Ciro Santilli @cirosantilli 34

 Incoming links: Planck units

D'alembert operator in Einstein notation  Updated 2024-12-15  +Created 1970-01-01

Given the function

ψ

:

ψ : R^{4} \to C

the operator can be written in Planck units as:

\partial_{i} \partial^{i} ψ (x_{0}, x_{1}, x_{2}, x_{3}) - m^{2} ψ (x_{0}, x_{1}, x_{2}, x_{3}) = 0

often written without function arguments as:

\partial_{i} \partial^{i} ψ

Note how this looks just like the Laplacian in Einstein notation, since the D'alembert operator is just a generalization of the laplace operator to Minkowski space.

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Dirac equation  Updated 2024-12-15  +Created 1970-01-01

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

spontaneous emission coefficients.
fine structure, notably for example Dirac equation solution for the hydrogen atom
antimatter
particle creation and annihilation

Experiments not explained: those that quantum electrodynamics explains like:

Lamb shift
TODO: quantization of the electromagnetic field as photons?

See also: Dirac equation vs quantum electrodynamics.

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:

i \partial_{t} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} ψ_{3} ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤ = - i \partial_{x} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{4} ψ_{3} ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ + \partial_{y} ⎣ ⎢ ⎢ ⎢ ⎡ - ψ_{4} ψ_{3} - ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ - i \partial_{z} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{3} - ψ_{4} ψ_{1} - ψ_{2} ⎦ ⎥ ⎥ ⎥ ⎤ + m ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} - ψ_{3} - ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤

Expanded Dirac equation in Planck units

.

Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)

PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)

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Klein-Gordon equation  Updated 2024-12-15  +Created 1970-01-01

A relativistic version of the Schrödinger equation.

Correctly describes spin 0 particles.

The most memorable version of the equation can be written as shown at Section "Klein-Gordon equation in Einstein notation" with Einstein notation and Planck units:

\partial_{i} \partial^{i} ψ - m^{2} ψ = 0

Has some issues which are solved by the Dirac equation:

it has a second time derivative of the wave function. Therefore, to solve it we must specify not only the initial value of the wave equation, but also the derivative of the wave equation,
As mentioned at Advanced quantum mechanics by Freeman Dyson (1951) and further clarified at: physics.stackexchange.com/questions/340023/cant-the-negative-probabilities-of-klein-gordon-equation-be-avoided, this would lead to negative probabilities.
the modulus of the wave function is not constant and therefore not always one, and therefore cannot be interpreted as a probability density anymore
since we are working with the square of the energy, we have both positive and negative value solutions. This is also a features of the Dirac equation however.

Bibliography:

Video "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=600
An Introduction to QED and QCD by Jeff Forshaw (1997) 1.2 "Relativistic Wave Equations" and 1.4 "The Klein Gordon Equation" gives some key ideas
2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta at around 7:30
www.youtube.com/watch?v=WqoIW85xwoU&list=PL54DF0652B30D99A4&index=65 "L2. The Klein-Gordon Equation" by doctorphys
sites.ualberta.ca/~gingrich/courses/phys512/node21.html from Advanced quantum mechanics II by Douglas Gingrich (2004)
- sites.ualberta.ca/~gingrich/courses/phys512/node23.html gives Lorentz invariance

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