98 2 Quantum Mechanics – I
2.47 The^9 Be+ion has a nucleus with spinI = 3 /2. What values are possible for
the hyperfine quantum number F for the^2 S 1 / 2 electronic level?
[Aligarh University]
2.48 Obtain an expression for the Doppler linewidth for a spectral line of wave-
lengthλemitted by an atom of massmat a temperatureT
2.49 For the 2P 3 / 2 →2S 1 / 2 transition of an alkali atom, sketch the splitting of the
energy levels and the resulting Zeeman spectrum for atoms in a weak exter-
nal magnetic field (Express your results in terms of the frequencyv 0 of the
transition, in the absence of an applied magnetic field)
[
The Lande g-factor is given byg= 1 +
j(j+1)+s(s+1)−l(l+1)
2 j(j+1)]
[Adapted from the University of London Holloway 2002]2.50 The spacings of adjacent energy levels of increasing energy in a calcium triplet
are 30× 10 −^4 and 60× 10 −^4 eV. What are the quantum numbers of the three
levels? Write down the levels using the appropriate spectroscopic notation.
[Adapted from the University of London, Royal Holloway 2003]
2.51 An atomic transition line with wavelength 350 nm is observed to be split into
three components, in a spectrum of light from a sun spot. Adjacent compo-
nents are separated by 1.7 pm. Determine the strength of the magnetic field in
the sun spot.μB= 9. 17 × 10 −^24 JT−^1
[Adapted from the University of London, Royal Holloway 2003]
2.52 Calculate the energy spacing between the components of the ground state
energy level of hydrogen when split by a magnetic field of 1.0 T. What fre-
quency of electromagnetic radiation could cause a transition between these
levels? What is the specific name given to this effect.
[Adapted from the University of London, Royal Holloway 2003]
2.53 Consider the transition 2P 1 / 2 →2S 1 / 2 , for sodium in the magnetic field of
1.0 T, given that the energy splittingΔE=gμBBmj, whereμBis the Bohr
magneton. Draw the sketch.
[Adapted from the University of London, Royal Holloway 2004]
2.54 To excite the mercury line 5,461A an excitation potential of 7.69 V is required. ̊
If the deepest term in the mercury spectrum lies at 84,181 cm−^1 , calculate the
numerical values of the two energy levels involved in the emission of 5,461A. ̊
[The University of Durham 1963]
2.55 The mean time for a spontaneous 2p→ 1 stransition is 1. 6 × 10 −^9 swhile
the mean time for a spontaneous 2s → 1 stransition is as long as 0.14 s.
Explain.