Download Lecture 26: Quantum Mechanics (Continued)

Lecture 26: Quantum Mechanics (Continued)
Review
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Summary
Heisenberg’s Uncertainty principle
Waves and their mathematical description
Wave equation: Harmonic oscillator
Application to quantum mechanics
Concept of operators
Schrodinger’s equation (continued)
By partial differentiating  with respect to position.
Note the last equation defines entirely new concept,
momentum as an operator. Schrodinger made use of this
connection to describe the total energy of electron system,
which is simply the sum of potential
and kinetic energy.
Whence, Schrodinger’s equation is written symbolically as:
Hˆ   E
Interpretation of 
In classical mechanics (i.e. Newtonian Mechanics),  has a
distinct connotation. It denotes the displacement of wave.
Unfortunately, such is not the case in quantum mechanics
due to wave-particle duality. It was therefore Max Born
who suggested that we should think of , in terms of
probability of finding electron. However, the probabilities
are real numbers and not complex numbers like . So the
suggestion was to look at *, as a measure of probability
of finding electron in a unit volume. However, we know if
the electron exists in a problem, then the sum of * over
the entire space must be unity,
1    * dxdydz    * d
Thus, * acts like a distribution function for the electron
distribution in space. This is like Maxwell-Boltzman we
have encountered before, for velocity distribution of gas.
Any physical property that depends on location of electron
can be calculated if we know the appropriate
wavefunction. Just as we have seen in the Bohr’s model
of hydrogen, we find that this distribution depends on the
electron energy. Electron in a given energy “level” will
have different distribution in space.
Solutions of Schrodinger’s equation
In order to solve the Schrodinger’s equation for electron,
we must specify the geometry and potentials involved in a
given problem. Three famous potentials we will employ (1)
particle in box (2) Harmonic Oscillator (i.e. 1/2kx2) and (3)
Coulombic 1/r (Hydrogen atom). In addition, we will place
several restrictions on the wave function:
1. Wave function must be a single valued function of
distance.
2. Where the potential is positive and very large, the
wave function must equal zero. This means particle or
wave will be reflected from such region.
3. For bound electron, such as those in the atoms, the
electron wavefunction must decay to zero further
away from the bound region
4. At the potential boundaries, the wave function or its
first derivative must be continuous. Only exception to
this rule is when potential jumps to +- infinity at the
boundary.
The particle in box Problem
This is a celebrated problem in quantum mechanics and
finds many applications in chemistry, physics and biology.
Many complex systems can be simplified using this particle
in box model. This example, also illustrates how abstract
quantum mechanics can be applied to a physical problem
yielding understandable solution. Furthermore, the solution
of the Schrodinger’s equation can be obtained analytically
without resorting to heavy numerical methods, that are
commonly used by quantum chemists.
Before we formally set out to solve the problem, let us
speculate about the result we may anticipate, by resorting to
analogy to the string problem we discussed previously. If
both the ends of the string are attached to fixed support and
a tension is established, we found that the waves are setup.
The wavelengths were simple multiple of the string length.
Perhaps the electron wave might act similarly!?
Now back to the real problem. We consider a particle of
mass m confined in a region space whose boundaries/walls
are defined by infinite potential as shown.
X=0
X=a
Particle in Box Problem (continued)
To set up Schrodinger equation, we note that the wave
function cannot exist for x<0 and x>0 due to highly
reflective potential constituting the walls of the box. So we
focus on its solution in the region 0<x<a. But, here the
electron has no potential energy, which means that total
energy of the electron is only kinetic:
Note mathematically, the problem is very much like the
classical harmonic oscillator problem. For solution of  ,
we may try following trial function composed of both sine
and cosine functions.
Where A and B are constants to be determined from the
boundary considerations. Note that the  must be zero at
x=0, but cos(0)=1 and Sin(0)=0. So the coefficient B must
be zero in order for the wavefunction to vanish at the
boundary.
Next we use normalization of wavefunction to determine
A. That is:
2
 nx 
1   * dx  A  sin 
.dx  A 
a
 a 
0
0
a
a
2
2
Particle in Box (continued)
Thus, our overall wavefunction can now be given as:
Note this is similar to what we observed for the harmonic
oscillator wavefunction. But more importantly, there is not
simply one plausible solution but actually infinitely large
number of solutions since n can assume any value from 1.
Now using the value of wavefunction, it’s easy to show that
the energies associated with different values of n are given
by:
The energy level spacing varies
quadratically with n. This is the
fundamental property of particle in
box problem.
Consider a electron localized in pi electron system of 
carotene molecule
Assuming a particle in box type of energy level diagram
and two electrons occupy each
energy level. We have for this 22
electron system. Upon excitation
with light we can envision
following situation. We will need to
calculate the energy difference
between the n=11 and n=12 states
and using E=hv, we can calculate
the largest wavelength of light for
excitation.
The actual spectrum of 
carotene molecule is shown to
the left and exhibits a maximum
absorption at 480nm. Since in the
above equation all other quantities
are known we can calculate the
length of box a, and find that the
effective length is 18A as opposed to
29A based on geometry.