Before two nuclei can react, they must come within $1 \mathrm{fm}$ or so of one another, and because of their Coulomb repulsion, this requires a lot of energy. (a) Calculate the energy needed to bring a proton from infinity to a separation of $1 \mathrm{fm}$ from an alpha particle. $(1 \mathrm{fm}=$ distance from surface to surface. $)$ (b) Do the same for an alpha particle and a uranium nucleus. [Use the above eqution to estimate the radii involved.]

Imagine a hydrogen atom in which the electron has no spin [so that the only magnetic moment is the orbital magnetic moment given by (9.35)]. The atom is placed in a magnetic field $B=1.5 \mathrm{~T}$ along the $z$ axis. (a) Explain the effect of the $B$ field on the $1 s$ and $2 p$ states of the hypothetical atom. Sketch the energy levels. (b) When $B=0$, there is a single spectral line corresponding to the $2 p \rightarrow 1 s$ transition. How many lines does this become when $B$ is switched on? (c) What is the fractional separation $\Delta f / f_0$ between adjacent lines?

We have said that a classical picture of the electron as a spinning ball of matter is unsatisfactory. To illustrate this, consider the following: Modern measurements show that the electron's radius is certainly less than $10^{-18} \mathrm{~m}$. Write an expression for the angular momentum of a uniform spinning ball of mass $m_{\mathrm{e}}$, radius $r$, and equatorial speed $v$. By equating this to the observed spin $\sqrt{3} \hbar / 2$, find the minimum possible value of $v$. What is $v / c$ ?

Calculate the total kinetic energy released in the $\beta$ decay of a single free neutron.

(a) Use the information in above Fig. to calculate the ground-state configurations of the following atoms: ${ }_{30} \mathrm{Zn},{ }_{31} \mathrm{Ga},{ }_{39} \mathrm{Y},{ }_{45} \mathrm{Cd},{ }_{53} \mathrm{I},{ }_{54} \mathrm{Xe},{ }_{55} \mathrm{Cs}$. (b) What is the total spin angular momentum for each of these atoms? (c) What are their total orbital angular momenta?

(a) Use the information in the above Fig. to calculate the ground-state configurations of the following atoms: ${ }_{30} \mathrm{Zn},{ }_{50} \mathrm{Hg},{ }_{37} \mathrm{Rb}$, ${ }_{55} \mathrm{Cs}$. (b) What is the total angular momentum of each of these atoms?

When Rutherford first observed the reaction , he assumed that the alpha particle simply knocked a proton out of the ${ }^{14} \mathrm{~N}$ in the process $\alpha+{ }^{14} \mathrm{~N} \rightarrow \alpha+\mathrm{p}+{ }^{13} \mathrm{C}$. Given that the incident alpha particles had about $5.6 \mathrm{MeV}$, was this reaction possible?

Use the above equation and the data of Appendix D to predict $\Delta K$, the kinetic energy released in the reaction $\mathrm{p}+{ }^7 \mathrm{Li} \rightarrow \alpha+\alpha$.

(a) It is known that a certain hydrogen atom has $n=5$ and $m=2$. How many different states are consistent with this information? (b) Answer the same question (in terms of $n$ and $m$ ) for arbitrary values of $n$ and $m$.

It is known that a certain hydrogen atom has a definite value of $l$. (a) What does this statement tell you about the angular momentum? (b) What are the allowed energies consistent with this information?

The energy levels of a particle in a cubical box can be found from tha above eq. by setting $a=b=c$. Calculate the lowest eight energy levels for a particle in a three-dimensional, rigid cubical box. Draw an energy-level diagram for these levels, showing their quantum numbers, energies, and degeneracies.

In most nuclear reactions the total nucleon number is conserved; that is, $\sum A_{\text {initial }}=\sum A_{\text {final }}$, where $A=Z+N$ for each nucleus, as usual. Show that, except in reactions involving $\beta$ decay, both $\sum Z$ and $\sum N$ are separately conserved. [Hint: Total charge is always conserved, and except in $\beta$ decay, all the charge is carried by protons.]

Consider the $\alpha$ decay of a nucleus $\mathrm{X}$ at rest, $\mathrm{X} \rightarrow \mathrm{Y}+\alpha$, where $\mathrm{Y}$ denotes the offspring nucleus. (a) Prove that because the $\alpha$ is much lighter than Y, almost all the available kinetic energy goes to the $\alpha$; specifically, prove that

$$K_\alpha=\frac{m_{\mathrm{Y}}}{m_\alpha} K_{\mathrm{Y}}$$

[Hint: Use conservation of momentum and treat all particles nonrelativistically.] (b) In the $\alpha$ decay of a typical naturally radioactive nucleus, what percentage of the available kinetic energy does the alpha particle take? (c) Assuming that $\beta$ decay has the form $\mathrm{X} \rightarrow \mathrm{Y}+\mathrm{e}$, use the relation corresponding to (17.86) to find what percentage of the available kinetic energy goes to the electron. (Your answer here is really an upper bound, since some energy can go to the neutrino. It is also only approximate since the electron can be fairly relativistic.)

Use the data in Appendix D to write down the whole of the radioactive series that begins with ${ }^{235} \mathrm{U}$. Draw the series on a plot of $Z$ against $N$ as in the above figure.