SOLUTION OF
DIRAC EQUATION
Discuss the general solution for Dirac’s free particle at
rest and find its eigenvalues and normalized
eigenvectors.
The Dirac equation is,
1
Where,
Dirac found that a and b do not represent basis of his
formulation. He studied properties of a and b and noted that
they are similar to Pauli spin matrices with some differences.
Dirac expressed a and b in terms of Pauli spin matrices in a
space of 4 x 4 matrices. These matrices satisfies all relations
involving a and b.
The covariant form of above equation is,
2
Where,
He found 4 gamma matrices. These
matrices could not form basis.
The combinations of these gamma matrices along with
identity matrix is found to form a complete basis of Dirac
formulation.
The solutions of Dirac equation
are the four component Dirac
spinors, given by,
Here,
Dirac equation is relativistic wave equation which describes fields
for fermions as a vector of four complex numbers (bispinors) in
contrast to the Schrodinger equation which describes field of any
one complex value.
For a free Dirac particle, equation (1) can be written as,
3
Let us consider a plane wave solution for a free Dirac particle
at rest,
4
P=0
Substituting (4) in (3),
Never vanish!
5
Here ua represents bispinor amplitudes, which are
independent of space-time coordinates.
Writing equation (5) in matrix form:
6
This matrix equation provides four sets of linear equations.
For non-trivial solution,
On solving,
>>
>>
7
Thus, there is possibility of negative energy states. That state
represents energy state of antiparticle.
To find the eigenstates, we use identity:
P = 0, for free Dirac particle at rest
>>
Writing this equation in matrix form:
8
H-E
H+E
Comparing the matrix (6) and (8), we observe that each
column of (H+E) matrix gives zero when operated by (H-E)
matrix.
6
Therefore, the columns of (H+E) matrix is our desired bispinor
amplitudes! Therefore we have,
Bispinor amplitudes are
The spinors u1 and u2 corresponds to energy E+ = mc2 with
spin-up and spin-down. Therefore,
Normalizing u1 spinor,
Similarly, we get,
Therefore, spinors u1 and u2 corresponds to positive energy
with spin-up (↑) and spin-down (↓) are given by,
9
The corresponding eigenstates are,
10
Similarly, spinor amplitudes of negative energy state
E- = -mc2 are given by,
Normalizing this, we get,
Therefore spinor amplitude becomes,
11
11
The corresponding eigenstates are,
12
Therefore (9) and (11) represent bispinor amplitudes of Dirac
free particle at rest. Similarly equation (10) and (12) represent
eigenstates of that particle.
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