Answer to Question #125309 in Quantum Mechanics for Komal sardar

Question #125309

. In Dirac’s theory, the probability current density is defined by the relation j(r, t) = CΨ* αΨ ,

where Ψ is the four component wave vector. Write the relations for jx, jy, jz in terms of the

component of Ψ i.e.

J(r, t) = Ψ*α Ψ ; jx = C Ψ* αx Ψ


1
Expert's answer
2020-07-06T17:02:17-0400

The answer is in the next formula:

"j_1(x_\\mu)=\\Psi^*_0\\Psi_3+\\Psi^*_1\\Psi_2+\\Psi^*_2\\Psi_1+\\Psi^*_3\\Psi_0"


"j_2(x_\\mu)=i\\Psi^*_0\\Psi_3+i\\Psi^*_1\\Psi_2+i\\Psi^*_2\\Psi_1-i\\Psi^*_3\\Psi_0"


"j_3(x_\\mu)=-\\Psi^*_1\\Psi_3+\\Psi^*_0\\Psi_2-\\Psi^*_3\\Psi_1+\\Psi^*_2\\Psi_0"

Note"\\Psi^*_i" is the complex conjugated component of the 4 vector of Dirac.

Notably;

"\\Psi_i=\\Psi_i(x\\mu),\\Psi^*_i=\\Psi^*_i(x_\\mu), i=0..3"


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