A mathematical solution of the azimuthal equation (7-22) is $\mathbf{\Phi }\left(\phi \right)\mathbf{=}{\mathbf{Ae}}^{\sqrt{\mathbf{-}\mathbf{D\phi }}}\mathbf{+}{\mathbf{Be}}^{{}^{\sqrt{\mathbf{-}\mathbf{D\phi }}}}$ , which applies when $\mathbf{D}$ is negative, (a) Show that this simply cannot meet itself smoothly when it finishes a round trip about the z-axis. The simplest approach is to consider $\mathbf{\phi }\mathbf{=}\mathbf{0}$ and $\mathbf{\phi }\mathbf{=}\mathbf{2}\mathbf{\pi }$. (b) If $\mathbf{D}$ were $\mathbf{0}$, equation (7-22) would say simply that the second derivative $\mathbf{\Phi }\left(\phi \right)$of is $\mathbf{0}$ . Argue than this too leads to physically unacceptable solution, except in the special case of $\mathbf{\Phi }\mathbf{\left(}\mathbf{\phi }\mathbf{\right)}$ being constant, which is covered by the ${\mathbf{m}}_{\mathbf{l}}\mathbf{=}\mathbf{0}$ , case of solutions (7-24).

An electron confined to a cubic 3D infinite well 1 nu on a side.

1. What are the three lowest different energies?
2. To how many different states do these three energies correspond?

To conserve momentum, an atom emitting a photon must recoil, meaning that not all of the energy made available in the downward jump goes to the photon. (a) Find a hydrogen atom's recoil energy when it emits a photon in a n = 2 to n = 1 transition. (Note: The calculation is easiest to carry out if it is assumed that the photon carries essentially all the transition energy, which thus determines its momentum. The result justifies the assumption.) (b) What fraction of the transition energy is the recoil energy?

(a) For one-dimensional particle in a box, what is the meaning of n? Specifically, what does knowing n tell us? (b) What is the meaning of n for a hydrogen atom? (c) For a hydrogen atom. What is the meaning of l and ${\mathit{m}}_{\mathbf{l}}$?

Consider two particles that experience a mutual force but no external forces. The classical equation of motion for particle 1 is ${\dot{\mathbf{v}}}_{\mathbf{1}}\mathbf{=}{\mathbf{F}}_{\mathbf{2}\mathbf{on}\mathbf{1}}\mathbf{/}{\mathbf{m}}_{\mathbf{1}}$, and for particle 2 is ${\dot{\mathbf{v}}}_{\mathbf{2}}\mathbf{=}{\mathbf{F}}_{\mathbf{1}\mathbf{on}\mathbf{2}}\mathbf{/}{\mathbf{m}}_{\mathbf{2}}$, where the dot means a time derivative. Show that these are equivalent to ${\dot{\mathbf{v}}}_{\mathbf{cm}}\mathbf{=}\mathbf{constant}$, and ${\dot{\mathbf{v}}}_{\mathbf{rel}}\mathbf{=}{\mathbf{F}}_{\mathbf{Mutual}}\mathbf{/}\mathbf{\mu }$ . Where, ${\dot{\mathbf{v}}}_{\mathbf{cm}}\mathbf{=}\left(m_1{\dot{v}}_1+m_2{\dot{v}}_2\right)\mathbf{/}\left(m_1{m}_2\right)\mathbf{,}{\mathbf{F}}_{\mathbf{Mutual}}\mathbf{=}\mathbf{-}{\mathbf{F}}_{\mathbf{ion}\mathbf{2}}\mathbf{}\mathbf{and}\mathbf{}\mathbf{\mu }\mathbf{=}\frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{m}}_{\mathbf{2}}}{\left(m_1+m_2\right)}$.

In other words, the motion can be analyzed into two pieces the center of mass motion, at constant velocity and the relative motion, but in terms of a one-particle equation where that particle experiences the mutual force and has the “reduced mass” $\mathbf{\mu }$.

Can the transition $\mathbf{2}\mathbf{s}\to \mathbf{1}\mathbf{s}$ in the hydrogen atom occur by electric dipole radiation? The lifetime of the 2 s is known to be unusual. Is it unusually short or long?