Problem 2
Model a hydrogen atom as an electron in a cubical box with side length \(L\). Set the value of \(L\) so that the volume of the box equals the volume of a sphere of radius \(a\) = 5.29 \(\times\) 10\(^{-11}\) m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.
Problem 3
A photon is emitted when an electron in a threedimensional cubical box of side length 8.00 \(\times\) 10\(^{-11}\) m makes a transition from the n\(_X\) = 2, n\(_Y\) = 2, n\(_Z\) = 1 state to the n\(_X\) = 1, n\(_Y\) = 1, n\(_Z\) = 1 state. What is the wavelength of this photon?
Problem 6
What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length 1.00 \(\times\) 10\(^{-14}\) m, the approximate diameter of a nucleus?
Problem 7
Consider an electron in the \(N\) shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of \(\hslash\) and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of \(\hslash\) and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of \(\hslash\) and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the z-direction to its orbital angular momentum in the z-direction?
Problem 8
An electron is in the hydrogen atom with \(n\) = 5. (a) Find the possible values of \(L\) and \(L$$_z\) for this electron, in units of \(\hslash\). (b) For each value of \(L\), find all the possible angles between \(\vec{L}\) and the z-axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(L\) S and the z-axis?
Problem 9
The orbital angular momentum of an electron has a magnitude of 4.716 \(\times\) 10\(^{-34}\) {kg\(\cdot\) m\(^2\)/s. What is the angular momentum quantum number \(l\) for this electron?
Problem 13
Calculate, in units of \(\hslash\), the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of 2, 20, and 200. Compare each with the value of n\(\hslash\) postulated in the Bohr model. What trend do you see?
Problem 14
(a) Make a chart showing all possible sets of quantum numbers \(l\) and \(m$$_l\) for the states of the electron in the hydrogen atom when n = 4. How many combinations are there? (b) What are the energies of these states?
Problem 15
a) How many different 5\(g\) states does hydrogen have? (b) Which of the states in part (a) has the largest angle between \(\vec L\) and the z-axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between \(\vec L\) and the z-axis, and what is that angle?
Problem 17
Show that \(\Phi\)(\(\phi\)) = \(e$$^{im_l}$$^\phi\) = \(\Phi\)(\(\phi\) + 2\(\pi\)) (that is, show that \(\Phi\) (\(\phi\)) is periodic with period 2\(\pi\)) if and only if m\(_l\) is restricted to the values 0, \(\pm\)1, \(\pm\)2,.... (\(Hint\): Euler's formula states that \(e$$^i$$^\phi\) = cos \(\phi\) + \(i\) sin \(\phi\).)
