Quantum Mechanical Momentum and You!

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Quantum Mechanical Momentum and You!

• James G. OBrien
• New Paltz Physics Talk
• SUNY New Paltz
• May 5th, 2008
• Dedicated to the late Dr. Walker

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The Stated Problem

• Classical Physics
• Dynamics are governed by Newtons equations

Quantum Physics Dynamics are governed by
Schrodingers equations
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The Stated Problem

• Classical Physics
• Momentum defined by the classical value

Quantum Physics Momentum defined by the quantum
value
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History

• 1905

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

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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.
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Fourier Transforms

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

Also, using Debroglies 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!
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The Calculations

• Let us consider taking the fourier transform of
  the following

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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

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Solution

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

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Half of the S.E.

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

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Right Hand Side of S.E.

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

We can observe
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Full form of the S.E.

And this is the famed Shrodingers Equation!
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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.

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Thank You

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