What is the ratio of the longest wavelength of the Balmer series to the longest wavelength of the Lyman series?
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.
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.
Details of the calculation:
(a) En' - En = hν = hcRH(1/n2 - 1/n'2).
5.425978 * 1010 = 2.99792458 * 108RH(¼92 - 1/502), RH = 1.097373 * 107 m-1.
(b) ν = cRH(1/992 - 1/1002) = 6.679710 * 109 s-1.
(c) The line is Doppler shifted.
ν' = ν[(1 + v/c)/(1 - v/c)]½ ~ ν(1 + v/c) if v/c << 1.
Here v is the relative velocity of source and observer; v is positive if the source and the observer approach each other, and v is negative if the source and the receiver recede from each other.
-106/(3*108) < v/c < 106/(3*108), 6.657 * 109 s-1 < ν' < 6.702 * 109 s-1.
(d) Here are some factors worth considering:
The amount of radiation gathered by a telescope is proportional to the square of its diameter D.
The smallest angle that can be resolved is approximately θ = λ/D.
(For each photon we have ΔxΔpx ~ h, Δx ~ D, ΔPx ~ h/D, θ = Δpx/p = hc/(Dhν) = λ/D.)
If you assume that the radiation has a frequency of ~ 10-10 s-1, then λ = c/ν = 3 cm. For the small telescope we therefore have θ = 0.03/1.2 while for the large telescope we have θ = 0.03/11. It takes (11/1.2)2 = 84 times as long to gather the same amount of radiation with the small telescope as with the large telescope.
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+)?
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).
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.
Details of the calculation:
∫|ψ(r,θ,φ)|2d3r = 1, ψ(r,θ,φ) = A r exp(-r/(2a0)) cosθ.
A2∫02πdφ∫0πsinθ dθ∫0∞r2dr r2exp(-r/a0) cos2θ
= A22π∫0πsinθ cos2θ dθ∫0∞r4dr exp(-r/a0) = 1.
A22πa05∫-11x2dx∫0∞r'4dr' exp(-r') = A22πa05⅔4!' = 1.
A2 = 1/(32πa05).
ψ(r,θ,φ) = (32πa03)-½(r/a0) exp(-r/(2a0)) cosθ.
is the probability of finding the electron in an infinitesimal volume d3r
about r =
P(r,θ,φ) is the probability per unit volume.
∫02πdφ∫0πsinθ dθ |ψ(r,θ,φ)|2r2 = P(r).
P(r) is the probability per unit length.
P(r) = [2π/(32πa05)] r4exp(-r/a0)∫0πsinθ cos2θ dθ
= [r4/(16a05)] exp(-r/a0)⅔ = [r4/(24a05)] exp(-r/a0).
(c) We are looking for the maximum in P(r).
dP(r)/dr|(r = RMP) = [1/(24a05)] [4r3 exp(-r/a0) - (r4/a0)exp(-r/a0)]|(r = RMP) = 0.
RMP = 4a0.
∫0∞r P(r) dr = [1/(24a05)]
∫0∞r5 exp(-r/a0) dr
= [a0/24] 5! = 5a0.
<r> > RMP.
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?
Find the momentum space wave function Φ(p) for the electron in the 1s state of the hydrogen atom.
The normalized wave function of an electron in a hydrogen atom, neglecting
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
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?
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)
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?
Details of the calculation:
(a) L2 = Lx2 + Ly2 + Lz2 = l(l + 1)ħ2 = 6ħ2. The maximum value of Lz is 2ħ, so the minimum value of Lx2 + Ly2 = L2 - Lz2 = 2ħ2.
(b) The minimum value of Lz is 0, so the maximum value of Lx2 + Ly2 = 6ħ2.
(c) For l = 2 and ml = 1 Lz2 = ħ2, so Lx2 + Ly2 = 5ħ2. Neither Lx or Ly can be determined from this?
(d) For the hydrogen atom the allowed value for l are all non-negative integers equal to or less than n - 1. So the minimum value of n this state can have is 3.