Homework Problems

1	(a) A distance scale is shown below the wavefunctions and probability densities illustrated in section 5.3. Compare this with the order-of-magnitude estimate derived in section 5.4 for the radius r at which the wavefunction begins tailing off. Was the estimate in section 5.4 on the right order of magnitude? (b) Although we normally say the moon orbits the earth, actually they both orbit around their common center of mass, which is below the earth's surface but not at its center. The same is true of the hydrogen atom. Does the center of mass lie inside the proton or outside it?
2	The figure shows eight of the possible ways in which an electron in a hydrogen atom could drop from a higher energy state to a state of lower energy, releasing the difference in energy as a photon. Of these eight transitions, only D, E, and F produce photons with wavelengths in the visible spectrum. (a) Which of the visible transitions would be closest to the violet end of the spectrum, and which would be closest to the red end? Explain. (b) In what part of the electromagnetic spectrum would the photons from transitions A, B, and C lie? What about G and H? Explain. (c) Is there an upper limit to the wavelengths that could be emitted by a hydrogen atom going from one bound state to another bound state? Is there a lower limit? Explain.
3	Before the quantum theory, experimentalists noted that in many cases, they would find three lines in the spectrum of the same atom that satisfied the following mysterious rule: 1/λ1 = 1/λ2 + 1/λ3. Explain why this would occur. Do not use reasoning that only works for hydrogen - such combinations occur in the spectra of all elements. [Hint: Restate the equation in terms of the energies of photons.]
4	Find an equation for the wavelength of the photon emitted when the electron in a hydrogen atom makes a transition from energy level n1 to level n2. [You will need to have read optional section 5.4.]	√
5	(a) Verify that Planck's constant has the same units as angular momentum. (b) Estimate the angular momentum of a spinning basketball, in units of h[bar].
6	Assume that the kinetic energy of an electron in the n = 1 state of a hydrogen atom is on the same order of magnitude as the absolute value of its total energy, and estimate a typical speed at which it would be moving. (It cannot really have a single, definite speed, because its kinetic and potential energy trade off at different distances from the proton, but this is just a rough estimate of a typical speed.) Based on this speed, were we justified in assuming that the electron could be described nonrelativistically?
7	The wavefunction of the electron in the ground state of a hydrogen atom is where r is the distance from the proton, and a = h[bar]2/kme2 = 5.3 × 10-11 m is a constant that sets the size of the wave. (a) Calculate symbolically, without plugging in numbers, the probability that at any moment, the electron is inside the proton. Assume the proton is a sphere with a radius of b = 0.5 fm. [Hint: Does it matter if you plug in r = 0 or r = b in the equation for the wavefunction?] (b) Calculate the probability numerically. (c) Based on the equation for the wavefunction, is it valid to think of a hydrogen atom as having a finite size? Can a be interpreted as the size of the atom, beyond which there is nothing? Or is there any limit on how far the electron can be from the proton?
8	Use physical reasoning to explain how the equation for the energy levels of hydrogen, should be generalized to the case of a heavier atom with atomic number Z that has had all its electrons stripped away except for one.	√ *
9	This question requires that you read optional section 5.4. A muon is a subatomic particle that acts exactly like an electron except that its mass is 207 times greater. Muons can be created by cosmic rays, and it can happen that one of an atom's electrons is displaced by a muon, forming a muonic atom. If this happens to a hydrogen atom, the resulting system consists simply of a proton plus a muon. (a) How would the size of a muonic hydrogen atom in its ground state compare with the size of the normal atom? (b) If you were searching for muonic atoms in the sun or in the earth's atmosphere by spectroscopy, in what part of the electromagnetic spectrum would you expect to find the absorption lines?
10	Consider a classical model of the hydrogen atom in which the electron orbits the proton in a circle at constant speed. In this model, the electron and proton can have no intrinsic spin. Using the result of problem 17 from book 4, ch. 6, show that in this model, the atom's magnetic dipole moment Dm is related to its angular momentum by Dm = (-e/2m)L, regardless of the details of the orbital motion. Assume that the magnetic field is the same as would be produced by a circular current loop, even though there is really only a single charged particle. [Although the model is quantum-mechanically incorrect, the result turns out to give the correct quantum mechanical value for the contribution to the atom's dipole moment coming from the electron's orbital motion. There are other contributions, however, arising from the intrinsic spins of the electron and proton.]