148
3 Quantum Mechanics – II
3.65 Show that (a) the electron density in the hydrogen atom is maximum at r = a 0 ,
where a 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 functions for 2 p are normalized.
3.67 Show that the three 3d functions for H-atom are orthogonal to each other.
(Refer to Table 3.2)
3.68 What is the degree of degeneracy for n = 1, 2, 3 and 4 in hydrogen atom?
3.69 What is the parity of the 1s, 2 p and 3d states of hydrogen atom.
3.70 Show that the 3d functions of hydrogen atom are spherically symmetric
3.71 In the ground state of hydrogen atom show that the probability ( p) for the
electron to lie within a sphere of radius R is
P = 1 − exp (−2R/a 0 )
1 + 2R/a 0 + 2R
2
/a
2
0
3.72 Locate the position of maximum and minimum electron density in the 2S orbit
(n = 2 and l = 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 p state.
(b) Calculate the mean lifetimes of this mesic atoms in the 3d-state, considering that the mean life of a hydrogen atom in the 3d state 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 function
ψ(r) =
πa
3
0
−
1
2
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
a0
5
p 2 +
a0
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 8/3π a 0 , where a 0 is
the Bohr radius.
3.76 Calculate the radius R inside which the probability for finding the electron in
the ground state of hydrogen atom is 50%.
3 Quantum Mechanics – II
3.65 Show that (a) the electron density in the hydrogen atom is maximum at r = a 0 ,
where a 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 functions for 2 p are normalized.
3.67 Show that the three 3d functions for H-atom are orthogonal to each other.
(Refer to Table 3.2)
3.68 What is the degree of degeneracy for n = 1, 2, 3 and 4 in hydrogen atom?
3.69 What is the parity of the 1s, 2 p and 3d states of hydrogen atom.
3.70 Show that the 3d functions of hydrogen atom are spherically symmetric
3.71 In the ground state of hydrogen atom show that the probability ( p) for the
electron to lie within a sphere of radius R is
P = 1 − exp (−2R/a 0 )
1 + 2R/a 0 + 2R
2
/a
2
0
3.72 Locate the position of maximum and minimum electron density in the 2S orbit
(n = 2 and l = 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 p state.
(b) Calculate the mean lifetimes of this mesic atoms in the 3d-state, considering that the mean life of a hydrogen atom in the 3d state 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 function
ψ(r) =
πa
3
0
−
1
2
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
a0
5
p 2 +
a0
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 8/3π a 0 , where a 0 is
the Bohr radius.
3.76 Calculate the radius R inside which the probability for finding the electron in
the ground state of hydrogen atom is 50%.