Bootcamp to Relativistic Quantum Mechanics

In this video, we will solve Dirac’s equation for a free electron at rest. Considering an electron that is not moving and possesses no momentum allows us the freedom to disregard its position. Thus, for a free electron at rest, its exact location and the time do not matter, as the electron remains stationary.

While this might not seem like a very practical scenario at first, it provides an excellent foundation for understanding the quantum spin properties of an electron and its antimatter counterpart, the positron. This approach is a crucial stepping stone before delving into the hyperfine spectrum of the hydrogen atom, where we will explore a profoundly successful solution. The bi-spinor appearing in the wavefunction is discussed in this lecture.

Learning Outcomes

1. Recap to the derivation of K-G and Dirac’s equation from mass-shell

2. Dirac’s equation in 4-vectors/tensors form with intrinsic quantum spin

3. Dirac’s equation in Matrix form with Pauli’s matrices

4. Dirac’s equation for rest electron/positron (anti-matter)

5. The Solution of Dirac’s Equation unfolded and bi-spinor unfolded

In this video lecture, we will derive the Klein-Gordon equation and Dirac's equation using the Operators Approach. This method is well-established in Quantum Mechanics due to its mathematical elegance. We will start with the famous equation E=mc^2, which has cemented Einstein's legacy in the history of science. Leveraging the advantages of operators, we will extend Dirac's equation to describe a moving electron and express it in matrix form, revealing a set of four equations. Additionally, we will explore the wave function that represents the spin of an electron.

Learning Outcomes

1. Building Blocks of RQM

2. Slope & Curvature Vs Derivative

3. Quantum Plane wave equation

4. Energy & Momentum Op

5. Angular Momentum Op

6. Spin Op

7. Electron Wave Function with Spin

8. Klein Gordan Equation

9. Dirac’s Equation

10. Dirac’s Equation for free electron at rest

11. Does ℋ ̂,  L ̂ commute?

12. Dirac’s Equation for moving electron

13. Solving Dirac’s Eq for moving electron

14. Dirac’s Wave Function for electron

The Schrodinger equation predicts the same energy levels for all orbital types, s, p, d, etc. corresponding to the same n value. But all orbital types except s are observed to actually split into two very close energy levels. The exact hydrogen-atom energy expression that comes out of the Dirac equation solution is fairly complicated. Here we shall avoid its derivation as it doesn't look anything like the Schrodinger equation. In addition to the principle quantum number n, the energy explicitly depends on the angular momentum quantum number j .  The total angular momentum quantum number j plays a role that corresponds to the fine structure due to relativistic corrections. The predicted energy split for n=2 and j=1/2,3/2 is the tiny amount 0.00004535 electron-volts.

Learning outcomes

1. Dirac’s Equation in Heuristic Approach

2. Dirac’s Solution for Hydrogen Atom

3. Schrodinger VS Dirac’s for Hydrogen Atom

4. Dirac Equation Predictions

In this lecture we shall look at the struggle of Dirac in giving interpretation to negative energies which is one of its outcome. It could have been ignored, but Dirac's belief in correctness of his equation led him through a journey to antimatter being discussed here through following points.

Learning Outcomes

1. Dirac’s Struggle for NEGATIVE energies

2. Dirac’s Sea of Electrons

3. Pauli’s principle to rescue Dirac’s interpretation

4. Carl Anderson came up with evidence for positron

5. Dirac’s basis for QED and Particle Physics