Download Pauli matrices

PHY 301: Mathematical Methods I
2 × 2 Complex (Real) Matrices
Dr. Alok Kumar∗
Department of Physical Sciences
IISER, Bhopal
Abstract
For physicist the 2 × 2 matrices are of utmost importance and in
particular σ-matrices are very important. Spin- 21 system is one of the
important portions of Quantum Mechanics which demands controlled
knowledge 2 × 2 matrices. It is not to be stressed that whatever we
do in Quantum Mechanics, at the end of the day we are with some
matrices (both for discrete as well as continuous basis) to play around
and even in a experiment you are with certain set of numbers in the
form of matrices or others. Let us study it before we move to formal
Matrix Theory.
N.B.: Mathematics is best learnt by solving problems.
N.B.: Pauli matrices are one of the convenient choices of the
generators of the Lie algebra of SU (2) group.
N.B.: Please note that the main reference will be always [1] and other
references are just supplementary and complementary to that.
The following matrices are called the σ-matrices or Pauli matrices.
!
σx =
σy =
σz =
∗
0 1
1 0 !
0 −i
i 0 !
1 0
.
0 −1











P auli (Sigma) M atrices










e-mail address: [email protected]
1
(1)
The 2 × 2 identity matrix is
I=
1 0
0 1
!
.
(2)
How can we get Pauli matrices? The Pauli matrices arised in Pauli’s treatment of spin in quantum mechanics. Read any Quantum Mechanics book or
Feynman Lecture Volume 3 [2,3].
The trace and the determinant of a matrix are the scalar quantity.The trace
is equal to the sum principal diagonal elements of the matrix. The determinant is the product of the eigenvalues of the matrix. The importance of the
trace and the determinant lie in the fact that its value is basis independent.
Formally, for n × n matrix ’A’ the trace is defined as follows,
tr(A) =
Xn
i=1
Aii = sum of eigenvalues of A.
(3)
It is to be pointed out the trace and the determinant of a matrix are the
sum and the product of the eigenvalues of a matrix respectively and hence
both related to each other, find out what is the relation between the trace
and the determinant of a matrix - det(exp(A)) = exp(tr(A)) or log det(A) =
tr(log(A)). We notice that σ-matrices are complex and it is always a good
habit to check some properties if a matrix or set matrices given to you like
trace, determinant, hemiticity, commutation relations, eigenvalues, rank .
We notice
tr(σx ) = tr(σy ) = tr(σz ) = 0
det(σx ) = det(σy ) = det(σz ) = −1
σx † = σx
σy† = σy
σz† = σz .








P roperties of P auli M atrices (4)







N.B.:Pauli Matrices are traceless, Hermitian, with the determinant
’-1’. All Pauli matrices have two eigenvalues +1 and -1 and find
corresponding eigen vectors.
The next question should come to your mind, what will happen if we do
matrix-multiplication of two Pauli matrices ? Let us do it once and commit
2
it to memory for the future.
σx 2 = σy 2 = σz 2 = I
σx σy = −σy σx = iσz
σy σz = −σz σy = iσx
σz σx = −σx σz = iσy
σx σy σz = iI








P roperties of P auli M atrices with cyclic properties.







(5)
All the properties of equation (5) can be written in compact from by introducing the notation, σ1 = σx , σ2 = σy and σ3 = σz along with ijk and δij .
σi σj =
+
iijk σk
| {z }
anti−symmetric part
δij I
(6)
|{z}
symmetric part
where dummy indices mean sum over for k =1, 2, 3.
(
δij =
1 if i=j
0 if i6= j
(7)



1 f or ijk = 123, 231, 312 : even permutation with cyclic combination
−1 f or ijk = 213, 132, 321 : odd permutation with cyclic combination


0
otherwise
(8)
where ijk = −jik = −ikj is permutation tensor and is antisymmetric with
respect to all pairs of indices.
Let us consider the product of two matrices ’A’ and ’B’ i.e. C=AB and we
can write
ijk =
AB =
1
1
(AB − BA) + (AB + BA)
|2
{z
} |2
{z
}
anti−symmetric
symmetric part
1
1
[A, B] +
=
2 | {z }
2
commutator
(9)
{A, B}
.
| {z }
anti−commutator
Now what about σ1 σj in the light of equation (9). We can have safely the
following relation
σi σj =
1
1
[σi , σj ] +
2 | {z }
2
commutator
3
{σi , σj }
|
{z
}
anti−commutator
.
(10)
On comparing the equation (6) and equation (10), we have the commutation
and the anti-commutation relations for Pauli matrices.
[σi , σj ] = 2iijk σk
|
{z
(11)
}
commutation relation
{σi , σj } = 2δij I
|
{z
(12)
}
anti−commutation relation
Always remember the commutation relation of sigma matrices as in equation
(11) and the anti-commutation relation as in equation (12). Now in equation
(12) if i 6= j, then the anti-commutation of different Pauli matrices are zero
and hence we say Pauli matrices anticommutate with each other (read pages
208 - 210 and exercises 3.2.13 and 3.2.14 of [1]).
N.B.:Pauli Matrices are complex, traceless, Hermitian, unitary,
anti-commuting with the determinant ’-1’. All Pauli matrices have
two eigenvalues +1 and -1.
For the sake of completion, let us study the following identity and theorem.
~ and B
~ that commute with ~σ , we have
For two three-dimensional vectors A
the following identity
~ σ .B)
~ = A.
~ BI
~ + i~σ .(A
~ × B)
~
(~σ .A)(~
(13)
where ~σ = (σx , σy , σz ) = (σ1 , σ2 , σ3 ). For proof, check L.H.S.=R.H.S..
Theorem: Any arbitrary 2 × 2 complex matrix can be written as
the linear combination of four matrices I, σx , σy and σz . Therefore,
it can be proved that (I, σx , σy , σz ) form an orthogonal basis for the
complex Hilbert space of all 2 × 2 matrices .
This set (I, σx , σy , σz ) forms a colplete set.
Let M is an arbitrary 2 × 2 complex matrix, then it can expanded as
M = a0 I + a1 σx + a2 σy + a3 σz = a0 I + ~a.~σ .
For
M=
m11 m12
m21 m22
(14)
!
(15)
we have the following solution for the equation (14) [3]
M=
m11 − m22
m12 + m21
m12 − m21
m11 + m22
I+
σz +
σx + i
σy .
2
2
2
2
4
(16)
Now if you observe the equation (14) closely, the matrix M will be Hermitian
iff all (a0 , a1 , a2 , a3 ) or (a0 , ~a) are real. Can you express all coefficients of
expansion in equation (14) in terms of matrix M? If yes, How? The answer
is yes - take the trace both sides of equation (14) and use the property of
traceless nature of Pauli matrices and we will find a0 . For a1 multiply the
equation (14) both sides by σx , take the trace and using the properties of
Pauli matrices as in equation (5) we find a1 and similar story for a2 and a3 .
The results can written in compact form as follows
a0 = 12 tr(M )
~a = 12 tr(~σ M )
)
(17)
where we are using the notation ~a = (a1 , a2 , a3 ) and ~σ = (σx , σy , σz ). The
algebra generated by orthonormal Pauli matrices (σx , σy , σz ) and identity
matrix is called Pauli algebra of space and it is isomorphic to 3-space algebra,
for its geometrical significance, read [4]. Finally, find the eigen vectors for
Pauli Matrices which are
1
0
!
;
1
√
2
1
1
!
1
√
2
1
i
!
0
1
!)
σz with eigenvalues 1 and − 1
1
;√
2
1
−1
!)
1
;√
2
1
−i
!)
(18)
σx with eigenvalues 1 and − 1
(19)
σy with eigenvalues 1 and − 1
(20)
In conclusion, in 1927 P. A. M. Dirac extended this formalism (Pauli algebra) to fast-moving particles of spin- 21 , such as electrons (and neutrinos).
The generalization Pauli matrices to four dimension is γ matrices or we can
say the relativistic treatment of spin- 12 particle leads to 4 × 4 γ−matrices ,
while the spin- 21 of a non relativistic particle is described by the 2 × 2 σ−
matrices. The generalization of Pauli algebra to 16-dimensional algebra of
γ µ matrices is known as Clifford algebra [6] . The spin algebra generated
by the Pauli matrices is just a matrix representation of the four-dimensional
Clifford algebra [1]. Clifford algebras and generalized Clifford algebras are
well studied with their physical applications [5].
References
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1. George B. Arfken and Hans J. Weber, Mathematical Methods for
Physicists (sixth edition), Academic Press, (2005).
2. Richard P. Feynman, Robert B. Leighton and Matthew Sands
Lectures on Physics, Volume 3 (Indian edition), Narosa Publishing
House, (986).
3. Claude Cohen-Tannoudji, Bernard Diu and Franck Laloe, Quantum
Mechanics, Volume 1, John Wiley & Sons (asia) Pte. Ltd., (2005).
4. W. E. Baylis, J. Huschilt and Jiansu Wei,
(1992).
Am. J. Phys. 60: 788
5. R. Jagannathan, On Generalized Clifford Algebras and their Physical
Applications, The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Editors: Krishnaswami Alladi, John R. Klauder, and
Calyampudi R. Rao, Springer (2010).
6. D. Hestenes, Am. J. Phys. 39: 1013 (1971).
The following exercises are required to be submitted.
1. tr(σi σj )= 2δij and hence that tr(~σ .~a ~σ .~b)= 2~a.~b;
2. (~σ .~a)2 =k~ak2 I;
3. e(iθ~σ.n̂) = (cos θ)I + i~σ .n̂ sin θ
~
4. Calculate tr(ei~σ.~a ei~σ.b )
5. Show that if U is a unitary 2 × 2 matrix, it can always be expressed as
U = eiγ (I cos ω + in̂.~σ sin ω) where ω and γ are real angles, and n̂ is a
real unit vector. And using the identity as in equation (13 ), show that
eiγ (I cos ω + in̂.~σ sin ω) = exp(iγ + iωn̂.~σ ).
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