Quantum Mechanics starting point – The Schrodinger Equation

(1)   \begin{equation*} i\hbar\frac{\partial\psi}{\partial t} = H \psi \end{equation*}

All laws of physics are required to be Lorentz Invariant . The above equation is not. To try and make it Lorentz invariant, we start with looking at the total energy of a relativistic particle.

Relativity starting point – The Total Energy of a Relativistic Particle

(2)   \begin{equation*} H = \sqrt{p^2 + m^2 c^4} \end{equation*}

If we substitute this into the above equation, we get:

(3)   \begin{equation*} i\hbar\frac{\partial\psi}{\partial t} = \sqrt{p^2 + m^2 c^4} \psi \end{equation*}

We need to promote the momentum in the above equation to an operator – which is done by substituting  i\hbar \frac{\partial}{\partial x} for p

(4)   \begin{equation*} i\hbar\frac{\partial\psi}{\partial t} = \sqrt{\hbar^2 \nabla^2 + m^2 c^4} \psi \end{equation*}

This is the Klein Gordon equation. This equation is Lorentz Invariant – so that serves as a good check for our first step towards combining  Quantum Mechanics with Relativity.

Anuj holds professional certifications in Google Cloud, AWS as well as certifications in Docker and App Performance Tools such as New Relic. He specializes in Cloud Security, Data Encryption and Container Technologies.

Initial Consultation

Anuj Varma – who has written posts on Anuj Varma, Hands-On Technology Architect, Clean Air Activist.

 Twitter Linkedin