3.2 Problems
151
3.91 The normalized 2 p eigen functions of hydrogen atom are
1
√ π
1
(2a 0 ) 3/2 e
−r/2a0 r
2a 0
sin θ e
iΦ
,
1
√
π
1
(2a 0 ) 3/2 e
−r/2a0 r
2a 0
cos θ,
1
√ π
1
(2a 0 ) 3/2 e
−r/2a 0
r
2a 0
sin θe
−iΦ
, for m = +1, 0, −1 respectively.
Apply the raising operator L + = L x + iL y and lowering operator to show that
the states with m = ±2 do not exist.
3.92 How can nuclear spin be measured from the rotational spectra of diatomic
molecules?
3.93 An electron is described by the following angular wave function
u(θ, ϕ) =
1
4
15
π
sin
2
θ cos 2ϕ
Re-express u in terms of spherical harmonics given below. Hence give the
probability that a measurement will yield the eigen value of L
2 equal to 6
2
You may use the following:
Y 20 (θ, ϕ) =
5
16π
3 cos
2
θ − 1
Y 2±1 (θ, ϕ) =
15
8π
sinθ cos θ exp (±iϕ)
Y 2±2 (θ, ϕ) =
15
32π
sin
2
θ exp (±2iϕ)
[University College, London]
3.94 Given that the complete wave function of a hydrogen-like atom in a particular
state is ψ(r, θ, ϕ) = Nr
2 exp
−
Zr
3a 0
sin
2
θ e
2iϕ determine the eigen value of
L z , the third component of the angular momentum operator.
3.95 Consider an electron in a state described by the wave function
ψ =
1
√
4π
(cosθ + sinθe
iϕ ) f (r )
where
∞
0
| f (r )|
2 r
2 dr = 1
(a) Show that the possible values of L z are + and zero
(b) Show that the probability for the occurrence of the L z values in (a) is 2/3
and 1/3, respectively.
3.96 Show that (a) [J z , J + ] = J + (b) J + |jm >= C jm + | j, m + 1 > (c) [J x , J y ]
= iJ z
151
3.91 The normalized 2 p eigen functions of hydrogen atom are
1
√ π
1
(2a 0 ) 3/2 e
−r/2a0 r
2a 0
sin θ e
iΦ
,
1
√
π
1
(2a 0 ) 3/2 e
−r/2a0 r
2a 0
cos θ,
1
√ π
1
(2a 0 ) 3/2 e
−r/2a 0
r
2a 0
sin θe
−iΦ
, for m = +1, 0, −1 respectively.
Apply the raising operator L + = L x + iL y and lowering operator to show that
the states with m = ±2 do not exist.
3.92 How can nuclear spin be measured from the rotational spectra of diatomic
molecules?
3.93 An electron is described by the following angular wave function
u(θ, ϕ) =
1
4
15
π
sin
2
θ cos 2ϕ
Re-express u in terms of spherical harmonics given below. Hence give the
probability that a measurement will yield the eigen value of L
2 equal to 6
2
You may use the following:
Y 20 (θ, ϕ) =
5
16π
3 cos
2
θ − 1
Y 2±1 (θ, ϕ) =
15
8π
sinθ cos θ exp (±iϕ)
Y 2±2 (θ, ϕ) =
15
32π
sin
2
θ exp (±2iϕ)
[University College, London]
3.94 Given that the complete wave function of a hydrogen-like atom in a particular
state is ψ(r, θ, ϕ) = Nr
2 exp
−
Zr
3a 0
sin
2
θ e
2iϕ determine the eigen value of
L z , the third component of the angular momentum operator.
3.95 Consider an electron in a state described by the wave function
ψ =
1
√
4π
(cosθ + sinθe
iϕ ) f (r )
where
∞
0
| f (r )|
2 r
2 dr = 1
(a) Show that the possible values of L z are + and zero
(b) Show that the probability for the occurrence of the L z values in (a) is 2/3
and 1/3, respectively.
3.96 Show that (a) [J z , J + ] = J + (b) J + |jm >= C jm + | j, m + 1 > (c) [J x , J y ]
= iJ z