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3 Quantum Mechanics – II
3.8 (a) State and explain Laporte rule for light emission.
(b) What are metastable states?
3.9 Show that the eigen values of a hermitian operator Q are real
3.10 The state of a free particle is described by the following wave function
(Fig. 3.1)
ψ(x) = 0 for x < −3a
= c for − 3a < x < a
= 0 for x > a
(a) Determine c using the normalization condition
(b) Find the probability of finding the particle in the interval [0, a]
Fig. 3.1 Uniform distribution
of ψ
3.11 In Problem 3.10,
(a) Compute < x > and σ
2
(b) Calculate the momentum probability density.
3.12 Particle is described by the wavefunction
ψ = 0 x < 0
=
√
2e
−x/L x ≥ 0
where L = 1 nm. Calculate the probability of finding the particle in the region
x ≥ 1 nm.
3.2.2 Schrodinger Equation
3.13 The radial Schrodinger equation, in atomic units, for an electron in a hydrogen
atom for which the orbital angular momentum quantum number, l = 0, is
((d
2
/dr
2 ) + (2/r ) + (2E))F(r ) = 0,
where E is the total energy.
3 Quantum Mechanics – II
3.8 (a) State and explain Laporte rule for light emission.
(b) What are metastable states?
3.9 Show that the eigen values of a hermitian operator Q are real
3.10 The state of a free particle is described by the following wave function
(Fig. 3.1)
ψ(x) = 0 for x < −3a
= c for − 3a < x < a
= 0 for x > a
(a) Determine c using the normalization condition
(b) Find the probability of finding the particle in the interval [0, a]
Fig. 3.1 Uniform distribution
of ψ
3.11 In Problem 3.10,
(a) Compute < x > and σ
2
(b) Calculate the momentum probability density.
3.12 Particle is described by the wavefunction
ψ = 0 x < 0
=
√
2e
−x/L x ≥ 0
where L = 1 nm. Calculate the probability of finding the particle in the region
x ≥ 1 nm.
3.2.2 Schrodinger Equation
3.13 The radial Schrodinger equation, in atomic units, for an electron in a hydrogen
atom for which the orbital angular momentum quantum number, l = 0, is
((d
2
/dr
2 ) + (2/r ) + (2E))F(r ) = 0,
where E is the total energy.