Multiple integrals: Applications of multiple integrals
Application of triple integrals in quantum chemistry
Calculate the averge distance \(\langle r\rangle\) of the electron to the nucleus in the \(3p_z\) orbital of a hydrogen-like atom with nuclear charge \(Z\).
As a reminder: the wave function is defined in spherical coordinates as \[\psi_{3p_z}=\psi_{3p_z} = 2N_{3}\,(6-\rho)\,\rho\, e^{-\rho/3}\cos\theta\] where \[\rho=\frac{Z\,r}{a_0}\quad\text{and}\quad N_3=\frac{1}{81}\sqrt{\dfrac{Z^3}{2\pi a_0^3}}\] You may also use the following integral in the calculation: \[\int_{0}^{\infty} x^n\,e^{-\alpha\,x}\dd x=\dfrac{n!}{\alpha^{n+1}}, \quad n\in\mathbb{N}, \alpha>0.\]
| \(\langle r\rangle={}\) |
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