Download Chemistry 218 October 14, 2002

Chemistry 218
Problem set 2
October 14, 2002
R. Sultan
I. Uncertainty Principle
In an experimental verification of the uncertainty principle, we attempt to measure
simultaneously the x-coordinate and the x-component of the linear momentum p,
traveling in the y-direction.
Consider a beam of particles with momentum p falling on a narrow slit of width w as
shown in Fig. 1.
Fig. 1
The particles getting out of the slit are diffracted (remember that particles have wave
properties) and produce a “diffraction pattern” on a photographic plate. Call  the
angle between the y-axis (original direction of travel) and the trajectory of the
diffracted particles producing a minimum at the detector (see Fig. 1).
1. Estimation of x and px:
-
-
Uncertainty in position:
 x  width of the slit =
. (1)
Uncertainty in x-component of the linear momentum:
 px  x-component of the momentum of a particle producing the
minimum =
. (in terms of p and  ) (2)
-
Product of uncertainties:
 x.  px =
. (3)
2. Calculation of :
Consider the particles passing through the slit at its upper edge toward the
minimum (direction AD) and particles exiting the slit at its center toward the
minimum (direction BD) as shown in Fig. 2.
Fig. 2
-
The condition for the first minimum is that the waves traveling along
AD and BD be out-of phase. This requires:
BD – AD =
-
.
Now draw AC (see Fig. 2) such that AD = CD.
Hence, we have
BC = BD – CD = BD – AD =
-
.
(5)
Assuming that AD is nearly parallel to BD (long distances traveled),
the triangle ACB becomes a right-angled triangle (why?).
. (in terms of w and  )
This implies that BC =
-
(4)
The expressions for BC in equations (5) and (6) are equal. This yields:
sin  =
. (7)
3. Uncertainty relation:
Substituting for sin  in the product of uncertainties (eq. 3), and using
De Broglie’s relation, arrive at an expression of the uncertainty principle.
(6)
II. Law of Rayleigh and Jeans
To account for the intensity of blackbody radiation at low frequency ( long
wavelength) as predicted by the classical theory (the Rayleigh-Jeans law), we
consider the equations for the amplitudes of the oscillating electric field and magnetic
field, given respectively by:
E = Eosin (kx-  t)
;
H = Hosin(kx-  t)
In one-dimensional space, where k  wave vector and   angular frequency. For
waves traveling in three dimensions, the number of light waves in a frequency interval
1
between  and   d is given by dN  the volume of a spherical shell of inner
8
radius s (a vector radius) and outer radius s+ds (we divide by 8 to consider only one
octant).
Thus,
dN =
.
The number of allowed frequencies or normal vibrations between  and d is
obtained by setting s2 = 4L2 (k 2x + k 2y + k 2z ), where L is the distance between the
reflecting walls of the cavity. Express s2 and ds in terms of  and hence substitute in
the expression of dN (Note also that the number of light waves must be multiplied by
2 to account for the presence of two planes of polarization). This leads to:
dN =
.
Now use the theorem of equipartition of energy to calculate the energy density dU
(taking for simplicity a cubic box of side L). You obtain:
dU = (energy/oscillation) x
dN
=
V
.
This should give you the law of Rayleigh and Jeans. Now express dU in terms of
 (instead of  ). You get:
dU =
.
Finally, verify that this expression can be obtained from the Planck distribution in the
limit of low frequency (or large  ).
III. Time-dependent phenomena:
By writing the time-dependent wave function of a system as
 (x,t) =  (x) f(t)
where  (x) is the solution of the time-independent Schrödinger equation Ĥ (x) =
E (x), and by substituting  (x, t) in the time-dependent Schrödinger equation,
solve for the temporal function f(t) in terms of E.
+ Atkins, Exercises: 11.1,3,4,6,8,11,12,13,16.