One-particle Dirac and KG equations

[Palindrom]

So there's something I don't quite understand.

The density in the KG equation stands for charge density. Here are several questions:

1. For a KG particle, how do I (if it all) find the position probabilty density?
2. For a Dirac particle, what does the (now positive definite) density stand for?
3. For a now KG or Dirac field, is there no position probability density?

 

[Meir Achuz]

In the Dirac equation |\psi|^2 is the position probability, just as in the Schrodinger Eq. The KG equation has a problem. The one particle solution does not conserve probablility for \phi^*\phi. One (not too satisfactory) solution of this is to call it charge density. The real way out is to second quantize the KG equation, leadilng to QFT. Then, the KG
operator is an operator on a new wave function. The same second quantization can be applied to the Schrodingeer and the Dirac equations.
You need a book beyond the first level QM for this.
I like one by Halzen and Martin, but there are several.

 

[denian]

The one-particle Dirac and KG equations are fundamental equations in quantum mechanics that describe the behavior of a single particle in a relativistic framework. These equations have been extensively studied and have provided a basis for understanding the behavior of particles at high energies and speeds.

To address your questions:

1. In the KG equation, the charge density represents the probability of finding the particle at a certain position. This can be calculated by taking the square of the wavefunction, which gives the probability amplitude, and then integrating over all space. This will give you the position probability density.

2. In the Dirac equation, the positive definite density represents the probability of finding the particle in a specific spin state. This is because the Dirac equation incorporates spin into its formalism, unlike the KG equation which only describes spinless particles.

3. For a KG or Dirac field, the concept of position probability density still exists. However, the equations themselves are not applicable to fields, as they are meant to describe the behavior of a single particle. To describe fields, we use different equations such as the Klein-Gordon or Dirac field equations.

I hope this helps clarify your understanding of these equations. They are complex and require a deep understanding of quantum mechanics, but they have been crucial in our understanding of the behavior of particles in the universe.