Download Lecture 27: Quantum Mechanics (Continued)

Lecture 27: Quantum Mechanics (Continued)
Review
o Particle in Box
o Poor Man’s Particle in Box
o Application to beta Carotene
Today
o Tunneling
o STM
o Imaging Wavefunctions
o Harmonic Oscillator
o Hydrogen Atom
o Wavefunctions
o Eigen-values (energies)
Poor Man’s Particle in Box
We can use a very simple approach to calculate the energies of a
particle confined in a box using Bohr’s approach. In the region
of box i.e., 0<x<a, let us try to describe a particle with only
kinetic energy. Using the classical Newtonian approach we can
immediately write.
1 2 p2
E  mv 
2
2m
However, as we make the particle smaller and length of the box
submicroscopic, we know that the particle can act as a wave
with a wavelength given by de Broglie:

h
h

mv p
p2
h2
E 

2m 2m2
Then the question becomes what wavelengths are acceptable? If
we assume that the nodes of the wave must be present at the box
boundaries then it immediately implies not all wavelengths can
be accepted, i.e. wavelength is quantized just as in the case of a
standing wave problem. The acceptable wavelengths depend on
the size of the box. Mathematically, we must have n=2a. So E
becomes:
h2
h2n2
n2h2
E


2
2
2m
2m(2a)
8ma 2
This is the same result obtained from the solution of
Schrodinger’s equation! However, note we cannot determine the
nature of wavefunction using this approach.
Tunneling
Particle in box is a highly idealized case, developed to
illustrate the basic ideas of quantum mechanics. More
realistic cases involve potentials that are not infinite but
finite. The walls in the particle in box problem denote an
extremely hard or impenetrable barrier. If we reduce the
magnitude of potential, i.e. make the walls bit softer, the
strict requirement of wavefunction be zero at the wall
boundaries is lifted. In other words, the wavefunction can
penetrate the potential wall/barrier. In interesting case
arises when the spatial potential is not only finite but also it
is localized, i.e., the wall has certain thickness. To take a
concrete example:
We can set up this problem kinetically as shown in figure
to the left. In classical kinetics the reactants must overcome
the activation barrier to realize chemical reaction.
However, in quantum mechanics, the wavefunction of
electron/proton can penetrate the barrier. Therefore the
reaction may also occur as shown in the figure to right.
Thus measured kinetic rate constants can be enhanced. This
is commonly observed in reactions involving electrons or
protons (i.e., the oxidation/reduction reactions)
Scanning Tunneling Microscopy (STM)
Since electronic wavefunctions have nodes and anti-nodes, is it
possible to actually image the electronic wavefunction? The
answer is definitely yes, but it took almost 60 years to
experimentally demonstrate it. 1986 Nobel prize went to two
scientists at IBM to precisely demonstrate it. The experiment
involves measurement of current. More recently this method was
exploited to measure wavefunction on
a carbon nanotube surface.
Note, how clearly we can see the
atomic structure of Benzene rings.
Recently physicists at Delft have
imaged the wavefunctions near
HOMO. Remarkable similarity of the
wavefunctions to simple particle in box
type behavior should be obvious.
Electronic wavefunctions on Carbon Nanotubes
Harmonic Oscillator
Consider simple harmonic oscillator again. Here we consider
two masses m1 and m2 attached to an elastic spring. The
potential energy of the system depends on the degree of
stretching or compression:
We can then immediately write down the corresponding
Schrodinger’s equation.
where  is the reduced mass of the system given by:
mm
 1 2
m1  m2
The corresponding wavefunctions are sketched below
The energy levels are given by
Harmonic Oscillator
Although the above wavefunctions may appear sinusoidal they
are bit more complex
They have more of a bell-shaped or gaussian appearance. Also
note that for quantum number =0 we have a finite energy
known as zero point energy. The fundamental frequency of the
oscillator,0 is still given by classical value:
 0
1
2
k
m
However as before the energy values are quantized.