The following content has been provided by the University of Erlangen-Nürnberg.

So welcome to the final lecture of this series and today I would like to point out where the subject is taken or was actually taken a hundred years ago.

Because when quantum mechanics proper was born by Schrodinger and then Dirac and Heisenberg and so on, there was already the first revolution in physics of that century already had taken place name relativity.

And so what we will do today, we will look at quantum mechanics in a relativistic context.

And that is a huge subject, but in fact, it could be that only small modifications to the existing formalism and equations would be necessary.

We add an extra term, that extra term takes care of the relativistic effects.

Well, nothing could be further from the truth.

Quantum and relativistic quantum mechanics, it turns out, will be crucially different from non-relativistic quantum mechanics.

And today we'll discover also in a technical sense why.

And so if we follow that route, we first come to the Klein-Gordon equation, the M.A. Dirac equation, and so on.

That's all peanuts.

But then one realizes that these equations, so if you make the Schrödinger equation relativistic,

this equation is no longer, no longer has a probability interpretation.

So you know, one thing is to have the dynamical equation that carries the solution or the state forward.

And, but the other thing is you must be able to interpret that physically.

And that miserably fails.

And in the beginning, this was quite a puzzle, but of course we can pinpoint exactly where it fails and why it must fail.

And there is no relativistic quantum mechanics in the type of the Schrödinger equation, even with modifications.

The so-called Klein-Gordon equation and the Dirac equation, they are not solid quantum mechanical equations.

One is forced to go to quantum field theory.

So this is where the whole subject is going.

Now, let's however review a little.

So let's say start first with heuristic deviation of the Schrödinger equation.

So we go back to something we actually cast into an axiom.

Remember axiom four was essentially the Schrödinger equation expressed in terms of the, of a state, rho.

Now, it's easy to write down an axiom, but the question is how do you get there?

And the thinking goes as follows.

So Schrödinger recognized, now one can think about why that is true, but we're not going into that.

Schrödinger recognized that replacing the energy wherever it appears by my, by plus i h bar d by dt,

sorry I should say replace by, and replacing the momentum in the eighth direction,

so say in a three-dimensional space, a is one, two, three, replacing this by minus i h bar in the direction of the eighth component,

this is the t component of a wave function family, that replacing e by that and p by that,

where in the energy-momentum relation, well the energy-momentum relation of non-relativistic mechanics,

of non-relativistic mechanics, and what is that?

Well, you know this is the total energy e is the kinetic energy, so that's p squared divided by 2m plus the potential energy,

and making thereby a differential equation or differential operator yields the Schrödinger equation.

So you do that on the left-hand side, you get i h bar d by dt, well it needs to act on something, you write a psi,

and the right-hand side you replace the p that yields a minus h bar squared by 2m,

and then you have the Laplacian here, psi plus v of x acting on psi, and you know this is the Schrödinger equation,

and we talked about these Schrödinger operators and so on.

So that is how you can motivate this, of course you have to think about this here,

well this here the p's go to this and the q's are multiplication, that's the Stone-Fernoyman theorem,

and well this is the extension so you get the Schrödinger equation.

Now, for whatever reason he did that, now assume somebody suggests this equation and says,

well I want complex representations of that, something like this, okay, how would you interpret this?

And now of course again you can invoke another axiom, our axiom 5, and say,

well the interpretation is fixed to be such that, this is a different row, this is not our state row,

this is a probability density row, you just take the absolute value squared of this field,

you then get a real field, so the result would be something from r3 to r,

and notably to r0 plus, because the absolute value squared at every point x is of course a non-negative real number,