Dirac equation

  • #1
Hello there I have a problem about Dirac equation
36725c97374cea9194a5e035be2c2b60.png

So I wanna know what is matrices β,αk,pk value. And is it right that with Dirac equation we can calculate every particle spin and how we take dervitative of Ψ(x,t) and what is Ψ(x,t) value.
 

Answers and Replies

  • #2
481
55
The matrices ##\beta##, ##\alpha_k## are 4x4 Hermitian matrices and are defined by the Clifford algebra, which means that in this notation they satisfy ( {,} is the anticommutator and 1 is the unity matrix):

##\{\alpha^i,\alpha^j\} = 2 \delta^{ij}##
##\{\alpha^i,\beta\} = 0##
##\beta^2 = 1##

Their explicit matrix elements depend on which of these you want to make diagonal (Dirac representation, Chiral representation, Majorana representation...). The ##p^k## is not a matrix but the momentum operator which becomes a quantum operator in the form of a spatial derivative: so the one you posted is the free Dirac equation in which ##\psi## is the eigenfuction unknown, a bispinor (4 component spinor). Solving for ##\psi## in the Dirac representation you find (after a lengthy calculation) the normalized eigenfunctions for positive and negative energy eigenvalues. And then you can change representations using the appropriate transformation matrices.
 
  • Like
Likes Matterwave and AleksanderPhy
  • #3
Thanks(;
 

Related Threads on Dirac equation

  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
11
Views
771
  • Last Post
2
Replies
37
Views
6K
  • Last Post
Replies
4
Views
632
  • Last Post
Replies
1
Views
626
  • Last Post
Replies
5
Views
836
  • Last Post
Replies
6
Views
854
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
16
Views
4K
Top