∂??(?)[p,F(x)]=−iℏ∂∂xF(x)

 

 

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