148 3 Quantum Mechanics – II
3.65 Show that (a) the electron density in the hydrogen atom is maximum atr=a 0 ,
wherea 0 is the Bohr radius (b) the mean radius is 3a 0 / 2
3.66 Refer to the hydrogen wave functions given in Table 3.2. Show that the func-
tions for 2pare normalized.
3.67 Show that the three 3dfunctions for H-atom are orthogonal to each other.
(Refer to Table 3.2)
3.68 What is the degree of degeneracy forn= 1 , 2 ,3 and 4 in hydrogen atom?
3.69 What is the parity of the 1s, 2 pand 3dstates of hydrogen atom.
3.70 Show that the 3dfunctions of hydrogen atom are spherically symmetric
3.71 In the ground state of hydrogen atom show that the probability (p)forthe
electron to lie within a sphere of radiusRis
P= 1 −exp(− 2 R/a 0 )
(
1 + 2 R/a 0 + 2 R^2 /a 02)
3.72 Locate the position of maximum and minimum electron density in the 2Sorbit
(n=2 andl=0) of hydrogen atom
3.73 When a negative muon is captured by an atom of phosphorous (Z=15) in
a high principal quantum number, it cascades down to lower state. When it
reaches inside the electron cloud it forms a hydrogen-like mesic atom with the
phosphorous nucleus.
(a) Calculate the wavelength of the photon for the transition 3d→ 2 pstate.
(b) Calculate the mean lifetimes of this mesic atoms in the 3d-state, consider-
ing that the mean life of a hydrogen atom in the 3dstate is 1. 6 × 10 −^8 s.
(mass of muon=106 MeV)
3.74 The momentum distribution of a particle in three dimensions is given by
ψ(p)=[1/(2π)^3 /^2 ]
∫
e−p.r/ψ(r)dτ. Take the ground state eigen functionψ(r)=(√
πa 03)− (^12)
exp(−r/a 0 )
Show that for an electron in the ground state of the hydrogen atom the
momentum probability distribution is given by
ψ|(p)|^2 =
8
π^2(
a 0) 5
[
p^2 +(
a 0) 2 ]^4
3.75 In Problem 3.74 (a) show that the most probable magnitude of the momentum
of the electron is/(
√
3 a 0 ) and (b) its mean value is 8/ 3 πa 0 , wherea 0 is
the Bohr radius.3.76 Calculate the radiusRinside which the probability for finding the electron in
the ground state of hydrogen atom is 50%.