72
3. Relativistic Quantum Mechanics
3.3 Spin
Four-momentum is not the only physical property of a particle obeying the
Dirac equation. We must now interpret the column vector (Dirac spinor)
part, ω, of the solution (3.41). The particular properties of the σ-matrices,
appearing in the α-matrices, have already led us to think in terms of spin.
A further indication that this is correct comes when we consider the explicit
form of ω given in (3.47). In this equation the two-component spinor φ is
completely arbitrary. It may be chosen in just two linearly independent ways,
for example
( )
( )
1
0
φ ↑ =
φ ↓ =
(3.60)
0
1
1
2
which (as the notation of course indicates) are in fact eigenvectors of S z
σ z
=
1
2
with eigenvalues ± (‘up’ and ‘down’ along the z-axis). Remember that, in
quantum mechanics, linear combinations of wavefunctions can be formed using
complex numbers as superposition coefficients, in general; so the most general
φ can always be written as
( )
a
φ =
= aφ ↑ + bφ ↓
(3.61)
b
where a and b are complex numbers. Hence, there are precisely two linearly
independent solutions, for a given 4-momentum, just as we would expect for
1
2
a quantum system with j = (the multiplicity is 2j + 1, in general).
In the rest frame of the particle (p = 0) this interpretation is straightforward. In this case choosing (3.60) for the two independent φ’s, the solutions
(3.61) for E = m reduce to
( )
( )
1
0
| 0 |
| 1 |
−imt
−imt
( ) e
and
( ) e
.
0
0
(3.62)
0
0
(a)
(b)
Since we have degeneracy between these two solutions (both have E = m)
there must be some operator which commutes with the energy operator, and
whose eigenvalues would distinguish the solutions (3.62). In this case the
energy operator is just βm (from (3.54) setting −i∇ to zero, since p = 0) and
the required operator commuting with β is
(
)
σ z 0
Σ z =
(3.63)
0 σ z
which has eigenvalues 1 (twice) and −1 (twice). Our rest-frame spinors appearing in (3.62) are indeed eigenstates of Σ z , with eigenvalues ±1 as can be
easily verified.
3. Relativistic Quantum Mechanics
3.3 Spin
Four-momentum is not the only physical property of a particle obeying the
Dirac equation. We must now interpret the column vector (Dirac spinor)
part, ω, of the solution (3.41). The particular properties of the σ-matrices,
appearing in the α-matrices, have already led us to think in terms of spin.
A further indication that this is correct comes when we consider the explicit
form of ω given in (3.47). In this equation the two-component spinor φ is
completely arbitrary. It may be chosen in just two linearly independent ways,
for example
( )
( )
1
0
φ ↑ =
φ ↓ =
(3.60)
0
1
1
2
which (as the notation of course indicates) are in fact eigenvectors of S z
σ z
=
1
2
with eigenvalues ± (‘up’ and ‘down’ along the z-axis). Remember that, in
quantum mechanics, linear combinations of wavefunctions can be formed using
complex numbers as superposition coefficients, in general; so the most general
φ can always be written as
( )
a
φ =
= aφ ↑ + bφ ↓
(3.61)
b
where a and b are complex numbers. Hence, there are precisely two linearly
independent solutions, for a given 4-momentum, just as we would expect for
1
2
a quantum system with j = (the multiplicity is 2j + 1, in general).
In the rest frame of the particle (p = 0) this interpretation is straightforward. In this case choosing (3.60) for the two independent φ’s, the solutions
(3.61) for E = m reduce to
( )
( )
1
0
| 0 |
| 1 |
−imt
−imt
( ) e
and
( ) e
.
0
0
(3.62)
0
0
(a)
(b)
Since we have degeneracy between these two solutions (both have E = m)
there must be some operator which commutes with the energy operator, and
whose eigenvalues would distinguish the solutions (3.62). In this case the
energy operator is just βm (from (3.54) setting −i∇ to zero, since p = 0) and
the required operator commuting with β is
(
)
σ z 0
Σ z =
(3.63)
0 σ z
which has eigenvalues 1 (twice) and −1 (twice). Our rest-frame spinors appearing in (3.62) are indeed eigenstates of Σ z , with eigenvalues ±1 as can be
easily verified.