Discussion  Fr 10.6.2005  at 10:30am  lecture hall SN 20.2

Home work 7       (LCAO, harmonic oscillator, HCl)
 

Exercise 1:
During the lecture we calculated the energies E_I and E_II of a system given by two configurations  |1> and |2>,  where both configurations were equivalent, i.e. they are described by the same energy E_o = <1|H|1> = <2|H|2> . Now calculate the energy under the assumption that the energies are different, i.e. E₁ = <1|H|1> and E₂ = <2|H|2> with E₁¹E₂.

a) Show that in case of a small interaction energy (H₂₁ << E₁,E₂ ) between |1> and |2> the energetically lower level E₁ is shifted to lower energies while the energetically higher level E₂ is shifted to higher energies by about the same amount ΔE_w.
b) Show that the coefficients a₁^I and a₂^I, which describe the resulting lower energy level E_I, |I> = a₁^I |1> + a₂^I |2>, are given by:
                a₁^I_/a₂^I = (E₁−E₂)_/H₂₁             | I > = {H₂₁² + (E₁−E₂)²}^−½ · [(E₁−E₂) | 1 > + H₂₁ | 2 >]
We used the abbreviation  H₁₂ = <1|H|2> and H₁₂=H₂₁.

Exercise 2:
Show that the commutator of the creation a₊ = ¹/_Ö2(−^d/_dz + z) and the annihilation operator a_- = ¹/_Ö2(^d/_dz + z) is given by  [a₊,a_-] = -1.

Exercise 3:
The wavefunction of the harmonic oscillator for the ground state and the first excited state are given by :
Y₀(z) = c₀ e^−z²/2 and Y₁(z) = c₁ z e^−z²/2 , respectively. z² = x² (mw/h)^½
a) Determine the constants c₀ and c₁ by using: _-¥ò^+∞e^−az²dz = (π/a)^½  and _-¥ò^+∞ z² e^−az²dz = ½ (π/a³)^½
b) What is the parity of the wavefunctions Y₀(z) and Y₁(z).
Parity characterizes the system under the symmetry operation of inversion. For a one dimensional wavefunction Y₀(z) and Y₁(z) it is the replacement of z by -z. The parity  of a wavefunction is called gerade (g), if Y^g(-z) = + Y^g(z), and ungerade (u), if Y^u(-z) = − Y^u(z).  

Exercise 4:

a) The distance between two neighboring levels of vibration in the HCl molecule is 2991 cm^-1, where 1 cm^-1 = 1.986·10^-23 J. What is the value of the zero point energy ?
b) Determine the maximum relativ velocity of the atoms in the HCl molecule in the vibrational ground state v = 0.
Use the reduced mass of HCl m_HCl = m_Hm_{Cl /}(m_H+m_Cl) with m_Cl = 35 u.
c) Determine the value of the force constant k.
d) What is the zero point energy and the distance between neighboring vibrational states in the DCl molekule ?

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.