Application of triple integrals in quantum chemistry

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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: $\int_{0}^{\infty} x^n\,e^{-\alpha\,x}\dd x=\dfrac{n!}{\alpha^{n+1}}, \quad n\in\mathbb{N}, \alpha>0.$

$\langle r\rangle={}$

$\Box\cdot \frac{a_0}{Z}$

Multiple integrals: Applications of multiple integrals

Application of triple integrals in quantum chemistry