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Chapter 7. Exercises, Exams of Quantum Mechanics

Georgetown University (GU)Quantum Mechanics

Determine the maximum of the radial distribution function for the ground state of hydrogen atom. Compare this value with the corresponding radius in the Bohr ...

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Chapter 7. Exercises
1. Assume that each circular Bohr orbit for an electron in a hydrogen
atom contains an integer number of de Broglie wavelengths, n= 1, 2, . . ..
Show that the orbital angular momentum must then be quantized. Bohr’s
formula for the hydrogen energy levels follows from this.
2. Based on your knowledge of the first few hydrogenic eigenfunctions,
deduce general formulas, in terms of nand `, for: (i) the number of radial
nodes in an atomic orbital; (ii) the number of angular nodes; (iii) the total
number of nodes.
3. Calculate the wavelength of the Lyman alpha transition (1s←2p) in
atomic hydrogen and in He+. Express the results in both nm and cm−1.
4. Determine the maximum of the radial distribution function for the
ground state of hydrogen atom. Compare this value with the corresponding
radius in the Bohr theory.
5. The following reaction might occur in the interior of a star:
He++ + H →He++ H+
Calculate the electronic energy change (in eV). Assume all species in their
ground states.
6. Which of the following operators is not equal to the other four: (i)
∂2/∂r2(ii) r−2∂/∂r r2∂/∂r (iii) r−1∂2/∂r2r(iv) (r−1∂/∂r r)2
(v) ∂2/∂r2+ 2r−1∂/∂r.
7. Calculate the expectation values of r,r2and of r−1in the ground state
of the hydrogen atom. Give results in atomic units.
8. Calculate the expectation values of potential and kinetic energies for the
1sstate of of a hydrogenlike atom.
9. Verify that the 3dxy orbital given in the table is a normalized eigenfunc-
tion of the hydrogenlike Schr¨odinger equation.
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Chapter 7. Exercises

  1. Assume that each circular Bohr orbit for an electron in a hydrogen atom contains an integer number of de Broglie wavelengths, n = 1, 2,.. .. Show that the orbital angular momentum must then be quantized. Bohr’s formula for the hydrogen energy levels follows from this.
  2. Based on your knowledge of the first few hydrogenic eigenfunctions, deduce general formulas, in terms of n and `, for: (i) the number of radial nodes in an atomic orbital; (ii) the number of angular nodes; (iii) the total number of nodes.
  3. Calculate the wavelength of the Lyman alpha transition (1s ← 2 p) in atomic hydrogen and in He+. Express the results in both nm and cm−^1.
  4. Determine the maximum of the radial distribution function for the ground state of hydrogen atom. Compare this value with the corresponding radius in the Bohr theory.
  5. The following reaction might occur in the interior of a star:

He++^ + H → He+^ + H+

Calculate the electronic energy change (in eV). Assume all species in their ground states.

  1. Which of the following operators is not equal to the other four: (i) ∂^2 /∂r^2 (ii) r−^2 ∂/∂r r^2 ∂/∂r (iii) r−^1 ∂^2 /∂r^2 r (iv) (r−^1 ∂/∂r r)^2 (v) ∂^2 /∂r^2 + 2r−^1 ∂/∂r.
  2. Calculate the expectation values of r, r^2 and of r−^1 in the ground state of the hydrogen atom. Give results in atomic units.
  3. Calculate the expectation values of potential and kinetic energies for the 1 s state of of a hydrogenlike atom.
  4. Verify that the 3dxy orbital given in the table is a normalized eigenfunc- tion of the hydrogenlike Schr¨odinger equation.
  1. Show that the function

ψ(r, θ, φ) = const

[

1 − r sin^2 (θ/2)

]

e−r/^2

is a solution of the Schr¨odinger equation for the hydrogen atom and find the corresponding eigenvalue (in atomic units).

  1. For the ground state of a hydrogenlike atom, calculate the radius of the sphere enclosing 90% of the electron probability in the 1s state of hydrogen atom. (This involves a numerical computation with successive approxima- tions.)
  2. Consider as a variational approximation to the ground state of the hydrogen atom the wavefunction ψ(r) = e−αr. Calculate the corresponding energy E(α) then optimize with respect to the parameter α. Compare with the exact solution.
  3. The electron-spin resonance hyperfine splitting for atomic hydrogen is given by

∆ν = 532. 65

8 π 3

|ψ(0)|^2 +

3 cos^2 θ − 1 r^3

MHz

Calculate ∆ν for the 1s and for the 2p 0 states. The result is in MHz when the bracketed terms are expressed in atomic units. (Hint: In the expectation value, do the integral over angles first.)

〈r−^1 〉 =

0

ψ1s(r) r−^1 ψ1s(r) 4πr^2 dr = 1

a 0

  1. Average potential energy:

< V >=

0

ψ1s(r)

Z

r

ψ1s(r) 4πr^2 dr = −Z^2

Average kinetic energy:

< T >=

0

ψ1s(r)

∇^2

ψ1s(r) 4πr^2 dr = Z^2 / 2

More simply, since total energy E1s = −Z^2 /2, 〈T 〉 = E1s − 〈V 〉. Note that 〈V 〉 = − 2 〈T 〉, consistent with the virial theorem.

  1. For an easier exercise, do the 2pz orbital instead.
  2. You should find that this function solves the Schr¨odinger equation with E = −Z^2 /8, i.e., n = 2. For normalization

const =

Z^3 /^2

π

Noting that sin^2 (θ/2) = (1 − cos θ)/2, the function is found to be an s-p hybrid orbital:

ψ =

(ψ 2 s + ψ 2 pz )

  1. Solve for R: (^) ∫ R

0

|ψ 1 s(r)|^2 4 πr^2 dr = 0. 9

or easier (^) ∫ (^) ∞

R

|ψ 1 s(r)|^2 4 πr^2 dr = 0. 1

We find, using integral table,

R

r^2 e−^2 r^ dr = e^2 R^ (1 + 2R + 2R^2 ) = 0. 1

Solving numerically, R = 2. 6612 a 0 = 1.41 ˚A.

  1. Let ψ(r) = e−αr^. Then

E(α) =

0 e

−αr (− 1 2 ∇

(^2) − Z/r)^ e−αr 4 πr (^2) dr ∫ (^) ∞ 0 e

− 2 αr 4 πr (^2) dr =

α^2 − Zα

E′(α) = 0 for minimum, giving α = Z. Thus ψ(r) = e−Zr^ and E = −Z^2 /2, which in this exceptional case equal the exact eigenfunction and eigenvalue.