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Relativistic Quantum Mechanics
Dr.K.P.SatheeshPrincipal
Government College, Ambalapuzha
November 5, 2010
(following is a lecture on elementary level introduction to
Relativisticquantum mechanics which has been used in various
seminars for studentsdoing MSc. in Physics. Main reference is the
text book on Advanced quan-tum Mechanics - Zakurai)
Relativistic notation - Initial attempts in Relativistic Quantum
mechanics- Klein Gordon Equation - Probability conservation -
Derivation of Dirac
Equation - Conserved current - Representation independence
1 Introduction
It is well known that free particle Hamiltonian is notp2
2m
when rela-
tivistic effects are taken into consideration. Consequently
Schrodinger equa-
tion obtained from this Hamiltonian by operator replacement has
to be mod-ified. Attempts to modify the Schrodinger equation by
using the relativisticexpression for energy (ie) E = (p2c2+
m2c4)
1
2 possess certain undesirable fea-tures. One unattractive
feature has been the appearance of different typesof space and time
operators which is against the spirit of relativity theory in
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which space and time are considered to be the components of same
four vec-
tor (ie) E i t , p i would lead to it = (2c22 + m2c4)1
2 .Moreover the meaning of the operator (2c22 + m2c4)
1
2 itself is not quiteclear. Even though the interpretation of
square root operator can be madetransparent by transforming to
momentum space the procedure fails if wewere to consider p (p
eA
c) which is required in the presence of elec-
tromagnetic field. attempts to develop a relativistic quantum
mechanics wasinitiated by Schrodinger himself immediately after his
development of wavemechanics. In order to avoid the difficulties
associated with the square rootof a linear operator he tried to
develop the basic equation by effecting theoperator replacement in
the expression for E2. Consequently he developed
the equation now known as Klein Gordon equation.Actually
Schrodinger con-sidered the relativistic equation unsatisfactory
and did not pursue it further.Physical implications of this
equation was taken up by Klein and Gordon.Meanwhile Dirac succeeded
in obtaining a first order wave equation elimi-nating many of the
apparent defects attributed to Klein Gordon equation.All these
interim attempts later lead to a full fledged Relativistic
QuantumMechanics which gradually took a satisfactory shape now
known as Rela-tivistic Quantum Field Theory. In this theory both
Klein Gordon and Diracequations are perfectly respectable
equations.
2 Klein Gordon Equation
Operator replacement is made in the relativistic energy equation
toobtain the following
E2 = p2c2 + m2c4 (1)
2
2m
2
t2= (2c22 + m2c4) (2)
1
c22
t2 2 +
mc
2 = 0 (3)
This equation is known as Klein Gordon equation. This equation
can bewritten in a compact and convenient form using the
relativistic notationdescribed below.
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3 Relativistic Notation
In the four vector notation (that is going to be used
throughout) four vector
b with = 1, 2, 3, 4 stands for b = (b1, b2, b3, b4) = (
b , ib0), b1, b2 and b3are real and b4 = ib0 is purely
imaginary. In general Greek indices , , . . .run from 1 to 4 where
as i, j, k . . . run from 1 to 3. The coordinate vector
x = (x1, x2, x3, x4) = (x,ict). Symbol x, y, z may also be used
in the place
ofx1, x2 and x3. All the repeated indices are summed unless
otherwise statedexplicitly.Under a Lorentz transformation we have x
= ax (repeated indices aresummed)a satisfy the following
relations:
aa = (4)
(a1) = a (5)
a = (a1)x
= ax
(6)
The DAlembertian operator (Box operator) is defined as
2 1
c22
t2(7)
Also = with =
xIn this notation the Klein Gordon equation
can be written as =
mc
2 (8)
In natural units = c = 1 and hence
( m2) = 0 (9)
is the Klein Gordon equation.
4 Defects of Klein Gordon equation
When Klein Gordon equation is treated as the field equation of
ascalar particle (like a meson) it is a perfectly respectable and
valid equa-tion. More over it is manifestly Lorentz covariant. But
in single particlerelativistic quantum mechanics it was treated
unsatisfactory. These defectswere historically the main motivation
for the development of Dirac equationin relativistic quantum
mechanics. Main defects are listed below.
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1. According to the dynamical postulate of quantum mechanics an
equa-
tion of motion should be first order in time. Klein Gordon
equation isnot.
2. Equation gives the possibility of negative energy states
which is notproperly accounted for.
3. The equation is not capable of predicting or handling the
spin of aparticle
4. Unlike Schrodinger equation the definition of a positive
definite proba-bility density and probability current density is
not normally possible
in Klein Gordon Equation
Apparent defects of Klein Gordon equation listed above motivated
Dirac todevelop a first order relativistic wave equation because he
identified that theabove mentioned defects are due to to the root
cause that wave equation issecond order. Attempt of Dirac met with
tremendous success theoreticallyand experimentally.
5 Probability Conservation in RQM
In conventional quantum mechanics ||2
d3
x is interpreted as the prob-ability of finding the particle
with wave function in the volume elementd3x. The probability
density P and the flux density S are given by
P = ||2 > 0 (10)
S =
i
2m
( ) (11)
By the elementary methods using Schrodinger equation the
continuity equa-tion is obtained as
P
t+ .S = 0 (12)
By proper normalization of we can set
P d3x = 1. This integral overall space is a constant of motion.
Now the question is how to construct arelativistic analogue of the
non relativistic equation of continuity? Followingare the main
requirements.
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Construction of bilinear forms which can be interpreted as the
proba-
bility density and the flux density satisfying the equation of
continuity
According to the fundamental hypothesis of quantum mechanics
theprobability density must be positive definite
According to the requirements of Special theory of relativity
ProbabilityP should transform as the fourth component of a four
vector density.
In the covariant form the continuity equation has to be written
asS = 0 with S (S, icP)
We use all the above ideas in RQM based on Klein Gordon
equation. The
four vector current density is defined asS A(
) (13)
Where is the solution of the Klein Gordon equation. Taking the
derivativewe get
Sx
S = A( +
)
= A[ [(
) ] ] (14)
According to the Klein Gordon equation this quantity vanishes
giving
S = 0 (15)Now consider a particle obeying Klein Gordon equation
which is
moving with non-relativistic velocities with energy E mc2. the
solu-tion of the Klein Gordon equation and Schrodinger equation are
related by emc
2t/. and are respectively the solutions of Schrodinger andKlein
Gordon equation.The components of the four vector density s cannow
be determined using equation [13].
s0 = is4
2imc
A||2 (16)
s = A[ ()] (17)s and s0 will respectively be the flux density
and c times the probabilitydensity of Schrodinger theory if we set
A = i
2m. Thus a four vector den-
sity obtained from the solution of Klein Gordon equation has the
followingproperties.
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1. The current density satisfies the continuity equation
2. The components of the current density coincide with the flux
densityand c times the probability density in the non relativistic
limit.
Using s = (S, icP) and x = (x,ict) the probability density P can
be writtenas
P =i
2mc2
t
t
(18)
In contrast to Schrodinger theory Klein Gordon theory has a
seriousdifficulty in interpreting the above equation. Since Klein
Gordon equationis second order in time both and d
dtcould be arbitrarily fixed. this spoils
the positive definite feature of P given by eqn.[18]. Clearly
this difficulty canbe avoided by writing a wave equation which is
linear in time. This is whatDirac did in 1928. Using the equation
which is linear in
tDirac succeeded in
deriving the zeroth component of current density which is
positive definite.the difficulty associated with probability
density was thus removed in Diracstheory. Because of this reason
Dirac equation was considered to be the onlycorrect relativistic
wave equation during the period 1928 to 1934. In 1934Pauli and
Weisskopf revived Klein Gordon equation and interpreted s asthe
charge density (instead of probability current density) which need
not bepositive definite. The interpretation is more appropriate
when the solution
of Klein Gordon equation is interpreted as a quantized field
operator insteadof a single particle wave function.
6 Derivation of Dirac Equation
As indicated earlier the derivation of Dirac1 equation was
motivatedby the apparent defects of Klein Gordon equation. Klein
Gordon equationdo not naturally incorporate spin. A study of the
method of incorporationof spin in non relativistic quantum
mechanics can be used as a spring boardfor finding the procedure
for incorporating spin in relativistic quantum me-
chanics. The method developed by Pauli to account for the
interaction ofelectron magnetic moment with the magnetic field in
non relativistic quantummechanics is to add term H(spin) =
e2mc
.B to the usual Hamiltonian.
1This approach based on the necessity of including spin is
different from the original
approach of Dirac who tried to make the equation linear so that
probability function
becomes positive definite.
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Where B is the applied magnetic field and are the Pauli spin
matrices.
This ad-hoc procedure appears unsatisfactory since we believe
that funda-mental electromagnetic interactions are those which can
be generated by thereplacement of mechanical momentum by
electromagnetic momentum (ie)p p
eAc
. this fact motivates the introduction of another procedure
for
the introduction of spin. Instead of using kinetic energy
operator H(KE) = p2
2m
in the absence of a vector potential we use H(KE) = (p)(p)2m
which is essen-tially same when there is no vector potential by
virtue of the equation
( A)( B) = A B + i (A B) (19)
But the equation is different if we make the substitution p p
eAc
Then we have1
2m
p
eA
c
p
eA
c
=1
2m
p
eA
c
2+
i
2m
p
eA
c
p
eA
c
=1
2m
p
eA
c
2
e
2mc B (20)
Note:Whenever operator product involving derivatives are used
remember
this. Always imagine that the product is applied to a quantity
and apply theproduct rule. Therefore we have p (A) = i[( A) + A
].Therefore p A = i( A) A p. (In deriving [eqn. 20] use of thisidea
is made)
The procedure used for incorporating electron spin in a non
relativistictheory can be used for developing relativistic theory
of a spin half particle.(ie) the replacement p p is made in the
relativistic energy equation
E2
c2
= p2 + m2c2 E(op)
c
pE(op)
c
+ p = mc2 (21)
The operators are E(op) = i t
= ic x0
;p = i This operator replace-ments gives a second order
differential equation first written by B.L. van derWaerden for a
free electron.
i
x0+ i
i
x0 i
= mc2 (22)
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In the above equation is a two component wave function.
In order to convert [eqn.22] to a first order differential
equation we definebelow two two component wave functions (R) and
(L)
(R) =1
mc
i
x0
i( )
, (L) = (23)
We can now consider [eqn.22] as equivalent to two first order
equations
[i( ) i (/x0)] (L) = mc(R)
[i( ) i (/x0)] (R) = mc(L) (24)
For massive particles the equations given above couple (L) and
(R). Eventhough these equations are equivalent to Dirac equation we
can write it inthe form originally written by Dirac by adding and
subtracting them.
i( )((R) (L)) i (/x0) ((R) + (L)) = mc((R) + (L))(25)
i( )((R) + (L)) + i (/x0) ((R) (L)) = mc((R) (L))(26)
Let us now define ((R) + (L)) A and ((R) (L)) B and write
the
above set of coupled differential equations in the form
i (/x0) i i i (/x0)
AB
= mc
AB
(27)
Where the four component wave function (Diracs four component
spinor)is defined by the relation
AB
=
(R) + (L)
(R) + (L)
(28)
By defining the Dirac matrices and using the relativistic
notation we canwrite the above matrix equation in a more compact
manner.
+ 4
(ix0)
+ mc
= 0 (29)
or using the summation convention
(x) +
mc
= 0 (30)
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The matrices with = 1, 2, 3, 4 are the 4 4 matrices usually
known
as Dirac matrices and can be written as a 2 2 matrix using the
Puli spinmatrices defined as follows.Note:The standard form of
Pauli 2 2 spin matrices will be used throughout.
1
0 11 0
, 2
0 ii 0
, 3
1 00 1
, I
1 00 1
I is the unit matrix. The space component (k) and the time
component (4)of the Dirac Gamma matrices are given by
k
0 ikik 0
, 4
I 00 I
(31)
For clarity we write explicitly two of the gamma matrices
below.
3
0 0 i 00 0 0 ii 0 0 00 i 0 0
, 4
1 0 0 00 1 0 00 0 1 00 0 0 1
, etc. (32)
[eqn.30] represents the celebrated Dirac equation which can be
more com-
pactly written in the natural units as
( + m) = 0 (33)
It should be noted that the Dirac equation is actually a set of
four differentialequations that couple the four components of the
bispinor or dirac spinor which can be represented by a column
matrix
=
123
4
(34)
The entity is called a spinor because its transformation
properties under theLorentz transformation is different from that
of an ordinary four vector.
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7 Properties of Gamma Matrices
The 4 4 matrices called the Gamma matrices or Dirac
matricespossess the following genral properties.
1. The square of Gamma matrices are unity (ie) 2 = 1
24 =
I 00 I
2=
I 00 I
2k =
0 ik
ik 0
0 ik
ik 0
=
2k 00 2k
=
I 00 I
2. and satisfy the following anticommutation relations
{, } = + = 2 (35)
For example
12 + 21 =
0 i1
i1 0
0 i2
i2 0
+
0 i2
i2 0
0 i1
i1 0
=
12 + 21 0
0 12 + 21
= 0
3. Gamma matrices are Hermitian (ie) =
4. Gamma matrices are traceless (ie) the sum of the diagonal
elementsvanish as can be easily verified.
8 Dirac equation in Hamiltonian form
Originally Dirac wrote the equation satisfied by a relativistic
particlein the Hamiltonian form H = i(/t) with the Hamiltonian
called DiracHamiltonian satisfying the relation H = ic + mc2 where
and ,are 4 4 matrices. They are defined by the relations
= 4 =
I 00 I
, k = i4k =
0 k
k 0
Alpha and Beta matrices are also traceless hermitian matrices
having proper-ties similar to gamma matrices {k, } = 0,
2 = 1, {k, l} = 2kl Conservedcurrent-Representation
independence
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9 Conserved current
The main motivation behind the derivation of Dirac equation
hasbeen the fact the Klein Gordon equation is not giving a positive
definiteprobability density as required by the basic postulates of
quantum mechanics.In the following we show that Dirac equation
actually gives a positive definiteprobability density. An adjoint
Dirac spinor () in contrast to the Hermitianconjugate spinor () is
defined by = 4. They can be represented asthe following row
matrices.
= (1, 2,
3,
4), = (
1,
2,
3,
4) (36)
Normally is considered to be an independent spinor different
from . Toobtain its wave equation first of all the Hermitian
conjugate of Dirac equationis taken.
xkk +
xk4 +
mc
= 0 (37)
The adjoint Dirac equation is obtained by multiplying the above
equationfrom the right by 4
x +
mc
= 0 (38)
As in the case of non-relativistic quantum mechanics we multiply
the original
Dirac equation from the left by and the adjoint equation from
the rightby and subtract to get
x
() = 0 (39)
The flux density four vector in Dirac theory can therefore be
written as
s = ic = (c, ic) (40)
which satisfies the continuity equation
sx
= s = 0. (41)
It is clear that the quantity
= =
4=1
(42)
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is positive definite unlike the case of Klein Gordon theory.
Thus in Dirac
theory the problem of negative probability density is
eliminated. The prob-ability density is 4 =
and the quantity sk = ick = ck is
the flux density.
10 Representation independence
Representation independence of Dirac equation means that Dirac
equa-tions with different form of gamma matrices are equivalent if
the gammamatrices satisfy the defining property (ie.)
anti-commutation. Let the Diracequation be written in the form
x+
mc
= 0 (43)
In the above equation satisfies the defining property of gamma
matrices(ie.)they are 4 4 matrices satisfying {,
} = 2 with = 1 . . . 4. By
representation independence we mean the assertion that the
equation given
above is equivalent to
x+ mc
= 0 which is the original Dirac equa-
tion. Different possible sets of 44 matrices satisfying the
anti-commutationrelation given above are said to be sets of gamma
matrices in different repre-
sentations. The representation independence can be proved by
using Paulisfundamental theorem which states that two sets of 4 4
matrices satisfyingthe anti-commutation relations {, } = 2 and
{
,
} = 2 with
, = 1 . . . 4 there exists a nonsingular 4 4 matrix S such
that
SS1 = (44)
where the matrix S is unique (up to a multiplicative constant).
The Diracequation in the new representation can now be written
as
SS
1
x +
mc
SS
1
= 0 (45)
Multiplying from the left by S1 we get
x+
mc
S1 = 0 (46)
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which is same as the original Dirac equation with the solution
given by S.
therefor equation [8] is equivalent to the Dirac equation and
the wave func-tions in both representations are related by
= S (47)
According to Paulis fundamental theorem S can be chosen to be
unitary if are Hermitian. The unitarity of the matrix S implies the
equivalence ofprobability density and flux density in both
representations as seen below.
= 4
= SS4S1S
S1S
= (48)
This clearly shows that all the physical consequences are same
in both repre-sentations. But the wave functions in different
representations look different.Most common gamma matrices used are
the following
1. The standard Dirac Pauli representation used in the present
treatment.
2. The Weyl representation in which k and 4 are off diagonal
matrices.
3. The Majorana representation in which k are purely real and 4
is
purely imaginary.
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