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Chapter 3: Problem 2

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: $$\int_{0}^{\infty} x^{n} \mathrm{c}^{-\alpha x} \mathrm{d} x=\frac{n !}{a^{n+1}}$$

Short Answer

Expert verified
The mean radius, mean square radius, and the most probable radius can be calculated with the wavefunctions of the electron and understanding of the radial distribution function. The exact values are dependent on the specifics of the wavefunctions, which differ according to each atomic orbital. Mathematical integration is used for both the mean radius and mean square radius, whereas the most probable radius is found by looking for the maximum of the radial distribution function.

Step by step solution

01

Calculating the Mean Radius

To find the mean radius, we integrate over all possible radii, which means integrating from zero to infinity. The mean radius formula for an electron in a hydrogen atom is \( = \frac{\int_0^\infty r * r^2*R^2_{nl} * dr}{\int_0^\infty r^2*R^2_{nl}*dr}\) where \(R_{nl}\) is radial part of the wavefunction. Since \( R_{nl}\) is different for each quantum number, the mean radius varies.
02

Calculating the Mean Square Radius

The formula to calculate the mean square radius is quite similar to the one for mean radius. The only difference is in the numerator, which now is \(r^2\), thus \( = \frac{\int_0^\infty r^2 * r^2*R^2_{nl} * dr}{\int_0^\infty r^2*R^2_{nl}* dr}\) just like with the mean radius, the value of the mean square radius changes for different quantum numbers.
03

Finding the Most Probable Radius

To find the most probable radius, we need to find the maximum of the radial distribution function \( |R_{nl}(r)|^2*r^2 \). This function represents the probability density of finding an electron at a certain distance from the nucleus, so its maximum indicates the most probable radius.

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Most popular questions from this chapter

(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 \(^{2} / s M R^{2}\)

(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 \(1 / 2 a\)

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
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