Problem 37
An electron in a hydrogen atom is in the \(2 s\) state. Calculate the probability of finding the electron within a Bohr radius \(\left(a_{0}=0.05295 \mathrm{nm}\right)\) of the proton. The ground-state wave function for hydrogen is: $$ \psi_{2 s}(r)=\frac{1}{4 \sqrt{2 \pi a_{0}^{3}}}\left(2-\frac{r}{a_{0}}\right) e^{-r / 2 a_{0}}. $$ The integral is a bit tedious, so you may want consider using mathematical programs such as Mathcad, Mathematica, etc., or doing the integral online at http://integrals.wolfram.com/index.jsp.
Problem 38
Calculate the energy needed to change a single ionized helium atom into a double ionized helium atom (that is, change it from \(\mathrm{He}^{+}\) into \(\mathrm{He}^{2+}\) ). Compare it to the energy needed to ionize the hydrogen atom. Assume that both atoms are in their fundamental state.
Problem 39
A \(\mathrm{He}^{+}\) ion consists of a nucleus ( containing two protons and two neutrons) and a single electron. Obtain the Bohr radius for this system.
Problem 41
The binding energy of an extra electron when As atoms are doped in a Si crystal may be approximately calculated by considering the Bohr model of a hydrogen atom. a) Show the ground energy of hydrogen-like atoms in terms of the dielectric constant and the ground state energy of a hydrogen atom. b) Calculate the binding energy of the extra electron in a Si crystal. (The dielectric constant of Si is about 10.0 , and the effective mass of extra electrons in a Si crystal is about \(20.0 \%\) of that of free electrons.)
Problem 42
What is the wavelength of the first visible line in the spectrum of doubly ionized lithium? Begin by writing down the formula for the energy levels of the electron in doubly ionized lithium - then consider energy-level differences that give energies in the appropriate (visible) range. Express the answer in terms of "the transition from state \(n\) to state \(n^{\prime}\) is the first visible, with wavelength \(X\)."
Problem 43
Following the steps outlined in our treatment of the hydrogen atom, apply the Bohr model of the atom to derive an expression for a) the radius of the \(n\) th orbit, b) the speed of the electron in the \(n\) th orbit, and c) the energy levels in a hydrogen-like ionized atom of charge number \(Z\) that has lost all of its electrons except for one electron. Compare the results with the corresponding ones for the hydrogen atom.
Problem 45
Consider an electron in the hydrogen atom. If you are able to excite its electron from the \(n=1\) shell to the \(n=2\) shell with a given laser, what kind of a laser (that is, compare wavelengths) will you need to excite that electron again from the \(n=2\) to the \(n=3\) shell? Explain.
Problem 46
A low-power laser has a power of \(0.50 \mathrm{~mW}\) and a beam diameter of \(3.0 \mathrm{~mm}\). a) Calculate the average light intensity of the laser beam, and b) compare it to the intensity of a 100 -W light bulb producing light viewed from \(2.0 \mathrm{~m}\).
Problem 47
A ruby laser consists mostly of alumina \(\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)\) and a small amount of chromium ions, responsible for its red color. One such laser of power \(3.00 \mathrm{~kW}\) emits light pulse of duration \(10.0 \mathrm{~ns}\) and of wavelength \(685 \mathrm{nm}\). a) What is the energy of the photons in the pulse? b) Determine the number of chromium atoms undergoing stimulated emission to produce this pulse.
Problem 48
You hold in your hands both a green \(543-n m, 5.00-m W\) laser and a red, \(633-\mathrm{nm}, 4.00\) -mW laser. Which one will produce more photons per second, and why?
