Average Radius for 2s
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Exercise:
Calculate the expectation value for the radius operator hat r in a s orbital of the hydrogen atom. The corresponding wave function is psi_sr fracsqrtpia_^/left-fracra_right e^-r/a_
Solution:
The expectation value is langle hat r rangle _V psi_s^*r r psi_sr textrmdV pi^ _^infty |psi_sr|^ r r^ textrmdr fracpipi a_^_^infty left-fracra_right^ e^-r/a_ r^ textrmdr frac a_^_^infty left-fracra_+fracr^a_^right e^-r/a_ r^ textrmdr frac a_^_^infty left I_-fraca_ I_ + fraca_^ I_ right with our usual definition I_n _^infty r^n e^-r/a textrmdr n! a^n+ We thus find langle hat r rangle frac a_^ left a_^ - fraca_ a_^+fraca_^ a_^ right frac a_^ a_^ a_ The calculation can be verified or exted to other states using the Mathematica file linked to from this exercise.
Calculate the expectation value for the radius operator hat r in a s orbital of the hydrogen atom. The corresponding wave function is psi_sr fracsqrtpia_^/left-fracra_right e^-r/a_
Solution:
The expectation value is langle hat r rangle _V psi_s^*r r psi_sr textrmdV pi^ _^infty |psi_sr|^ r r^ textrmdr fracpipi a_^_^infty left-fracra_right^ e^-r/a_ r^ textrmdr frac a_^_^infty left-fracra_+fracr^a_^right e^-r/a_ r^ textrmdr frac a_^_^infty left I_-fraca_ I_ + fraca_^ I_ right with our usual definition I_n _^infty r^n e^-r/a textrmdr n! a^n+ We thus find langle hat r rangle frac a_^ left a_^ - fraca_ a_^+fraca_^ a_^ right frac a_^ a_^ a_ The calculation can be verified or exted to other states using the Mathematica file linked to from this exercise.