96 2 Quantum Mechanics – I
2.33 Moseley’s law for characteristic x-rays is of the form
√
v=a(z−b). Calculate
the value of a forKα2.34 TheKαline has a wavelengthλfor an element with atomic numberZ=19.
What is the atomic number of an element which has a wavelengthλ/4forthe
Kαline?
Experiment 2.3.4 Spin andμand Quantum Numbers – Stern–Gerlah’s
Experiment
2.35 Evidence for the electron spin was provided by the Sterrn–Gerlah experiment.
Sketch and briefly describe the key features of the experiment. Explain what
was observed and how this observation may be interpreted in terms of electron
spin.
[Adapted from University of London 2006]
2.36 (i) Write down the allowed values of the total angular momentum quantum
number j, for an atom with spinsandl, respectively (ii) Write down the
quantum numbers for the states described as^2 S 1 / 2 ,^3 D 2 and^5 P 3 (iii) Determine
if any of these states are impossible, and if so explain why.
[Adapted from the University of London, Royal Holloway 2003]
2.37 (a) show that an electron in a classical circular orbit of angular momentum
Laround a nucleus has magnetic dipole moment given byμ=−eL/ 2 me
(b) State the quantum mechanical values for the magnitude and the z-compo-
nent of the magnetic moment of the hydrogen atom associated with (i) electron
orbital angular momentum (ii) electron spin
[Adapted from the University of London, Royal Holloway 2004]
2.38 In a Stern-Gerlach experiment a collimated beam of hydrogen atoms emitted
from an oven at a temperature of 600 K, passes between the poles of a magnet
for a distance of 0.6 m before being detected at a photographic plate a further
1.0 m away. Derive the expression for the observed mean beam separation, and
determine its value given that the magnetic field gradient is 20 Tm−^1 (Assume
the atoms to be in the ground state and their mean kinetic energy to be 2 kT;
Bohr magnetonμB=^9.^27 ×^10 −^24 JT−^1
[Adapted from the University of London, Royal Holloway 2004]
2.39 State the ground state electron configuration and magnetic dipole moment of
hydrogen (Z=1) and sodium (Z=11)
2.40 In a Stern–Gerlah experiment a collimated beam of sodium atoms, emit-
ted from an oven at a temperature of 400 K, passes between the poles of
a magnet for a distance of 1.00 m before being detected on a screen a fur-
ther 0.5 m away. The mean deflection detected was 0. 14 ◦. Assuming that
the magnetic field gradient was 6.0Tm−^1 and that the atoms were in the