Deriving the continuity equation from the Dirac equation (Relativistic Quantum)

In summary, the continuity equation is Derivation of the continuity equation:- The Dirac equation governs the behavior of particles in a quantum system.- The equation states that the momentum of a particle is a function of its position and energy.- The equation can be solved for the position and momentum of a particle at any given time.- The continuity equation states that the momentum of a particle at any given time is a function of the momentum of the particle at previous time.
  • #1
toam
13
0
So I am trying to derive the continuity equation:

[tex]\frac{\partial}{\partial x^{\mu}}J^{\mu} = 0[/tex]

From the Dirac equation:

[tex]i\gamma^{\mu} \frac{\partial}{\partial x^{\mu}}\Psi - \mu\Psi = 0[/tex]

And its Hermitian adjoint:

[tex]i\frac{\partial}{\partial x^{\mu}}\overline{\Psi}\gamma^{\mu} - \mu\overline{\Psi} = 0[/tex]

Where:

[tex]\overline{\Psi}=\Psi^{+}\gamma^{0}[/tex] (Dirac conjugate)



The Attempt at a Solution


By multiplying the Dirac equation on the right by [tex]\overline{\Psi}[/tex] and the adjoint on the right by [tex]\Psi[/tex] I get:

[tex]i(\frac{\partial}{\partial x^{\mu}}(\gamma^{\mu}\Psi)\overline{\Psi} + \frac{\partial}{\partial x^{\mu}}(\overline{\Psi})\gamma^{\mu}\Psi) - \mu(\Psi\overline{\Psi} - \overline{\Psi}\Psi)=0[/tex]

The first term is basically what I am after (except I am not 100% sure I can simply apply the product rule - what is the correct order?) which means I shoudl expect the second term to go to zero:

[tex]\Psi\overline{\Psi} - \overline{\Psi}\Psi =0[/tex]

But because [tex]\gamma^{0}[/tex] is a 4x4 matrix, [tex]\Psi[/tex] is a 4x1 and [tex]\overline{\Psi}[/tex] is a 1x4, I should also expect the second term to be multiplied by the 4x4 identity matrix (so that the subtraction makes sense). However the first term is NOT a constant multiplied by the identity so I don't see how this works.



Any help would be greatly appreciated...
 
Physics news on Phys.org
  • #2
toam said:
By multiplying the Dirac equation on the right by [tex]\overline{\Psi}[/tex] and the adjoint on the right by [tex]\Psi[/tex] …

Hi toam! :smile:

Don't you have to multiply one of them on the left? :confused:
 
  • #3
I tried that and got something else that didn't work. However I will try again because I was surprised that it didn't work so I may have made a mistake or missed something obvious...
 
  • #4
Ok so it turned out I had multiplied the wrong function on the left. It worked out quite simply when I fixed that. The lecture notes had erroneously shown both functions multiplied on the right.

Thanks, tiny-tim.
 

What is the Dirac equation?

The Dirac equation is a relativistic quantum mechanical equation that describes the behavior of spin-1/2 particles, such as electrons, in a quantum field theory. It was developed by physicist Paul Dirac in 1928.

What is the continuity equation?

The continuity equation is a fundamental principle in physics that states that the total amount of a quantity, such as energy or charge, must remain constant in a closed system. It is a consequence of the law of conservation of energy and can be applied to various physical systems, including quantum systems.

How is the continuity equation derived from the Dirac equation?

The continuity equation can be derived from the Dirac equation by considering the time evolution of the probability density of a particle described by the Dirac equation. This probability density is related to the continuity equation through the concept of probability conservation. By applying mathematical operators to the Dirac equation, the continuity equation can be obtained.

What are the assumptions made when deriving the continuity equation from the Dirac equation?

The derivation of the continuity equation from the Dirac equation assumes that the quantum system is in a stationary state, meaning that it is not changing over time. It also assumes that the system is in a vacuum with no external forces acting on it.

What are the implications of the continuity equation in quantum mechanics?

The continuity equation has important implications in quantum mechanics, as it describes the behavior of quantum systems and their conservation laws. It allows for the prediction of the behavior of particles and their interactions with the surrounding environment. It also plays a crucial role in understanding the behavior of quantum fields and their energy distributions.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
823
  • Advanced Physics Homework Help
Replies
0
Views
618
  • Advanced Physics Homework Help
Replies
10
Views
461
  • Advanced Physics Homework Help
3
Replies
95
Views
5K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
926
  • Advanced Physics Homework Help
Replies
25
Views
3K
  • Advanced Physics Homework Help
Replies
4
Views
891
  • Advanced Physics Homework Help
Replies
1
Views
1K
Back
Top