Chapter 3

Calculate (a) the mean radius, (b) the mean square radius, and (c) the most probable radius of the 1 s- \(, 2\) sand 3 s-orbitals of a hydrogenic atom of atomic number \(Z\) Hint. For the most probable radius look for the principal maximum of the radial distribution function. You will find the following integral useful: $$\int_{0}^{\infty} x^{n} \mathrm{c}^{-\alpha x} \mathrm{d} x=\frac{n !}{a^{n+1}}$$

(a) Construct a wavepacket \(\Psi=\mathrm{N} \sum_{m_{\mathrm{r}}=0}^{\infty}\left(1 / m_{l} !\right) \mathrm{e}^{\mathrm{in} \varphi}\) and normalize it to unity, Sketch the form of \(|\Psi|^{2}\) for \(0 \leq \varphi \leq 2 \pi\) (b) Calculate \(\langle\varphi\rangle,\langle\sin \varphi\rangle,\) and \(\left\langle l_{z}\right\rangle\) (c) Why is \(\left\langle l_{z}\right\rangle \leq \hbar ?\) Hint. Draw on a variety of pieces of information, including \(\sum_{n=0}^{\infty} x^{n} / n !=\mathrm{e}^{x}\) and the following integrals: $$\int_{0}^{2 \pi} \mathrm{e}^{\operatorname{scos} \phi} \mathrm{d} \varphi=2 \pi I_{0}(z) \int_{0}^{2 \pi} \cos \varphi \mathrm{e}^{\sec \varphi} \mathrm{d} \varphi=2 \pi I_{1}(z)$$ with \(I_{0}(2)=2.280 \ldots, I_{1}(2)=1.591 \ldots ;\) the \(I(z)\) are modificd Bessel functions.

Evaluate the probability that a particle of mass \(m\) in the ground state of a circular square well of radius \(a\) will be found within the circular area of radius \(1 / 2 a\)

(a) Confirm that the radius of gyration of a solid uniform sphere of radius \(R\) is \(r=\left(\frac{2}{5}\right)^{1 / 2} R .\) (b) What is the radius of gyration of a solid uniform cylinder of radius \(R\) and length \(l\) about an axis perpendicular to its principal axis? Hint. For part (a), the moment of incrtia of a sphere is \(^{2} / s M R^{2}\)

Calculate the angle that the angular momentum vector makes with the \(z\) -axis when the system is described by the wavefunction \(\psi_{l m_{i}}\). Show that the minimum angle approaches zero as \(l\) approaches infinity. Calculate the allowed angles when \(l\) is \(1,2,\) and 3