Density Matrix Description of NMR BCMB/CHEM 8190 Operators in Matrix Notation • It will be important, and convenient, to express the commonly used operators in matrix form • Consider the operator Iz and the single spin functions α and β - recall ˆ ˆ ˆ ˆ ˆ ˆ Ix α = 1 2 β Ix β = 1 2α Iy α = 1 2 iβ Iy β = −1 2 iα Iz α = +1 2α Iz β = −1 2 β α α = β β =1 α β = β α = 0 - recall the expectation value for an observable Q = ψ Qˆ ψ = ∫ ψ∗Qˆψ dτ Qˆ - some operator ψ - some wavefunction - the matrix representation is the possible expectation values for the basis functions α β α ⎡ α Iˆ α α Iˆ β ⎤ ⎢ z z ⎥ ⎢ ˆ ˆ ⎥ β ⎣ β Iz α β Iz β ⎦ ⎡ α Iˆ α α Iˆ β ⎤ ⎡1 2 α α −1 2 α β ⎤ ⎡1 2 0 ⎤ ⎡1 0 ⎤ ˆ ⎢ z z ⎥ 1 Iz = = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⎢ ˆ ˆ ⎥ 1 2 β α −1 2 β β 0 −1 2 2 0 −1 ⎣ β Iz α β Iz β ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎣ ⎦ • This is convenient, as the operator is just expressed as a matrix of numbers – no need to derive it again, just store it in computer Operators in Matrix Notation • The matrices for Ix, Iy,and Iz are called the Pauli spin matrices ˆ ⎡ 0 1 2⎤ 1 ⎡0 1 ⎤ ˆ ⎡0 −1 2 i⎤ 1 ⎡0 −i ⎤ ˆ ⎡1 2 0 ⎤ 1 ⎡1 0 ⎤ Ix = ⎢ ⎥ = ⎢ ⎥ Iy = ⎢ ⎥ = ⎢ ⎥ Iz = ⎢ ⎥ = ⎢ ⎥ ⎣1 2 0 ⎦ 2 ⎣1 0⎦ ⎣1 2 i 0 ⎦ 2 ⎣i 0 ⎦ ⎣ 0 −1 2⎦ 2 ⎣0 −1 ⎦ • Express α , β , α and β as 1×2 column and 2×1 row vectors ⎡1⎤ ⎡0⎤ α = ⎢ ⎥ β = ⎢ ⎥ α = [1 0] β = [0 1] ⎣0⎦ ⎣1⎦ • Using matrices, the operations of Ix, Iy, and Iz on α and β , and the orthonormality relationships, are shown below ˆ 1 ⎡0 1⎤⎡1⎤ 1 ⎡0⎤ 1 ˆ 1 ⎡0 1⎤⎡0⎤ 1 ⎡1⎤ 1 Ix α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = β Ix β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = α 2⎣1 0⎦⎣0⎦ 2⎣1⎦ 2 2⎣1 0⎦⎣1⎦ 2⎣0⎦ 2 ⎡1⎤ ⎡0⎤ α α = [1 0]⎢ ⎥ =1 β β = [0 1]⎢ ⎥ =1 0 1 ˆ 1 ⎡0 −i⎤⎡1⎤ 1 ⎡0⎤ 1 ˆ 1 ⎡0 −i⎤⎡0⎤ 1 ⎡−i⎤ 1 ⎣ ⎦ ⎣ ⎦ Iy α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = i β Iy β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = −i α 2⎣ i 0⎦⎣0⎦ 2⎣i ⎦ 2 2⎣ i 0⎦⎣1⎦ 2⎣ 0⎦ 2 ⎡0⎤ ⎡1⎤ α β = [1 0]⎢ ⎥ = 0 β α = [0 1]⎢ ⎥ = 0 ˆ 1 ⎡1 0 ⎤⎡1⎤ 1 ⎡1⎤ 1 ˆ 1 ⎡1 0 ⎤⎡0⎤ 1 ⎡ 0 ⎤ 1 ⎣1⎦ ⎣0⎦ Iz α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = α Iz β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = − β 2⎣0 −1⎦⎣0⎦ 2⎣0⎦ 2 2⎣0 −1⎦⎣1⎦ 2⎣−1⎦ 2 Operators in Matrix Notation • Likewise, we can express αα αβ βα ββ αα ⎡ αα Iˆ αα αα Iˆ αβ αα Iˆ βα αα Iˆ ββ ⎤ the two-spin operators in ⎢ Ax Ax Ax Ax ⎥ ⎢ ˆ ˆ ˆ ˆ ⎥ matrix form αβ αβ IAx αα αβ IAx αβ αβ IAx βα αβ IAx ββ ⎢ ⎥ - example: consider the ˆ ˆ ˆ ˆ βα ⎢ βα IAx αα βα IAx αβ βα IAx βα βα IAx ββ ⎥ expectation value for the Ix ⎢ ⎥ Iˆ Iˆ Iˆ Iˆ operator on the A spin in an ββ ⎣⎢ ββ Ax αα ββ Ax αβ ββ Ax βα ββ Ax ββ ⎦⎥ AX system in the two spin basis ˆ ˆ ˆ ˆ ˆ ˆ (recall I x α = 1 2 β I x β = 1 2 α I y α = 1 2 i β I y β = − 1 2 iα I z α = 1 2 α I z β = − 1 2 β ) ⎡1 2 αα βα 1 2 αα ββ 1 2 αα αα 1 2 αα αβ ⎤ ⎡ 0 0 1 2 0 ⎤ ⎡0 0 1 0⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 2 αβ βα 1 2 αβ ββ 1 2 αβ αα 1 2 αβ αβ ˆ ⎢ ⎥ ⎢ 0 0 0 1 2⎥ 1 ⎢0 0 0 1⎥ IAx = ⎢ ⎥ = = 1 2 βα βα 1 2 βα ββ 1 2 βα αα 1 2 βα αβ ⎢ 1 2 0 0 0 ⎥ 2⎢1 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 1 2 0 0 0 1 0 0 ⎣1 2 ββ βα 1 2 ββ ββ 1 2 ββ αα 1 2 ββ αβ ⎦ ⎣ ⎦ ⎣ ⎦ • The matrices for Ix, Iy,and Iz are ⎡0 0 1 0⎤ ⎡0 1 0 0⎤ ⎡0 0 −i 0⎤ ⎡0 −i 0 0⎤ ⎡1 0 0 0 ⎤ ⎡1 0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 0 0 0 1 1 1 0 0 0 1 0 0 0 −i 1 i 0 0 0 1 0 1 0 0 1 0 −1 0 0 Iˆ = ⎢ ⎥ Iˆ = ⎢ ⎥ Iˆ = ⎢ ⎥ Iˆ = ⎢ ⎥ Iˆ = ⎢ ⎥ Iˆ = ⎢ ⎥ Ax 2⎢1 0 0 0⎥ Xx 2⎢0 0 0 1⎥ Ay 2⎢i 0 0 0⎥ Xy 2⎢0 0 0 −i⎥ Az 2⎢0 0 −1 0⎥ Xz 2⎢0 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 1 0 0⎦ ⎣0 0 1 0⎦ ⎣0 i 0 0⎦ ⎣0 0 i 0⎦ ⎣0 0 0 −1⎦ ⎣0 0 0 −1⎦ Density Matrices and Observables for an Ensemble of Spins • Expectation values give average results for an observable for a single spin or spin system - i.e. for a group of spins that all have the same wavefunction • The density matrix/operator provides an avenue for average or net behavior of an ensemble of spins • First, consider a single spin - write a wavefunction for a linear combination of the members of the basis set ( and ) α β ∗ ∗ ∗ ψ = c1 α + c2 β = ∑cjφ j ψ = c1 α + c2 β = ∑c jφ j j j • We can substitute the vector equivalents for α , β , α and β to get vector expressions for ψ and ψ ⎡1⎤ ⎡0⎤ α = ⎢ ⎥ β = ⎢ ⎥ α = [1 0] β = [0 1] ⎣0⎦ ⎣1⎦ ⎡1⎤ ⎡0⎤ ⎡c1 ⎤ ∗ ∗ ∗ ∗ ∗ ∗ ψ = c1 α + c2 β = c1 ⎢ ⎥+ c2 ⎢ ⎥ = ⎢ ⎥ ψ = c1 α + c2 β = c1 [1 0]+ c2[0 1] = [c1 c2 ] ⎣0⎦ ⎣1⎦ ⎣c2 ⎦ • So, the wavefunction (and its complex conjugate) can be written in terms of vectors of the coefficients only Density Matrices and Observables for an Ensemble of Spins • The expectation value for some observable represented by the operator Q, operating on a wavefunction ψ can be written as Q = ψ Qˆ ψ ∗ ∗ ∗ ψ = c1 α + c2 β = ∑cjφ j ψ = c1 α + c2 β = ∑c jφ j j j ⎡ ∗ ∗ ⎤ ˆ ⎡ ⎤ = ⎣c1 α + c2 β ⎦ Q ⎣c1 α + c2 β ⎦ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ = c1 c1 α Q α + c1 c2 α Q β + c2c1 β Q α + c2c2 β Q β = ∑c jck ψ j Q ψk j • Substitute the matrix/vector equivalents ⎡c ⎤ ˆ 1 ∗ ∗ Q = ψ Q ψ ψ = ⎢ ⎥ ψ = [c1 c2 ] ⎣c2 ⎦ ⎡ α Qˆ α α Qˆ β ⎤⎡c ⎤ ⎡c α Qˆ α + c α Qˆ β ⎤ ∗ ∗ ⎢ ⎥ 1 ∗ ∗ ⎢ 1 2 ⎥ = [c1 c2 ] ⎢ ⎥ = [c1 c2 ] ⎡ ˆ ˆ ⎤ α Qz α α Qz β ⎢ ˆ ˆ ⎥⎣c2 ⎦ ⎢ ˆ ˆ ⎥ ⎣ β Q α β Q β ⎦ ⎣c1 β Q α + c2 β Q β ⎦ Qˆ = ⎢ ⎥ ⎢ ˆ ˆ ⎥ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ ⎣ β Qz α β Qz β ⎦ = c1 c1 α Q α + c1 c2 α Q β + c2c1 β Q α + c2c2 β Q β = ∑c jck ψ j Q ψk j • Note the expression for Q ˆ includes products of the coefficients, rather than the coefficients themselves, suggesting a way to represent the spin state (wavefunction) in terms of the products ⎡c ⎤ ⎡c c∗ c c∗ ⎤ ψ ψ = 1 [c∗ c∗ ] = ⎢ 1 1 1 2 ⎥ - note: NOT ψ ψ = 1 ! ⎢ ⎥ 1 2 ∗ ∗ ⎣c2 ⎦ ⎣c2c1 c2c2 ⎦ Density Matrices and Observables for an Ensemble of Spins • This leads to a different way to formulate the expectation value - i.e. for a group of spins that all have the same wavefunction Qˆ = Tr{ ψ ψ Qˆ} ⎧ ∗ ∗ ⎡ ˆ ˆ ⎤⎫ ⎡ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ ⎤ ⎪⎡c c c c ⎤ α Q α α Q β ⎪ c1c1 α Q α + c1c2 β Q α c1c1 α Q β + c1c2 β Q β = Tr⎨⎢ 1 1 1 2 ⎥⎢ ⎥⎬ = Tr⎢ ⎥ c c∗ c c∗ ⎢ ˆ ˆ ⎥ ⎢ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ ⎥ ⎩⎪⎣ 2 1 2 2 ⎦⎣ β Q α β Q β ⎦⎭⎪ ⎣c2c1 α Q α + c2c2 β Q α c2c1 α Q β + c2c2 β Q β ⎦ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ = c1c1 α Q α + c1c2 β Q α + c2c1 α Q β + c2c2 β Q β ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ = c1 c1 α Q α + c1 c2 α Q β + c2c1 β Q α + c2c2 β Q β = ∑c jck ψ j Q ψk j • Now….for an ensemble of spins, the magnitude of an observable is the sum of the expectation values for each spin ˆ ˆ ˆ ˆ Q = ψ1 Q ψ1 + ψ2 Q ψ2 + ψ3 Q ψ3 +....... ψN Q ψN Tr ...... Qˆ = {( ψ1 ψ1 + ψ2 ψ2 + ψ3 ψ3 + ψN ψN ) } • The average value of an observable for an ensemble is N ˆ Q = (1 N)∑ ψi Q ψi i=1 Tr ⎡ 1 N ...... ⎤Qˆ = {⎣( )( ψ1 ψ1 + ψ2 ψ2 + ψ3 ψ3 + ψN ψN )⎦ } - the overbar indicating the average over the ensemble Density Matrices and Observables for an Ensemble of Spins • To simplify this expression, we'll define the density operator ˆ ρ = (1 N)( ψ1 ψ1 + ψ2 ψ2 + ψ3 ψ3 ) +...... ψN ψN = ψ ψ • For an ensemble, the average value of the observable, and the average macroscopic observable, then are ˆ ˆ ˆ ˆ Q = Tr{ρQ} Qmacro = N Tr{ρQ} • What does this mean? - any observable can be ascertained from two spin operators - one of these represents the observable - the other represents the ensemble average of the wavefunctions (not necessary to know the wavefunctions for all N members of the ensemble) • For two non-interacting spins, the matrix representation of the density operator (density matrix) is ⎡ ∗ ∗ ⎤ ⎡ ∗ ∗ ⎤ c1c1 c1c2 ⎡ρ11 ρ12 ⎤ c1c1 c1c2 ρˆ = ψ ψ = ⎢ ⎥ = ⎢ ⎥ recall ψ ψ = ⎢ ⎥ ∗ ∗ ∗ ∗ ⎢ ⎥ ⎣ρ21 ρ22 ⎦ c c c c ⎣c2c1 c2c2 ⎦ ⎣ 2 1 2 2 ⎦ - ρ11 and ρ22 are the populations of states 1 and 2 (i.e.