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The momentum space wave function for the ground state of hydro
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The momentum space wave function for the ground state of hydrogen.

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ψ=1πa3e−r/a⇒ϕ(p)=1(2πℏ)3/21πa3∫e−ip⋅r/ℏe−r/ar2sinθdrdθdϕ

Suggested axes, p⋅r=prcosθ.

Integral

ϕ(p)=2π(2πaℏ)3/21π∫0∞r2e−r/a[∫0πe−iprcosθ/ℏsinθdθ]dr

∫0πe−iprcosθ/ℏsinθdθ=ℏipre−iprcosθ/ℏ|0π                                       =ℏipr(eipr/ℏ−e−ipr/ℏ)                                       =2ℏprsin(prℏ)

ϕ(p)=1π21(aℏ

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Question Text
The momentum space wave function for the ground state of hydrogen.
TopicAll topics
SubjectPhysics
ClassClass 12
Answer TypeText solution:1