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

Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as

$\frac{\mathbf{1}}{{\left(2a_0\right)}^{\mathbf{3}\mathbf{/}\mathbf{2}}\sqrt{\mathbf{3}}{\mathbf{a}}_{\mathbf{0}}}$

Classically, what happens when a moving object has a head-on elastic collision with a stationary object of exactly equal mass? What if it strikes an object of smaller mass? Of larger mass? How do these ideas relate to Rutherford’s conclusion about the nature of the atom?

A hydrogen atom electron is in a 2p state. If no experiment has been done to establish a z-component of angular momentum, the atom is equally likely to be found with any allowed value of $L_Z$. Show that if the probability densities for these different possible states are added (with equal weighting), the result is independent of both $\mathit{\varphi }$and$\theta$

A wave function with a non-infinite wavelength-however approximate it might be- has nonzero momentum and thus nonzero kinetic energy. Even a single "bump" has kinetic energy. In either case, we can say that the function has kinetic energy because it has curvature- a second derivative. Indeed, the kinetic energy operator in any coordinate system involves a second derivative. The only function without kinetic energy would be a straight line. As a special case, this includes a constant, which may be thought of as a function with an infinite wavelength. By looking at the curvature in the appropriate dimension(s). answer the following: For a givenn,isthe kinetic energy solely

(a) radial in the state of lowest l- that is, l=0; and

(b) rotational in the state of highest l-that is, l=n-1?

We have noted that for a given energy, as lincreases, the motion is more like a circle at a constant radius, with the rotational energy increasing as the radial energy correspondingly decreases. But is the radial kinetic energy 0 for the largest lvalues? Calculate the ratio of expectation values, radial energy to rotational energy, for the$\mathbf{(}\mathit{n}\mathbf{,}\mathit{l}\mathbf{,}{\mathit{m}}_{\mathbf{t}}\mathbf{)}\mathbf{=}\left(2.1,+1\right)$state. Use the operators

$$\begin{gathered} {\overbrace{\mathbf{K}\mathbf{E}}}_{\mathbf{r}\mathbf{a}\mathbf{d}}\mathbf{=}\frac{\mathbf{-}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{2}}}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{r}}\left(r\frac{\partial }{\partial r}\right) \\ {\overbrace{\mathbf{K}\mathbf{E}}}_{\mathbf{r}\mathbf{a}\mathbf{d}}\mathbf{=}\frac{{\mathbf{h}}^{\mathbf{2}}\mathbf{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{2}\mathbf{m}{\mathbf{r}}^{\mathbf{2}}} \end{gathered}$$

Which we deduce from equation (7-30).

An electron is in the 3d state of a hydrogen atom. The most probable distance of the electron from the proton is$\mathbf{9}{\mathit{a}}_{\mathbf{o}}$. What is the probability that the electron would be found between$\mathbf{8}{\mathit{a}}_{\mathbf{o}}\mathit{a}\mathit{n}\mathit{d}\mathbf{10}{\mathit{a}}_{\mathbf{o}}$?

Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen $\mathbf{(}\mathit{i}\mathbf{.}\mathit{e}\mathbf{.}\mathbf{,}\mathit{l}\mathbf{=}\mathit{n}\mathbf{-}\mathbf{1}\mathbf{)}$, the radial probability is of the form

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

Show that the most probable radius is given by

${\mathit{r}}_{\mathbf{m}\mathbf{o}\mathbf{s}\mathbf{t}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{b}\mathbf{a}\mathbf{b}\mathbf{l}\mathbf{e}}\mathbf{=}{\mathit{n}}^{\mathbf{2}}{\mathit{a}}_{\mathbf{o}}$

For states where l = n - t the radial probability assumes the general form given in Exercise 54. The proportionality constant that normalizes this radial probability is given in Exercise 64.

(a) Show that the expectation value of the hydrogen atom potential energy is exactly twice the total energy. (It turns out that this holds no matter what l may be)

(b) Argue that the expectation value of the kinetic energy must be the negative of the total energy.

For a hydrogen atom in the ground state. determine (a) the most probable location at which to find the electron and (b) the most probable radius at which to find the electron, (c) Comment on the relationship between your answers in parts (a) and (b).

Consider an electron in the ground state of a hydrogen atom. (a) Sketch plots of E and U(r) on the same axes (b) Show that, classically, an electron with this energy should not be able to get farther than $\mathbf{2}{\mathit{a}}_{\mathbf{0}}$from the proton. (c) What is the probability of the electron being found in the classically forbidden region?