5.61 Physical Chemistry 24 Pauli Spin Matrices Page 1. Pauli Spin Matrices

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1 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 1 Pauli Spin Matrices It is a bit awkward to picture the wavefunctions for electron spin because the electron isn t spinning in normal 3D space, but in some internal dimension that is rolled up inside the electron. We have invented abstract states α and β that represent the two possible orientations of the electron spin, but because there isn t a classical analog for spin we can t draw α and β wavefunctions. This situation comes up frequentl in chemistr. We will often deal with molecules that are large and involve man atoms, each of which has a nucleus and man electrons and it will simpl be impossible for us to picture the wavefunction describing all the particles at once. Visualizing one particle has been hard enough! In these situations, it is most useful to have an abstract wa of manipulating operators and wavefunctions without looking eplicitl at what the wavefunction or operator looks like in real space. The wonderful tool that we use to do this is called Matri Mechanics (as opposed to the wave mechanics we have been using so far). We will use the simple eample of spin to illustrate how matri mechanics works. The basic idea is that we can write an electron spin state as a linear combination of the two states α and β: ψ c α α c β β Note that, for now, we are ignoring the spatial part of the electron wavefunction (e.g. the angular and radial parts of ψ). You might ask how we can be sure that ever state can be written in this fashion. To assure ourself that this is true, note that for this state, the probabilit of finding the electron with spin up ( down ) is c α ( c β ). If there was a state that could not be written in this fashion, there would have to be some other spin state, γ, so that ψ c α α c β β c γ γ In this case, however, there would be a probabilit c γ of observing the electron with spin γ which we know eperimentall is impossible, as the electron onl has two observable spin states. The basic idea of matri mechanics is then to replace the wavefunction with a vector:

2 Phsical Chemistr 4 Pauli Spin Matrices Page c α ψ c α α c β β ψ c β Note that this is not a vector in phsical (,,z) space but just a convenient wa to arrange the coefficients that define ψ. In particular, this is a nice wa to put a wavefunction into a computer, as computers are ver adept at dealing with vectors. Now, our goal is to translate everthing that we might want to do with the wavefunction ψ into something we can do to the vector ψ. B going through this step b step, we arrive at a few rules Integrals are replaced with dot products. We note that the overlap between an two wavefunctions can be written as a modified dot product between the vectors. For eample, if φ d α α d β β then: * * * * * * * * * φ ψ dσ = d α c α α α dσ d α c β α β dσ d β c α β α dσ d β c β β β dσ * * = d α c α d β c β c = (d * d * ) α α β c β φ iψ Where, on the last line, we defined the adjoint of a vector: 1 * * ( 1 ). Thus, comple conjugation of the wavefunction is replaced b taking the adjoint of a vector. Note that we must take the transpose of the vector and not just its comple conjugate. Otherwise when we took the overlap we would have (column) (column) and the number of rows and columns would not match. These rules lead us to natural definitions of normalization and orthogonalit of the wavefunctions in terms of the vectors: * * * ( α β ) c α ψ ψ dσ = 1 ψiψ = c c = c c α c β = 1 β and

3 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 3 c * * α * * φ ψ dσ = 0 φ iψ = (dα d β ) = dα c α dβ c β = 0 c β * Finall, we d like to be able to act operators on our states in matri mechanics, so that we can compute average values, solve eigenvalue equations, etc. To understand how we should represent operators, we note that in wave mechanics operators turn a wavefunction into another wavefunction. For eample, the momentum operator takes one wavefunction and returns a new wavefunction that is the derivative of the original one: ψ ( ) pˆ dψ ( ) d Thus, in order for operators to have the analogous behavior in matri mechanics, operators must turn vectors into vectors. As it turns out this is the most basic propert of a matri: it turns vectors into vectors. Thus we have our final rule: Operators are represented b matrices. As an illustration, the Ŝ z operator will be represented b a matri in spin space:?? Ŝ z S z?? We have et to determine what the elements of this matri are (hence the question marks) but we can see that this has all the right properties: vectors are mapped into vectors:?? c α c α ' Ŝ z ψ =ψ =?? c β c β ' and average values are mapped into numbers ψ * Ŝ z ψ dσ * * (c α c β )?? c α?? cβ Now, the one challenge is that we have to determine which matri belongs to a given operator that is, we need to fill in the question marks above. As an eercise, we will show how this is done b working out the matri representations of Ŝ, Ŝ z, Ŝ and Ŝ. We ll start with Ŝ. We know how Ŝ acts on the α and β wavefunctions: 3 3 Ŝ α = α Ŝ β = β 4 4

4 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 4 Now represent Ŝ as a matri with unknown elements. c d S = e f In wave mechanics, operating Ŝ on α gives us an eigenvalue back, because α is and eigenfunction of Ŝ 3 (with eigenvalue 4 ). Translating this into matri mechanics, when we multipl the matri S times the vector α, we should get the same eigenvalue back times α : Ŝ α = 3 3 α S α = α 4 4 In this wa, instead of talking about eigenfunctions, in matri mechanics we talk about eigenvectors and we sa that α is an eigenvector of S. We can use this information to obtain the unknown constants in the matri S : 3 3 c d Ŝ α = α S α = α = e f 0 0 so Operating on β Thus: So for S we have c 3 = 4 e c = 3 e = c d Ŝ β = β S α = α = e f 1 1 d 0 = 3 f 4 d = 0 f = 3 4 S = We can derive S z in a similar manner. We know its eigenstates as well: Ŝ α = α Ŝ β = z z β So that the matri must satisf:

5 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 5 which gives us Analogous operations on β give c d 1 1 e f 0 0 Ŝ α = α z = c = e = 0 c = e 0 c d 0 0 e f 1 1 Ŝ β = β z = d 0 = f d = 0 f = Compiling these results, we arrive at the matri representation of S z : 1 0 S z = 0 1 Now, we need to obtain, which turns out to be a bit more trick. We remember from our operator derivation of angular momentum that we can re write the S in terms of raising and lowering operators: 1 1 = (S S - ) S = (S S - ) i where we know that Ŝ β = c α Ŝ α = 0 and Ŝ α = c β Ŝ β = 0 where c and c are constants to be determined. Therefore for the raising operator we have c d 0 c Ŝ β = c α = c d 1 0 Ŝ α = 0 = e f 1 e f 0 0 d = c f = 0 c = 0 e = 0 And for the lowering operator c d 1 0 c d 0 0 Ŝ α = c β = Ŝ β = 0 = e f 0 c e f 1 0 c = 0 e = c d = 0 f = 0 Thus S = c S - = c

6 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 6 Therefore we find for S 1 0 c 1 0 ic = 1 1 (S S ) = S = i (S S ) = c ic Thus, we just need to determine the constants: c and c. We do this b using two bits of information. First, we note that are observable, so the must be Hermitian. Thus, α * Ŝ β dσ = ( β * Ŝ αdσ ) (1 0) 1 c = * * c 0 0 c = (0 1) 1 c 0 0 * c = c This result reduces our work to finding one constant (c ) in terms of which we have 1 0 c 1 0 ic = * S = * c ic Note that S have the interesting propert that the are equal to their adjoints: 1 0 c 1 0 c = * = * = c c 1 0 ic 1 0 ic S = * = * = S ic ic This propert turns out to be true in general: Hermitian operators are represented b matrices that are equal to their own adjoint. These selfadjoint matrices are tpicall called Hermitian matrices for this reason, and the adjoint operation is sometimes called Hermitian conjugation. To determine the remaining constant, we use the fact that S = S S z. Plugging in our matri representations for, S z we find: c 0 c 1 0 ic 0 ic = * * * * c c 4 ic ic 1 0 c 1 0 c 1 0 = c c = c c =

7 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 7 where on the last line, we have made an arbitrar choice of the sign of c. Thus we arrive at the final epressions for S i = S = 1 0 i In summar, then, the matri representations of our spin operators are: i = S = S z = S = 1 0 i Given our representations of z now we can work out an propert we like b doing some linear algebra. For eample, we can work out the commutator of S : i 0 i 0 1 S,S = SS S = 1 0 i i i 0 i 0 1 = i i 1 0 i 0 i 0 i 0 = = 4 0 i 0 i 0 i,s = is z It is comforting to see that the matrices that represent the angular momentum operators obe the commutation relations for angular momentum! It is similarl possible to work out the eigenvalues of Ŝ, Ŝ z and Ŝ b computing the eigenvalues of z ; average values can be obtained b taking (row)*(matri)(column) products. We are thus in a position to compute anthing we want for these operators. As it turns out, an operator in this space can be written as a linear combination of, S z. So, in some sense, we now have all the information we could possibl want about spin ½ sstems. In honor of Pauli, it is conventional to define the dimensionless versions of, S z (i.e. matrices without the leading factor of / ) as Pauli Spin matrices: i 1 0 σ σ σ z 1 0 i 0 1 One of the reasons the Pauli spin matrices are useful (or equivalentl the matri representations of z ) is that the are the simplest

8 5.61 Phsical Chemistr 4 Pauli Spin Matrices Page 8 possible place to practice our skills at using matri mechanics. We will see later on that more complicated sstems can also be described b matrivector operations. However, in these cases the matrices will have man more elements sometimes even infinite numbers of elements! Therefore, it is best to get some practice on these simple matrices before taking on the more complicated cases.