Chapter 7: Quantum Mechanics in Three Dimensions and the Hydrogen Atom

(a) What is the expectation value of the distance from the proton of an electron in a 3p state? (b) How does this compare with the expectation value in the 3 d state, calculated in Example 7.7? Discuss any differences.

Imagine two classical charges of -q, each bound to a central charge of. +q One -q charge is in a circular orbit of radius R about its +q charge. The other oscillates in an extreme ellipse, essentially a straight line from it’s +q charge out to a maximum distance ${\mathit{r}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$.The two orbits have the same energy. (a) Show that${\mathit{r}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{=}\mathbf{2}\mathit{r}$. (b) Considering the time spent at each orbit radius, in which orbit is the -q charge farther from its +q charge on average?

In hydrogen’s characteristic spectra, each series - the Lyman, the Balmer, and so on – has a “series limit,” where the wavelengths at one end of the series tend to bunch up, approaching a single limiting value. Is it at the short-wavelength or the long-wavelength end of the series that the series limit occurs, and what is it about hydrogen’s allowed energies that leads to this phenomenon? Does the infinite well have series limits?

Question: Consider an electron in the ground state of a hydrogen atom. (a) Calculate the expectation value of its potential energy. (b) What is the expectation value of its kinetic energy? (Hint: What is the expectation value of the total energy?)

Question: The kinetic energy of hydrogen atom wave functions for which lis its minimum value of 0 is all radial. This is the case for the 1s and 2s states. The 2 p state has some rotational kinetic energy and some radial. Show that for very large n, the states of largest allowed lhave essentially no radial kinetic energy. Exercise 55 notes that the expectation value of the kinetic energy (including both rotational and radial) equals the magnitude of the total energy. Compare this magnitude with the rotational energy alone,$L^2/2m{r}^2$
,assuming that n is large. That lis as large as it can be, and that$r\equiv n^2{a}_0$.

For the more circular orbits, $\mathit{\ell }\mathbf{=}\mathit{n}\mathbf{-}\mathbf{1}$and

$\mathit{P}\mathbf{\left(}\mathit{r}\mathbf{\right)}\mathbf{\propto }{\mathit{r}}^{\mathbf{2}\mathbf{n}}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{r}\mathbf{/}\mathbf{n}{\mathbf{a}}_{\mathbf{0}}}$

a) Show that the coefficient that normalizes this probability is

localid="1660047077408" ${\left(\frac{2}{n{a}_0}\right)}^{\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}}\frac{\mathbf{1}}{\mathbf{\left(}\mathbf{2}\mathbf{n}\mathbf{\right)}\mathbf{!}}$

b) Show that the expectation value of the radius is given by

$\overline{\mathbf{r}}\mathbf{=}\mathit{n}\left(n+\frac{1}{2}\right){\mathit{a}}_{\mathbf{0}}$

and the uncertainty by

$\mathit{\Delta }\mathit{r}\mathbf{=}\mathit{n}{\mathit{a}}_{\mathbf{0}}\sqrt{\frac{\mathbf{n}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{4}}}$

c) What happens to the ratio$\mathit{\Delta }\mathit{r}\mathbf{/}\overline{\mathbf{r}}$in the limit of large n? Is this large-n limit what would be expected classically?

Which electron transitions in singly ionized helium yield photon in the 450 - 500 nm(blue) portion of the visible range, and what are their wavelengths?

Doubly ionized lithium, $\mathit{L}{\mathit{i}}^{\mathbf{2}\mathbf{+}}$absorbs a photon and jumps from the ground state to its n=2level. What was the wavelength of the photon?

Roughly, how does the size of a triply ionized beryllium ion compare with hydrogen?

For an electron in the$\left(n,l,m_{\mathcal{l}}\right)\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}$ state in a hydrogen atom, (a) write the solution of the time-independent Schrodinger equation,

(b) verify explicitly that it is a solution with the expected angular momentum and energy.