Download Quantum Mechanical Energy and You!

Quantum Mechanical
Momentum and You!
James G. O’Brien
New Paltz Physics Talk
SUNY New Paltz
May 5th, 2008
Dedicated to the late Dr. Walker
The Stated Problem
Classical Physics:
Dynamics are governed by Newton’s equations:
Quantum Physics:
Dynamics are governed by Schrodinger’s
equations:
The Stated Problem
Classical Physics:
Momentum defined by the classical value:
Quantum Physics:
Momentum defined by the quantum value:
History
1905:
Einstein Publishes the Photoelectric effect relating the energy of a
photon to its frequency, establishing the wave particle duality of
light.
History
1924:
Louis De Broglie publishes his Ph.D. Thesis on the wave particle
duality of matter, establishing the theoretical existence of matter
waves, thus changing the viewpoint of how we think of particles in
nature.
History
Classically:
=
Classically macroscopic objects are thought of as point like
particles. Particles are highly localized and the dynamics can be
understood using 2 quantities, space and time.
History
Quantum:
=
In the new realm of thought, macroscopic objects now be thought
of in terms of their matter waves or Debroglie waves. Waves are
quantities that are highly non local and spread through space in
time. Thus, the idea of how we talk of the fundamental quantity
position must be re-invented, as well as all secondary quantities.
Changing of thought
 Classically we have the fundamental quantites x
and t which govern the dynamics of the system.
 For waves, x loses some of its meaning, so
instead we talk of things such as the wave
number, frequency, wavelength, etc…
Ground Work
Classically we begin from the very simple
equation:
Now in the new quantum thought, the momentum must
be reconsidered since it is derived from position. Instead
we must make use of some other, more instructive
quantity for waves, namely, the wave number.
Fourier Transforms
Luckily, there exists a mathematical function for relating these two, the
Fourier transforms:
Also, using Debroglie’s wavelength formula, we can derive the formula
for the Fourier transform to relate position and momentum, by noting:
Thus, we now have a way to rewrite the kinetic energy term in
the original classical energy equation while following quantum
thought!
The Calculations
Let us consider taking the fourier transform
of the following:
Important Conclusion
Relating the first and last line of the previous
relation, as well as making use of the
DeBroglie wavelength, we come to the
following conclusions:
Solution
So momentum changed due to the implantation of
wave analysis, thus the way we view momentum
had to be changed.
Half of the S.E.
 Using the new definition of momentum, we
can solve for the left hand side of the
classical energy equation.
Right Hand Side of S.E.
Using another F.T. between the time and
frequency,
We can observe:
Full form of the S.E.
+
And this is the famed Shrodinger’s Equation!
Conclusions
 All the physics that we know and love are
the same.
 Sometimes we have to re-evaluate how
we look at things.
 Small changes to perception, lead to
massive changes in consequence.
Thank You
Thank you all for listening. I would like to
espescially thank Dr. Halpern, Dr. Biswas
and Dr. Nunes.