Home work Physical and Theoretical Chemistry I II

- Structure of Matter -

Discussion Fr 3.6.2005 at 10:30am lecture hall SN 20.2

Home work 6 (Standard deviation, Eigenfunction, Tunneling)

Exercise 1:

The standard deviation ΔA for a quantity A is given by

ΔA² = <(A − <A>)²> = <A²>- 2<A><A> + <A>² = <A²>-<A>²

ΔA = (<A²>-<A>²)½

Calculate the product Dx · Δp for a particle in a box being in the n=2 eigenstate with the wave function Y = Ö(2/a)·sin(np x/a). Calculate first the expectation values of <x²>, <x>², <p²>, and <p>².

Exercise 2:

a) Which of the following functions are eigenfunctions to the operator of the kinetic energy (T = -h²/2md²/dx²) ?

y = sin wx; y = w4x4 + w²x²; f = e−iwx; F = e−ax²

b) Which eigenvalues belong to these eigenfunctions ?

Exercise 3:

Determine a general expression for the tunneling probability T of a particle having the same amount of energy E as the potential barrier V: E=V?

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# Exercise 4:

The two H atoms of H2O2 are almost perpendicular to the O-O bond and form an torsional angle of 120°. Calculate the probability for a H atom in the vibrational ground state to tunnel through the trans or through the cis barrier. The width a of the barrier is a quarter of the circumference of a circle formed by an OH bond of r = 100 pm, i.e. = 2r/4 = 157 pm. The potential energy of the trans barrier is Vtrans = 0.029 eV, of the cis barrier is Vcis = 0.22 eV, and the energy E of the vibrational ground state of the torsional motion is E = 0.02 eV. For both barriers the approximate equation for calculating the probability of tunneling is sufficient.