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

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Quantum Mechanical Quantum Mechanical Momentum and You! Momentum and You! James G. O’Brien James G. O’Brien New Paltz Physics Talk New Paltz Physics Talk SUNY New Paltz SUNY New Paltz May 5 May 5 th th , 2008 , 2008 Dedicated to the late Dr. Dedicated to the late Dr. Walker Walker
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Transcript of Quantum Mechanical Momentum and You! James G. O’Brien New Paltz Physics Talk SUNY New Paltz May 5...

Page 1: 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.

Quantum Mechanical Quantum Mechanical Momentum and You!Momentum and You!

James G. O’BrienJames G. O’BrienNew Paltz Physics TalkNew Paltz Physics Talk

SUNY New PaltzSUNY New PaltzMay 5May 5thth, 2008, 2008

Dedicated to the late Dr. WalkerDedicated to the late Dr. Walker

Page 2: 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 ProblemThe Stated Problem

Classical PhysicsClassical Physics::

Dynamics are governed by Newton’s equations:Dynamics are governed by Newton’s equations:

Quantum PhysicsQuantum Physics::Dynamics are governed by Schrodinger’s Dynamics are governed by Schrodinger’s equations:equations:

Page 3: 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 ProblemThe Stated Problem

Classical PhysicsClassical Physics::

Momentum defined by the classical value:Momentum defined by the classical value:

Quantum PhysicsQuantum Physics::Momentum defined by the quantum value:Momentum defined by the quantum value:

Page 4: 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.

HistoryHistory

1905:1905:

Einstein Publishes the Photoelectric effect relating the energy of a photon to its frequency, establishing the wave particle duality of light.

Page 5: 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.

HistoryHistory

1924:1924:

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.

Page 6: 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.

HistoryHistory

Classically: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.

=

Page 7: 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.

HistoryHistory

Quantum: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.

=

Page 8: 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.

Changing of thoughtChanging of thought

Classically we have the fundamental quantites x Classically we have the fundamental quantites x and t which govern the dynamics of the system.and t which govern the dynamics of the system.

For waves, x loses some of its meaning, so For waves, x loses some of its meaning, so instead we talk of things such as the wave instead we talk of things such as the wave number, frequency, wavelength, etc…number, frequency, wavelength, etc…

Page 9: 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.

Ground WorkGround Work

Classically we begin from the very simple Classically we begin from the very simple equation:equation:

Now in the new quantum thought, the momentum must Now in the new quantum thought, the momentum must be reconsidered since it is derived from position. Instead be reconsidered since it is derived from position. Instead we must make use of some other, more instructive we must make use of some other, more instructive quantity for waves, namely, the wave number.quantity for waves, namely, the wave number.

Page 10: 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.

Fourier TransformsFourier Transforms

Luckily, there exists a mathematical function for relating these two, the Luckily, there exists a mathematical function for relating these two, the Fourier transforms: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!

Page 11: 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 CalculationsThe Calculations

Let us consider taking the fourier transform Let us consider taking the fourier transform of the following:of the following:

Page 12: 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.

Important ConclusionImportant Conclusion

Relating the first and last line of the previous Relating the first and last line of the previous relation, as well as making use of the relation, as well as making use of the DeBroglie wavelength, we come to the DeBroglie wavelength, we come to the following conclusions:following conclusions:

Page 13: 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.

SolutionSolution

So momentum changed due to the implantation of So momentum changed due to the implantation of wave analysis, thus the way we view momentum wave analysis, thus the way we view momentum had to be changed.had to be changed.

Page 14: 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.

Half of the S.E.Half of the S.E.

Using the new definition of momentum, we Using the new definition of momentum, we can solve for the left hand side of the can solve for the left hand side of the classical energy equation.classical energy equation.

Page 15: 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.

Right Hand Side of S.E.Right Hand Side of S.E.

Using another F.T. between the time and Using another F.T. between the time and frequency, frequency,

We can observe:

Page 16: 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.

Full form of the S.E.Full form of the S.E.

+

And this is the famed Shrodinger’s Equation!

Page 17: 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.

ConclusionsConclusions

All the physics that we know and love are All the physics that we know and love are the same.the same.

Sometimes we have to re-evaluate how Sometimes we have to re-evaluate how we look at things.we look at things.

Small changes to perception, lead to Small changes to perception, lead to massive changes in consequence.massive changes in consequence.

Page 18: 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.

Thank YouThank You

Thank you all for listening. I would like to Thank you all for listening. I would like to espescially thank Dr. Halpern, Dr. Biswas espescially thank Dr. Halpern, Dr. Biswas and Dr. Nunes.and Dr. Nunes.