PHY 301: Mathematical Methods I 2 2 Complex (Real) Matrices

Transcription

1 PHY 30: Mathematical Methods I Complex Real Matrices Dr. Alok Kumar Department of Physical Sciences IISER, Bhopal Abstract For physicist the matrices are of utmost importance and in particular σ-matrices are very important. Spin- system is one of the important portions of Quantum Mechanics which demands controlled knowledge 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 group. N.B.: Please note that the main reference will be always [] and other references are just supplementary and complementary to that. The following matrices are called the σ-matrices or Pauli matrices. 0 σ x = 0 0 i σ y = P auli Sigma M atrices i 0 0 σ z =. 0 address: alok@iiserbhopal.ac.in

2 The identity matrix is I = 0 0. 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 [,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, tra = n i= A ii = 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 - detexpa = exptra or log deta = trloga. 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 trσ x = trσ y = trσ z = 0 detσ x = detσ y = detσ z = σ x = σ x σ y = σ y σ z = σ z. P roperties of P auli Matrices 4 N.B.:Pauli Matrices are traceless, Hermitian, with the determinant -. All Pauli matrices have two eigenvalues + and - 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

3 it to memory for the future. σ x = σ y = σ z = I σ x σ y = σ y σ x = iσ z σ y σ z = σ z σ y = iσ x P roperties of P auli M atrices with cyclic properties. σ z σ x = σ x σ z = iσ y σ x σ y σ z = ii 5 All the properties of equation 5 can be written in compact from by introducing the notation, σ = σ x, σ = σ y and σ 3 = σ z along with ɛ ijk and δ ij. σ i σ j = iɛ ijk σ k + δ ij I anti symmetric part symmetric part where dummy indices mean sum over for k =,, 3. 6 δ ij = { if i=j 0 if i j for ijk = 3, 3, 3 : even permutation with cyclic combination ɛ ijk = for ijk = 3, 3, 3 : odd permutation with cyclic combination 0 otherwise 8 where ɛ ijk = ɛ jik = ɛ 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 7 AB = AB BA + AB + BA } {{}} {{} anti symmetric symmetric part 9 = [A, B] + {A, B}. commutator anti commutator Now what about σ σ j in the light of equation 9. We can have safely the following relation σ i σ j = [σ i, σ j ] + {σ i, σ j }. 0 commutator anti commutator 3

4 On comparing the equation 6 and equation 0, we have the commutation and the anti-commutation relations for Pauli matrices. [σ i, σ j ] = iɛ ijk σ k commutation relation {σ i, σ j } = δ ij I anti commutation relation Always remember the commutation relation of sigma matrices as in equation and the anti-commutation relation as in equation. Now in equation if i j, then the anti-commutation of different Pauli matrices are zero and hence we say Pauli matrices anticommutate with each other read pages 08-0 and exercises 3..3 and 3..4 of []. N.B.:Pauli Matrices are complex, traceless, Hermitian, unitary, anti-commuting with the determinant -. All Pauli matrices have two eigenvalues + and -. For the sake of completion, let us study the following identity and theorem. For two three-dimensional vectors A and B that commute with σ, we have the following identity σ. A σ. B = A. BI + i σ. A B 3 where σ = σ x, σ y, σ z = σ, σ, σ 3. For proof, check L.H.S.=R.H.S.. Theorem: Any arbitrary complex matrix can be written as the linear combination of four matrices I, σ x, σ y and σ z. Therefore, it can be proved that I, σ x, σ y, σ z form an orthogonal basis for the complex Hilbert space of all matrices. This set I, σ x, σ y, σ z forms a colplete set. Let M is an arbitrary complex matrix, then it can expanded as M = a 0 I + a σ x + a σ y + a 3 σ z = a 0 I + a. σ. 4 For m m M = m m we have the following solution for the equation 4 [3] 5 M = m + m I + m m σ z + m + m 4 σ x + i m m σ y. 6

5 Now if you observe the equation 4 closely, the matrix M will be Hermitian iff all a 0, a, a, a 3 or a 0, a are real. Can you express all coefficients of expansion in equation 4 in terms of matrix M? If yes, How? The answer is yes - take the trace both sides of equation 4 and use the property of traceless nature of Pauli matrices and we will find a 0. For a multiply the equation 4 both sides by σ x, take the trace and using the properties of Pauli matrices as in equation 5 we find a and similar story for a and a 3. The results can written in compact form as follows a 0 = trm a = tr σm } 7 where we are using the notation a = a, a, a 3 and σ = σ x, σ y, σ z. The algebra generated by orthonormal Pauli matrices σ x, σ y, σ z 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 0 i ; ; ; 0 } σ z with eigenvalues and 8 } σ x with eigenvalues and 9 } σ i y with eigenvalues and 0 In conclusion, in 97 P. A. M. Dirac extended this formalism Pauli algebra to fast-moving particles of spin-, such as electrons and neutrinos. The generalization Pauli matrices to four dimension is γ matrices or we can say the relativistic treatment of spin- particle leads to 4 4 γ matrices, while the spin- of a non relativistic particle is described by the σ matrices. The generalization of Pauli algebra to 6-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 []. Clifford algebras and generalized Clifford algebras are well studied with their physical applications [5]. References 5

6 . George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists sixth edition, Academic Press, Richard P. Feynman, Robert B. Leighton and Matthew Sands Lectures on Physics, Volume 3 Indian edition, Narosa Publishing House, Claude Cohen-Tannoudji, Bernard Diu and Franck Laloe, Quantum Mechanics, Volume, John Wiley & Sons asia Pte. Ltd., W. E. Baylis, J. Huschilt and Jiansu Wei, Am. J. Phys. 60: 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 D. Hestenes, Am. J. Phys. 39: The following exercises are required to be submitted.. trσ i σ j = δ ij and hence that tr σ. a σ. b= a. b;. σ. a = a I; 3. e iθ σ.ˆn = cos θi + i σ.ˆn sin θ 4. Calculate tre i σ. a e i σ. b 5. Show that if U is a unitary matrix, it can always be expressed as U = e iγ I cos ω + iˆn. σ sin ω where ω and γ are real angles, and ˆn is a real unit vector. And using the identity as in equation 3, show that e iγ I cos ω + iˆn. σ sin ω = expiγ + iωˆn. σ. 6