Chapter Questions
By substituting the wave function $\psi(x)=A x e^{-b x}$ into Eq. $7.2,$ show that a solution can be obtained only for $b=1 / a_{0}$, and find the ground-state energy.
Show that the probability density for the ground-state solution of the one-dimensional Coulomb potential energy has its maximum at $x=a_{0}$.
An electron in its ground state is trapped in the onedimensional Coulomb potential energy. What is the probability to find it in the region between $x=0.99 a_{0}$ and $x=1.01 a_{0} ?$
An electron is in an angular momentum state with $l=3$. (a) What is the length of the electron's angular momentum vector? (b) How many different possible $z$ connponents can the angular momentum vector have? List the possible $z$ components. ( $c$ ) What are the values of the angle that the $\overrightarrow{\mathbf{L}}$ vector makes with the $z$ axis?
What angles does the $\overrightarrow{\mathbf{L}}$ vector make with the $z$ axis when $l=2 ?$
List the 16 possible sets of quantum numbers $n, l, m_{l}$ of the $n=4$ level of hydrogen (as in Figure 7.6 ).
(a) What are the possible values of $l$ for $n=6 ?$ (b) What are the possible values of $m_{l}$ for $l=6 ?(c)$ What is the smallest possible value of $n$ for which $l$ can be $4 ?(d)$ What is the smallest possible $l$ that can have a $z$ component of $4 \hbar ?$
Show that the (1,0,0) and (2,0,0) wave functions listed in Table 7.1 are properly normalized.
Show by direct substitution that the $n=2, l=0, m_{l}=0$ and $n=2, l=1, m_{l}=0$ wave functions of Table 7.1 are both solutions of Eq. 7.10 corresponding to the energy of the first excited state of hydrogen.
Show by direct substitution that the wave function corresponding to $n=1, l=0, m_{l}=0$ is a solution of Eq. 7.10 corresponding to the ground-state energy of hydrogen.
Consider a thin spherical shell located between $r=0.49 a_{0}$ and $0.51 a_{0} .$ For the $n=2, l=1$ state of hydrogen, find the probability for the electron to be found in a small volume element that subtends a polar angle of $0.11^{\circ}$ and an azimuthal angle of $0.25^{\circ}$ if the center of the volume element is located at: $(a) \theta=0, \phi=0 ;$ (b) $\theta=90^{\circ}, \phi=0$; (c) $\theta=90^{\circ}, \phi=90^{\circ} ;$ (d) $\theta=45^{\circ}, \phi=0 .$ Do the calculation for all possible $m_{l}$ values.
Show that the radial probability density of the $1 s$ level has its maximum value at $r=a_{0}$.
Find the values of the radius where the $n=2, l=0$ radial probability density has its maximum values.
What is the probability of finding a $n=2, l=1$ electron between $a_{0}$ and $2 a_{0} ?$
For a hydrogen atom in the ground state, what is the probability to find the electron between $1.00 a_{0}$ and $1.01 a_{0} ?$
Find the directions in space where the angular probability density for the $l=2, m_{l}=\pm 1$ electron in hydrogen has its maxima and minima.
Find the directions in space where the angular probability density for the $l=2, m_{l}=0$ electron in hydrogen has its maxima and minima.
(a) Including the electron spin, what is the degeneracy of the $n=5$ energy level of hydrogen? (b) By adding up the number of states for each value of $l$ permitted for $n=5,$ show that the same degeneracy as part $(a)$ is obtained.
For each $l$ value, the number of possible states is $2(2 l+1)$. Show explicitly that the total number of states for each principal quantum number is $\sum_{l=0}^{n-1} 2(2 l+1)=2 n^{2} .$ This gives the degeneracy of each energy level.
Explain why each of the following sets of quantum numbers $\left(n, l, m_{l}, m_{s}\right)$ is not permitted for hydrogen. (a) $(2,2,-1,+1 / 2)$ (b) $(3,1,+2,-1 / 2)$ (c) $(4,1,+1,-3 / 2)$ (d) $(2,-1,+1,+1 / 2)$
List the excited states (in spectroscopic notation) to which the $4 p$ state can make downward transitions.
(a) A hydrogen atom is in an excited $5 g$ state, from which it makes a series of transitions by emitting photons, ending in the $1 s$ state. Show, on a diagram similar to Figure $7.19,$ the sequence of transitions that can occur. (b) Repeat part $(a)$ if the atom begins in the $5 d$ state.
(a) List in spectroscopic notation all levels with $n=7$. (b) An electron is initially in the state with $n=7, l=2$. List in spectroscopic notation all lower states to which transitions are allowed.
Consider the normal Zeeman effect applied to the $3 d$ to $2 p$ transition. (a) Sketch an energy-level diagram that shows the splitting of the $3 d$ and $2 p$ levels in an external magnetic field. Indicate all possible transitions from each $m_{l}$ state of the $3 d$ level to each $m_{l}$ state of the $2 p$ level. $(b)$ Which transitions satisfy the $\Delta m_{l}=\pm 1$ or 0 selection rule? (c) Show that there are only three different transition energies emitted.
A collection of hydrogen atoms is placed in a magnetic field of $3.50 \mathrm{~T}$. Ignoring the effects of electron spin, find the wavelengths of the three normal Zeeman components $(a)$ of the $3 d$ to $2 p$ transition; $(b)$ of the $3 s$ to $2 p$ transition.
Calculate the wavelengths of the components of the first line of the Lyman series, taking the fine structure of the $2 p$ level into account.
Calculate the energies and wavelengths of the $3 d$ to $2 p$ transition, taking into account the fine structure of both levels. How many component wavelengths might there be in the transition?
Show that the wave function $\psi(x)=A\left(x+c x^{2}\right) e^{-b x}$ gives a solution to the Schrödinger equation for the one-dimensional Coulomb potential energy. Evaluate the constants $A, b, c,$ and find the energy corresponding to this solution.
Find the probabilities for the $n=2, l=0$ and $n=2, l=1$ electron states in hydrogen to be further than $r=5 a_{0}$ from the nucleus. Which has the greater probability to be far from the nucleus?
The mean or average value of the radius $r$ can be found according to $r_{\mathrm{av}}=\int_{0}^{\infty} r P(r) d r .$ Show that the mean value of $r$ for the $1 s$ state of hydrogen is $\frac{3}{2} a_{0} .$ Why is this greater than the Bohr radius?
Find the value of $r_{\text {av }}$ (see Problem 30) for the $2 s$ and $2 p$ levels.
The mean or average value of the potential energy of the electron in a hydrogen atom can be found from $U_{\mathrm{av}}=\int_{0}^{\infty} U(r) P(r) d r .$ Find $U_{\mathrm{av}}$ in the $1 s$ state and compare with the potential energy computed with the Bohr model when $n=1$
Suppose the source of atoms in a Stern-Gerlach experiment were an oven of temperature $1000 \mathrm{~K}$. Assume the magnetic field gradient to be $10 \mathrm{~T} / \mathrm{m}$, and take the length of the magnetic field region and the field-free region between magnet and screen to be $1 \mathrm{~m}$ each. Make any other assumptions you may need and estimate the separation of the images observed on the screen.
For the $1 s, 2 s,$ and $2 p$ states of hydrogen, show that $\left(r^{-1}\right)_{\mathrm{av}}=1 / n^{2} a_{0} .$ This turns out to be a general result for any state of hydrogen. Based on this result, explain why the Bohr model gives such a good estimate for the finestructure splitting as well as for other magnetic effects due to the circulating electron.