For an electron in 1s orbital of He⁺, the average value of r, (r) is

Concept:

• The average value in Quantum Mechanics is the average result obtained when a large number of measurements are made on identical systems or systems in the same state.
• For an observable quantity A, the average value or expectation value is given by,

\(\left\langle {\rm{A}} \right\rangle {\rm{ = }}\int {{{\rm{\Psi }}^{\rm{*}}}{\rm{A\Psi d\tau }}} \)

Explanation:

• The normalized ground state wave function for a one-electron system like an H atom (He⁺) is:

\({\Psi _{1s}} = N{e^{ - {r \over {Z{a_ \circ }}}}}\)  where, N is the normalization constant and Z is the atomic number of the nuclei.

• The average value or expectation value of r is given by,

\(\left\langle r \right\rangle = \int {{\Psi ^ * }_nr{\Psi _n}} d\tau \)

• Now, for an electron in 1s orbital of He+, the average value of r, (r) is given by,

\(\left\langle r \right\rangle = \int {{\Psi ^ * }_{1s}r{\Psi _{1s}}} d\tau \), where \(d\tau = {r^2}dr\sin \theta d\theta d\phi \)

\( = N\int_0^ \propto {r3{e^{ - {{2r} \over {Z{a_ \circ }}}}}} dr\int_0^\pi {sin\theta d\theta } \int_0^{2\pi } {d\phi } \)

\( = {{3{a_ \circ }} \over {2Z}}\)

• Now, for the He+ ion the atomic number

(Z) = 2.

• Thus, the average value of r, \(\left\langle r \right\rangle \)  for He+ ion is

​\( = {3 \over 4}{a_ \circ }\).

Conclusion:

Hence, for an electron in 1s orbital of He+, the average value of r, (r) is \({3 \over 4}{a_ \circ }\)