- 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?
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.