98
2 Quantum Mechanics – I
2.47 The
9 Be
+ ion has a nucleus with spin I = 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 wavelength λ emitted by an atom of mass m at a temperature T
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 external magnetic field (Express your results in terms of the frequency v 0 of the
transition, in the absence of an applied magnetic field)
The Lande g-factor is given by g = 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 components are separated by 1.7 pm. Determine the strength of the magnetic field in
the sun spot. μ B = 9.17 × 10
−24 J T
−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 frequency 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 splitting ΔE = gμ B Bm j , where μ B is the Bohr
magneton. Draw the sketch.
[Adapted from the University of London, Royal Holloway 2004]
2.54 To excite the mercury line 5,461 ˚
A 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 5,461 ˚
A.
[The University of Durham 1963]
2.55 The mean time for a spontaneous 2 p → 1s transition is 1.6 × 10
−9 s while
the mean time for a spontaneous 2s → 1s transition is as long as 0.14 s.
Explain.
2 Quantum Mechanics – I
2.47 The
9 Be
+ ion has a nucleus with spin I = 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 wavelength λ emitted by an atom of mass m at a temperature T
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 external magnetic field (Express your results in terms of the frequency v 0 of the
transition, in the absence of an applied magnetic field)
The Lande g-factor is given by g = 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 components are separated by 1.7 pm. Determine the strength of the magnetic field in
the sun spot. μ B = 9.17 × 10
−24 J T
−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 frequency 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 splitting ΔE = gμ B Bm j , where μ B is the Bohr
magneton. Draw the sketch.
[Adapted from the University of London, Royal Holloway 2004]
2.54 To excite the mercury line 5,461 ˚
A 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 5,461 ˚
A.
[The University of Durham 1963]
2.55 The mean time for a spontaneous 2 p → 1s transition is 1.6 × 10
−9 s while
the mean time for a spontaneous 2s → 1s transition is as long as 0.14 s.
Explain.