Quantum Mechanical Momentum and You! James G. O’Brien New Paltz Physics Talk SUNY New Paltz May 5 th, 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.