94 2 Quantum Mechanics – I
2.14 Calculate the wavelengths of the first four lines of the Lyman series of the
positronium on the basis of the simple Bohr’s theory
[Saha Institute of Nuclear physics 1964]
2.15 (a) Show that the energyEnof positronium is given byEn=−α^2 mec^2 / 4 n^2
wheremeis the electron mass,nthe principal quantum number andαthe
fine structure constant
(b) the radii are expanded to double the corresponding radii of hydrogen atom
(c) the transition energies are halved compared to that of hydrogen atom.2.16 A non-relativistic particle of massmis held in a circular orbit around the
origin by an attractive forcef(r)=−krwherekis a positive constant
(a) Show that the potential energy can be written
U(r)=kr^2 / 2
AssumingU(r)=0 whenr= 0
(b) Assuming the Bohr quantization of the angular momentum of the particle,
show that the radiusrof the orbit of the particle and speedvof the particle
can be written
v^2 =(
n
m)(
k
m) 1 / 2
r^2 =(
n
k)(
k
m) 1 / 2
wherenis an integer
(c) Hence, show that the total energy of the particle isEn=n(
k
m) 1 / 2
(d) Ifm= 3 × 10 −^26 kg andk=1180 N m−^1 , determine the wavelength of
the photon in nm which will cause a transition between successive energy
levels.2.17 For high principle quantum number (n) for hydrogen atom show that the spac-
ing between the neighboring energy levels is proportional to 1/n^3.
2.18 In which transition of hydrogen atom is the wavelength of 486.1 nm produced?
To which series does it belong?
2.19 Show that for large quantum numbern, the mechanical orbital frequency
is equal to the frequency of the photon which is emitted between adjacent
levels.
2.20 A hydrogen-like ion has the wavelength difference between the first lines of
the Balmer and lyman series equal to 16.58 nm. What ion is it?