The hydrogen atom

Energy levels

Problem:

What is the ratio of the longest wavelength of the Balmer series to the longest wavelength of the Lyman series?

Solution:

Problem:

A hydrogen atom at rest in the laboratory emits the Lyman α radiation.
(a)  Express the measured wavelength in terms of the Rydberg constant RH = 1.097*107/m.
(b)  Compute the recoil kinetic energy of the atom.
(c)  What fraction of the excitation energy of the n = 2 state is carried by recoiling atom.

Solution:

Problem:

In the interstellar medium electrons may recombine with protons to form hydrogen atoms with high principal quantum numbers.  A transition between successive values of n gives rise to a recombination line.
(a)  A radio recombination line occurs at 5.425978 * 1010 Hz for a n = 50 to n = 49 transition.  Calculate the Rydberg constant for H.
(b)  Compute the frequency of the H recombination line corresponding to the transition n = 100 to n = 99.
(c)  Assume the mean speed in part (b) is 106 m/s.  At what frequency or frequencies would the recombination line be observable?
(d)  Consider that radio recombination lines may be observed at either of two facilities, the 11 meter telescope at Kitt Peak near Tuscon, Arizona, and the 1.2 meter radio telescope at Columbia University in New York.  Relative larger blocks of time are available on the smaller telescope, but its intrinsic noise is moderately high.  Where would you choose to map recombination radiation emanating from an external galaxy.  Discuss both technical and non technical aspects of your choice.

Solution:

Problem:

Determine the wavelengths of the first three transitions in the Balmer series for hydrogen.  How would the wavelengths change for an ion with Z = 2 (He+)?

Solution:

Problem:

A hydrogen atom is placed in a uniform electric field, E = -Ek.  Place the proton at the origin of your coordinate system.  An electron in the hydrogen atom then has potential energy U(r) = -kqe2/r – qeEz.  U(0,0,z) becomes increasingly positive for negative z .  For positive z the potential energy U(0,0,z) contains a “hill” and then decreases with increasing z.
Sketch the potential energy of the electron, U(0,0,z), as function of z and calculate the energy at the maximum at positive z.  Equate this energy to the energy of the unperturbed (zero field) hydrogen energy level and thereby determine the value of the field required to field-ionize a hydrogen atom with principle quantum number n (neglect tunneling).

Solution:


Wave functions

Problem:

An electron in the n = 2, l = 1, m = 0 state of a hydrogen atom has a wave function ψ(r,θ,φ) proportional to r exp(-r/(2a0)) cosθ, where a0 is the Bohr radius.
(a)  Find the normalized wave function ψ(r,θ,φ).
(b)  Find the probability per unit length of finding the electron a distance r from the nucleus.
(c)  Find the most probable distance RMP of the electron from the nucleus.
(d)  Find the average distance <r> from the nucleus.

Solution:

Problem:

Assume that the nucleus of a hydrogen atom is a sphere of radius b.  In the limit where
b << a0 (where a0 is the Bohr radius), what is the probability that an electron in the ground state of hydrogen will be found inside the nucleus?

Solution:

Problem:

Find the momentum space wave function Φ(p) for the electron in the 1s state of the hydrogen atom.

Solution:

Problem:

The normalized wave function of an electron in a hydrogen atom, neglecting spin, is
ψ(r,t) = ⅓½Φ100(r,θ,φ) + ⅔½Φ210(r,θ,φ),
where the Φnlm(r,θ,φ) are the usual normalized hydrogenic eigenfunctions.
(a)  If a single measurement is made of the energy, what results are possible?
(b)  What are the probabilities of obtaining each of the particular possible results?
(c)  What is the expectation value of the energy?
(d)  If a single measurement is made of the total angular momentum, what results are possible?
(e)  What are the probabilities of obtaining each of the particular possible results?
(f)  If the observable O is a constant of the motion, then it satisfies the equation

d/dt<O> = <∂O/∂t> + (i/ħ)<[H,O]> = 0

where H is the Hamiltonian.
For the hydrogen atom described above, is the z-component of angular momentum, Lz, a constant of the motion?
(g)  Is the z-component of linear momentum, pz, a constant of the motion?
(h)  How do the expectation values of Lz and pz depend on time?

Given:
Φ100(r,θ,φ) ∝ exp(-r/a0),      Φ210(r,θ,φ) ∝ (r/a0) exp(-r/(2a0)) cos(θ). 
∫Φ*100 (∂/∂z) Φ210 d3r  = (√2/a0)⅔4 exp(-i(E2 – E1)t/ħ) = A exp(-i(E2 – E1)t/ħ)
∫Φ*210 (∂/∂z) Φ100 d3r  = -(√2/a0)⅔4 exp(i(E2 – E1)t/ħ) = -A exp(i(E2 – E1)t/ħ)

Solution:

Problem:

For l = 2 in the hydrogen atom:
(a)  What is the minimum value of Lx2 + Ly2?
(b)  What is the maximum value of Lx2 + Ly2?
(c)  What is Lx2 + Ly2 for l = 2 and ml = 1?  Can either Lx or Ly be determined from this?
(d)  What is the minimum value of n this state can have?

Solution: