Saturday, 1 August 2015

Normalisation of relativistic wave functions. Is the Born interpretation viable in relativistic quantum mechanics ?

I have just been reminding myself about relativistic particle physics with the help of these wonderful lecture notes given to 4th year undergraduates (MSc level) at Cambridge university (scroll down)

Anyway a point of significance which is normally skated over, is that in relativistic quantum mechanics, the wavefunctions (solutions to either the Klein Gordan equation or the Dirac equation ) are normalised such that the Integral of the modulus squared of the wave function over a volume is eqaul to 2E where E is the energy of the particle, This is in contrast to the case in non relativistic quantum mechanics where the integral of the modulus squared of the wave function is put equal to 1 ie for Non relativistic quantum mechanics we have

$$\int \psi^*\psi dV = 1 $$ but in relativistic quantum mechanics we have

$$\int \psi^*\psi dV = 2E $$

The reason for this normalisation is that in relativity volumes contract in proportion to the energy of the particle and the factor of 2 is conventional. In non relativistuc quantum mechanics the normalisation effectively means that there is 1 particle per unit volume. Whereas in relativistic quantum mechanics there are now 2E particles per unit volume.

In non relativistic quantum mechanics the normalisation to unity, leads to the Born interpretation and despite all the hoo hah in the popular literature about how quantum mechanics is not understandable, The so called wave function (solution to Schrodinger's equation) has a fairly simple interpretation as effectively the square root of a probability density function, albeit in order to account for quantum phenomenon this square root of the probability density function is often a complex function and not a real one.

However given that probabilities are constant and not functions of energy, the direct link between the solution to Schrodinger's equation and a probability density function is no longer there in relativistic quantum mechanics, Those who spend all their time trying to understand the 'meaning' of the wave function in quantum mechanics are going to have change their understanding when it comes to relativistic quantum mechanics.

There is a solution in terms of particle currents which I will post a subsequent blog on but I do find it surprising given all the hoo hah there is about the interpretation of quantum mechanics and focusing on the solution to Schrodinger's equation that this difference hasn't been emphasised at all, At least superficially it would mean that the Born interpretation is no longer applicable to relativistic quantum mechanics,

No comments:

Post a Comment