Dirac current probablities

In summary, the student is trying to integrate the Dirac current eq. and is having trouble because the adjoint is not needed and the real part of the first Dirac eq is always larger than the real part of any of the other three.
  • #1
woodland
3
0
I'm having a problem writing the third Dirac current eq.
$$1 = \int ψ^t \gamma^0 \gamma^2 ψ$$
which should come out as
$$1 = \int i ψ^0 ψ^3 - i ψ^1 ψ^2 + i ψ^2 ψ^1 - i ψ^3 ψ^0$$
By inspection the first and last terms add to zero and the second and third terms add to zero, so the integral cannot equal one. I checked my matrix math, calculated new gammas but nothing works. Am I making a typo error somewhere or what? Thanks
 
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  • #2
Shouldn't the transpose actually be an adjoint?
 
  • #3
Thanks. I checked on the adjoint point. Adjoint means the conjugate transpose. I'm trying to use some curve fitting on the ψ's and this leads to problems. Such as the four terms in $$ψ^t γ^0 γ^0 ψ$$ would all ways be positive and this would make the first Dirac current always larger than the other three. I simulated this for 10000 tries and it was true every time. As such all four equations cannot be integrated to 1 simultaneously. This is if the adjoint is used. So checking around I cannot find anywhere that using it must be the case. It's just used. I remember my first college physics classes. A lot of the other students (myself included I suppose) had teachers who used conjugation just to get real answers rather than use another class time to explain what happens to the imaginary part. So I have to ask if the Dirac currents should be the adjoint where is the explanation for this need?
 
  • #4
Sorry my last entry should have had the first Dirac eq as the adjoint $$ψ^* γ^0 γ^0 ψ $$
I redid the idea of is real(Dirac eq #1) <> real(Dirac eq 1 to 4) a million times with random choices for the complex (about 1 to 0) $$ψ^μ$$ and the real part of the first Dirac eq is always larger than the real part of any of the other three. I cannot see how the integral for all four can be made 1. Remember all four ψ's occur in any current so any choice of a coefficient of one applies to all. That's why I was researching the Dirac current not being needed to be adjoint. To be fair this needs to be continued on another thread. I'll check back though, maybe someone might have some ideas.
 

What is Dirac current probability?

Dirac current probability, also known as the probability current density, is a concept in quantum mechanics that describes the flow of probability of a particle in a particular direction. It is represented by the symbol j, and is related to the wave function of a particle through the continuity equation.

How is Dirac current probability related to the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is closely related to Dirac current probability, as it is used to calculate the wave function of a particle, which in turn is used to calculate the probability current density.

What is the significance of Dirac current probability in quantum mechanics?

Dirac current probability is significant in quantum mechanics because it allows us to understand the behavior of particles at the quantum level. It is a fundamental concept that is used to calculate the probability of a particle being in a certain location at a given time, which is crucial for understanding the behavior of particles in quantum systems.

How is Dirac current probability different from classical current?

Dirac current probability is fundamentally different from classical current, as it describes the flow of probability rather than the flow of particles. In classical physics, current is described as the flow of charged particles, whereas in quantum mechanics, it is described as the flow of probability of a particle being in a particular location.

What is the relationship between Dirac current probability and quantum tunneling?

Quantum tunneling is a phenomenon where a particle can pass through a potential barrier even when it does not have enough energy to overcome it. Dirac current probability plays a crucial role in understanding this phenomenon, as it allows us to calculate the probability of a particle tunneling through a potential barrier, which is essential for many applications in quantum technology.

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