Combining Relativity and Quantum Mechanics – A first step

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.

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