A set $S \subset \mathbf{R}^n$ is called starlike if there is some base point $x \in S$ such that for every point $y \in S$, the line between $x$ and $y$ is contained in $S$.
(a) Prove that every convex subset of $\mathbf{R}^n$ is starlike.
(b) Prove that every starlike subset of $\mathbf{R}^n$ is connected.

(a) Prove that every convex subset of $\mathbf{R}^n$ is starlike.

Proof: Let $C$ be a convex subset of $\mathbf{R}^n$. We can select any point $x \in C$ as the base point. By the definition of convexity, for any point $y \in C$, the line segment between $x$ and $y$ is contained in $C$. This is because if we consider two points $x, y \in C$, then for every $t \in [0,1]$, the point $z = tx + (1-t)y$ is in $C$. Thus, the entire line between $x$ and $y$ is contained in $C$. Hence, by the definition of starlike, $C$ is starlike.

(b) Prove that every starlike subset of $\mathbf{R}^n$ is connected.

Proof: Let $S$ be a starlike subset of $\mathbf{R}^n$ with base point $x$. To prove that $S$ is connected, we must show that for any two points $y, z \in S$, there exists a path between $y$ and $z$ contained entirely within $S$.

Because $S$ is starlike, the line segments from $x$ to $y$ and from $x$ to $z$ are both contained in $S$. Therefore, we can create a path from $y$ to $z$ by first moving along the line segment from $y$ to $x$, and then moving along the line segment from $x$ to $z$. Since both of these line segments are contained in $S$, this entire path is contained in $S$.

Therefore, for any two points in $S$, there is a path between them that lies entirely within $S$. This is exactly the definition of a connected set, so $S$ is connected.

Express each of the following quantities in terms of $\hbar$, e, c, $m=$ electron mass, $M=$ proton mass. Also give a rough estimate of numerical size for each.
(a) Bohr radius $(\mathrm{cm})$.
(b) Binding energy of hydrogen (eV).
(c) Bohr magneton (choosing your own unit).
(d) Compton wavelength of an electron $(\mathrm{cm})$.
(e) Classical electron radius $(\mathrm{cm})$.
(f) Electron rest energy $(\mathrm{MeV})$.
(g) Proton rest energy $(\mathrm{MeV})$.
(h) Fine structure constant.
(i) Typical hydrogen fine-structure splitting $(\mathrm{eV})$.
(a) $\mathrm{a}=\hbar^2 / m e^2=5.29 \times 10^{-9} \mathrm{~cm}$.
(b) $\mathrm{E}=m e^4 / 2 \hbar^2=13.6 \mathrm{eV}$.
(c) $\mu_B=e \hbar / 2 m c=9.27 \times 10^{-21} \mathrm{erg} \cdot \mathrm{Gs}^{-1}$.
(d) $\lambda=2 \pi \hbar / m c=2.43 \times 10^{-10} \mathrm{~cm}$.
(e) $r_e=e^2 / m c^2=2.82 \times 10^{-13} \mathrm{~cm}$.
(f) $E_e=m c^2=0.511 \mathrm{MeV}$.
(g) $E_p=M c^2=938 \mathrm{MeV}$.
(h) $\alpha=e^2 / \hbar c=7.30 \times 10^{-3} \approx 1 / 137$.
(i) $\mathrm{AE}=e^8 m c^2 / 8 \hbar^2 c^4=\frac{1}{8} \alpha^4 m c^2=1.8 \times 10^{-4} \mathrm{eV}$.

(a) Bohr radius (in cm): $a_0 = \frac{\hbar^2}{me^2} = 5.29 \times 10^{-9} \, \text{cm}$

(b) Binding energy of hydrogen (in eV): $E = \frac{me^4}{2\hbar^2} = 13.6 \, \text{eV}$

(c) Bohr magneton (in erg·Gauss$^{-1}$): $\mu_B = \frac{e\hbar}{2mc} = 9.27 \times 10^{-21} \, \text{erg} \cdot \text{Gauss}^{-1}$

(d) Compton wavelength of an electron (in cm): $\lambda = \frac{2\pi\hbar}{mc} = 2.43 \times 10^{-10} \, \text{cm}$

(e) Classical electron radius (in cm): $r_e = \frac{e^2}{mc^2} = 2.82 \times 10^{-13} \, \text{cm}$

(f) Electron rest energy (in MeV): $E_e = mc^2 = 0.511 \, \text{MeV}$

(g) Proton rest energy (in MeV): $E_p = Mc^2 = 938 \, \text{MeV}$

(h) Fine structure constant: $\alpha = \frac{e^2}{\hbar c} = 7.30 \times 10^{-3} \approx \frac{1}{137}$

(i) Typical hydrogen fine-structure splitting (in eV): $\Delta E = \frac{e^8mc^2}{8\hbar^2c^4} = \frac{1}{8} \alpha^4 mc^2 = 1.8 \times 10^{-4} \, \text{eV}$

Each of these equations expresses a physical quantity in terms of fundamental constants: Planck’s constant ($\hbar$), the speed of light ($c$), the elementary charge ($e$), and the masses of the electron ($m$) and proton ($M$). The numerical estimates also provide a sense of the typical scales of these quantities.

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