The Relativistic Schrödinger Equation

By:

​

Justin Harrington

Jacob DesRoches

Cody Farragher

John Westerhof

​

Who is Schrödinger?

Erwin Schrödinger (12 August 1887 - 4 January 1961)

​

• Much of Schrödinger’s life was spent as an instructor in some capacity.

​

• In 1926 Schrödinger originally derived the equation that would make him as famous as he is today.
• He later shared a Nobel Prize with Paul Dirac for his work.

​

• Shortly following his discovery, Schrödinger became Planck’s successor in Berlin.

​

• 1933 saw the rise of the Nazi Party in Germany and caused Schrödinger to flee from the country

Cont.

• 3 years later, Schrödinger returned to Austria and took up a new teaching position

​

• With the annexation of Austria, Schrödinger was forced to again flee.

​

• Schrödinger eventually found himself, once again, teaching at Oxford University.

​

• Shortly after this, he was offered a chance to open a University for Sciences in Dublin and accepted.

​

• This led him to be the Director of the School for Theoretical Physics until his retirement in 1955.

​

• Throughout this entire period Schrödinger continued to work on papers focused on various topics.

​

• After his retirement Schrödinger returned to Vienna where he eventually died of illness on January 4th, 1961.

Schrödinger’s bust at the University of Vienna

What is Schrödinger’s Equation?

​

​

​

It is a Differential Equation that gives the behavior of a wave function in the terms used to describe a particle.

​

​

​

Goals:

​

• Derive the relativistic schrodinger equation, not accounting for spin.

​

• Obtain the electromagnetic spectrum by solving for m = 0.

​

• Solve the new equation for energy.

​

​

​

​

What about Relativity?!

Finding the New Equation

Step 1: Multiply by Ψ across the whole equation.

Step 2: Transform momentum (p) using a Fourier Transform.

Step 3: Solve for the second order derivative of Ψ.

Solving for Energy

We can now solve for the energy of our new relativistic equation.

Step 1: Divide everything by Ψ.

Step 2: Take the square root of each side.

What does ±E mean?

• The first person to propose the existence of a negative energy was Paul Dirac (August 8th, 1902 - October 20th, 1984) in 1928.

​

• Experimental evidence of this particle was found on August 2nd, 1930.

​

• It is now understood that all particles have an antiparticle.

​

• Although some particles, known as Majorana particles, are their own antiparticle.

​

• An antiparticle has both opposite charge and opposite quantum spin.

​

​

Solving for the New k Value

Start at where we divided Ψ out earlier.

Step 1: Arrange it such that k is on its own side.

Step 2: Divide by c²ћ² and take the square root.

The Wavevector k

• The wavevector k comes from the wave equation ψ. The variable is related to energy.

​

• Since there is a positive and negative energy there needs to be a positive and negative value for k.

​

​

When m = 0 ...

Step 1: Take the Fourier Transform of E and p. Multiply everything by (-1).

Step 2: Divide by c²ћ² and set m = 0. The term with m falls out.

Step 3: Move the derivative terms over so that the equation is equal to 0.

Further Advancing the Schrödinger Equation

The first Schrödinger equation was published in 1926 in order to describe the behavior of a wave function in the terms used to describe a particle.

​

In 1928 Paul Dirac made a new equation that not only accounted for relativity, but accounted for spin as well. This equation was accepted for many years until further advancements in quantum field theory required further derivations.

The second version of the schrodinger equation was derived by John, Jake, Justin, and Cody in 2015. The new equation accounted for relativistic momentum and potential energy, but neglected to include spin. Other scientists contributed to making similar equations around 1927. Some included Vladimir Fock, Johann Kudar, Théophile de Donder and Frans-H. van den Dungen.

Now that we have proven antimatter exists, what can we do with it?

• Are there any applications for antimatter?
• Will we be able to harness antimatter as an energy source?
• Would using the matter and antimatter explosive interaction provide energy that we could harness and use to power our world.

?

References

“What is Antimatter?. N.p., n.d. Web. 12 Apr. 2015.

​

“Erwin Schrödinger - Biographical.” Erwin Schrödinger - Biographical.

The Nobel Foundation, n.d. Web. 12 Apr. 2015.

​

“Why is There Something Instead of Nothing?” Starts With A Bang.

N.p., n.d. Web. 12 Apr. 2015.

​

Merzbacher, Eugen. Quantum Mechanics. New York: John Wiley &

Sons, Inc, 1998. Print.