Combining Relativity and Quantum Mechanics – A first step

Posted on September 27, 2011 by Anuj Varma

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.

Quantum Mechanics starting point – The Schrodinger Equation

Relativity starting point – The Total Energy of a Relativistic Particle

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