Problem 33
(a) The doubly charged ion \(N^{2+}\) is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the \(N^{2+}\) ion? (b) Estimate the energy of the least strongly bound level in the \(L\) shell of \(N^{2+}\). (c) The doubly charged ion \(P^{2+}\) is formed by removing two electrons from a phosphorus atom. What is the ground-state electron configuration for the \(P^{2+}\) ion? (d) Estimate the energy of the least strongly bound level in the \(M\) shell of \(P^{2+}\).
Problem 38
The energies for an electron in the \(K, L,\) and \(M\) shells of the tungsten atom are -69,500 eV, -12,000 eV, and -2200 eV, respectively. Calculate the wavelengths of the \(K_\alpha\) and \(K_\beta\) x rays of tungsten.
Problem 40
An electron is in a three-dimensional box with side lengths \(L_X =\) 0.600 nm and \(L_Y = L_Z = 2L_X\). What are the quantum numbers \(n_X, n_Y,\) and \(n_Z\) and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy due to spin)?
Problem 41
A particle is in the three-dimensional cubical box of Section 41.2. (a) Consider the cubical volume defined by \(0 \leq x \leq L/4, 0 \leq y \leq L/4\), and \(0 \leq z \leq L/4\). What fraction of the total volume of the box is this cubical volume? (b) If the particle is in the ground state \((n_X = 1, n_Y = 1, n_Z = 1)\), calculate the probability that the particle will be found in the cubical volume defined in part (a). (c) Repeat the calculation of part (b) when the particle is in the state \(n_X = 2, n_Y = 1, n_Z = 1\).
Problem 46
A particle is described by the normalized wave function \(\psi$$(x, y, z)\) = \(Axe$${^-}{^a}{^x}^2$$e$${^-}{^\beta}$${^y}^2$$e$${^-}{^y}^z$$^2\), where \(A\), \(\alpha\),\(\beta\), and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume \(dx\) \(dy\) \(dz\) centered at the point \((x_0\), \(y_0\), \(z_0\)) is \(\mid$$\psi$$(x_0\), \(y_0\), \(z_0\))\(\mid$$^2\) \(dx\) \(dy\) \(dz\). (a) At what value of \(x_0\) is the particle most likely to be found? (b) Are there values of \(x_0\) for which the probability of the particle being found is zero? If so,at what \(x_0$$?\)
Problem 47
(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number \(n\) is \(2n^2\). \([Hint\): The sum of the first \(N\) integers 1 + 2 + 3 + \(\cdots\) + \(N\) is equal to \(N$$(N + 1)\)/2.] (b) Which shell has 50 states?
Problem 48
(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is \(\sqrt{20}\) \(\hbar$$?\) (b) What are the largest and smallest values of the \(z\)-component of the orbital angular momentum (in terms of \(\hbar\)) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of \(\hbar\)) for the electron in part (a)\(?\) (d) What are the largest and smallest values of the orbital angular momentum (in terms of \(\hbar\)) for an electron in the \(M\) shell of hydrogen?
Problem 51
The normalized radial wave function for the \(2p\) state of the hydrogen atom is \(R_2{_p}\) = \(( 1/ \sqrt{24a^5}\))\(re$$^-{^r}{^/}{^2}{^a}\). After we average over the angular variables, the radial probability function becomes \(P$$(r)\) \(dr\) = \((R_2{_p}$$)^2\)r\(^2\) \(dr\). At what value of \(r\) is \(P$$(r)\) for the \(2p\) state a maximum? Compare your results to the radius of the \(n\) = 2 state in the Bohr model.
Problem 55
Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a \(p\) state to an \(s\) state. She finds that the line at 575.050 nm has actually split into three lines, with adjacent lines 0.0462 nm apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?
Problem 64
A hydrogen atom initially in an \(n\) = \(3,\) \(l\) = 1 state makes a transition to the \(n\) = \(2\), \(l\) = \(0\), \(j\) = \\(\frac{1}{2}\\) state. Find the difference in wavelength between the following two photons: one emitted in a transition that starts in the \(n\) = \(3\), \(l\) = \(1\), \(j\) = \\(\frac{3}{2}\\) state and one that starts instead in the \(n\) = \(3\), \(l\) = \(1\), \(j\) = \\(\frac{1}{2}\\) state. Which photon has the longer wavelength?
