Assigned problems for Chapter 10 on the particle in a box
-1
1. calculate the energy separation in J, kJ mol-1 , eV, and cm
,
between the levels (a) n = 2 and n = 1 , ( b ) n= 6 and n = 5
of an electron in a 1- D box of length 1,0 nm
n 2h
E=
8m e L2
E=
n 2h
(6.626x10 −34 Js) 2
=
= 6.02x10 −20 j
2
−31
−9
2
8m e L
(8)x(9.109x10 Kg)x(1.0x10 m)
The conversion factors required are
E/(kJ mol -1 ) =
Na
E/ J
10
1 eV = 1.602x10 -19 J; 1 cm-1 = 1.986x10 -23 J
h
−23
(a) E 2 − E1 = (4 - 1 )
J
2 = 3x(6.02x10
8m e L
= 18.06x10 -20 J = 1.81x10 -19 J; 110kJ mol -1 ; 1.1 eV; 9100 cm-1
2. Calculate the probability that a particle
will be found between 0.49 L and 0.51L in a 1 - D
box of length L
when it has (a) n = 1 , (b) n=2.
The the wavefunction to be a constant in this range
the wavefunctions are
n
=
2
 n x
sin

 L 
L
The propability is
0.51L
P= ∫
n
2
dx ≈
n
2
x
0.49L
(a)
P=
2
 n x
sin 2 
 x
 L 
L
For x =.5L n= 1
P=
2
 
sin 2   x0.02L = 0.04
 2
L
3.
(a) Write the Schrödinger equation for a particle in a 1-D square well
of length L
(b) Calculate the expectation value of p and p 2 for a particle in the state n=1
Answ:
(a)
h2 d 2
−
=E
2m dx 2
which has the solution
n
 n x
2
sin

L  L 
=
ˆp = −
h d
i dx
L
< pˆ >= ∫
0
n
 n x
2h L  n x  d
* pˆ n dx = 2 ∫ sin

sin
 dx
iL 0  L  dx  L 
2h L  n x 
 n x
= 2 ∫ sin
 cos
 dx = 0
 L 
iL 0  L 
4. What are the most likely locations of a particle in a
box of length L in the state n = 3 ?
3
=
P ( x )∝
2
 3 x
sin 

 L 
L
3
3
 3 x
∝ sin 2 

 L 
The maxima and minima in P(x) corresponds to
 3 x
 3 x
 6 x
dP(x)
∝ sin 
 cos
 ∝ sin 

 L 
 L 
 L 
dx
sin = 0 when
 6 x
=
 = n' , n = 0,1,2.....
 L 
which corresponds to x =
n' L
, n'< 6.
6
n' = 0,2,4, and 6 corresponds to minima in
n' = 1, 3, and 5 to maxima
dP(x)
=0
dx
5.
Set up the Schrödinger equation for a particle of mass m in a threedimensional square well with
sides L x ; Ly ; Lz . Show that the wavefunction is defined by three quantum numbers
and that the Schrödinger equantion is separable . Find the energy levels, and specialize
the results to a cubic box of side L
Please see handout on particle in 3-D box (Lecture-10)
6. To a crude first approximation, a electron in a linear polyene
my be considered to be a particle in a one dimensional box.
The polyene - carotene contains 22 conjugated C atoms, and the
average internuclear distance is 140 pm. Each state up to n =11 is
occupied by two electrons.
Calculate
(a) the separation in energy between the ground state and
the first excited state in which one electron occupies the state with n=12.
(b) The frequency of the radiation required to produce a transition between
these two states
(c) The total probability of finding an electron between C atoms 11 and 12 in the
groundsate of the 22-electron molecule
Answ:
( a ) : L =(21)x(1.40x10-10 m) = 2.94x10 −9 m
E = E12 − E11
=
h2
h2
= (2n + 1)x
= (2x11 + 1)x
8mL2
8mL2
(23)x(6.626x10 −34 Js)
= 1.603x10
(8)x(9.11x10 −31 Kg)x(2.94x10 −9 ) 2
−19
J = 1.60x10 −19 J
(b) :
E 1.603x10 −19 J
=
=
= 2.42x10 14 s −1 = 2.42x10 14 Hz
−34
h
6.626x10
Js
7. Consider a particle in a cubic box. What is the degeneracy of the levels
that has an energy three times that of the lowest level.