Concepts in Magnetic Resonance Part A
Volume 2024, Issue 1 9907579
Research Article
Open Access

Density Matrix of Two Spin-1/2 Particles: Pure and Mixed States

Eric R. Johnston, 

Corresponding Author

Eric R. Johnston

Department of Chemistry and Biochemistry , The University of North Carolina at Greensboro , Greensboro , North Carolina, 27412 , USA , uncg.edu

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First published: 14 November 2024
https://doi.org/10.1155/2024/9907579
Academic Editor: Michele Benedetti

Abstract

The density matrix of an arbitrary pure state of a system consisting of two spin-1/2 particles is derived from the Pauli spin angular momentum operators. Mixed singlet and triplet states are then formed from linear combinations of pure states and their corresponding density matrices constructed. Singlet and triplet states are exemplified by the spin isomers parahydrogen and orthohydrogen, respectively. Partial mixing is illustrated with the example of bilinear spin–spin coupling. Various properties of the density matrices of pure and mixed states are discussed, including idempotence, factoring, and spin correlation.

1. Introduction

The density matrix of a spin-1/2 particle (or a collection of them) is often presented as a starting point in NMR discussions without an adequate explanation of its origin. This work demonstrates its formation from the spin angular momentum operators of Pauli given by the author of [1].
()
each with eigenvalues +1 and −1. Strictly speaking for spin-1/2 particles, these matrices are multiplied by a factor of one-half and have half-integer eigenvalues. We omit this factor and introduce it below as needed. The theory is then extended to a system of two spins. The Dirac ket–bra formalism [2] is used throughout.

2. Theory and Results

2.1. One-Spin Density Matrix

The spin-1/2 system may be represented by the general linear Hermitian operator
()
with . The iα values are the projections of the spin angular momentum along a given axis. The asterisk denotes complex conjugate.
The unitary eigenvector matrix U that diagonalizes O is found to be [3].
()
with corresponding eigenvalues +1 and −1.
The density matrix is then given by the eigenvector outer product
()
observing the complex conjugation of ket–bra interchange. Equation (4) may be written as
()
where E is the unit matrix. The density matrix further simplifies if iα = ±1 α = x, y, z. The density matrix equation (5) is then the eigenvector outer product of the relevant operator:
()

The idempotent nature of σ is evident as σ2 = |1〉〈1|1〉〈1| = |1〉〈1| = σ.

2.2. Time Evolution of the Density Matrix

The density matrix evolves in time via the unitary transformation
()
where H is an appropriate Hamiltonian operator acting upon the spins. Equation (6) is the integrated solution of the differential Liouville–von Neumann equation of motion of the density matrix [4]. R is the unitary exponential operator calculated from the Hamiltonian, and |ψ(0)〉 is a state vector, in this case an eigenvector of the total spin angular momentum operator O, so that the initial density matrix in equation (6) is given by equations (5) or (5a). In the latter case, the initial density matrix is simply proportional to the relevant angular momentum operator [4]. Expectation values (x, y, and z magnetizations) are obtained from the transformed density matrix using the operators of equation (1) and the trace expression [5].
()

The use of equation (6) to calculate the motion of a spin-1/2 ensemble induced by a nonresonant radiofrequency field is demonstrated in [3].

2.3. Two-Spin Density Matrix: Pure States

Equation (3) can be extended to two spins I and S by means of direct products [5].

2.3.1. Two-Spin Operator O and Eigenvector Matrix U

For two spins, the relevant Hermitian operator O becomes
()
The operators of equation (8) are given by the direct products
()
()
()
Equation (8) may then be written in full as
()
Eigenvector |1〉 of the corresponding eigenvector matrix U is obtained as the direct product of one-spin eigenvectors of equation (3):
()
where c and s now bear I and S labels.
The remaining eigenvectors of U are formed from the direct products
()
The diagonal matrix of eigenvalues of O is
()
and accordingly O = UDU† = 2(|1〉〈1| − |4〉〈4|).

2.3.2. Two-Spin Density Matrix

The two-spin density matrix is given by the eigenvector outer product
()
with and . The two-spin density matrix is idempotent.
Equation (16) is equivalently expressed as the direct product of the respective one-spin density matrices of equation (5):
()
or as the direct product of sums of one-spin operators
()
in which the one-spin Pauli matrices of equation (1) have been labeled as I or S in equation (18) to reflect the fact that direct multiplication results in corresponding four-dimensional operators with the same I-S labels, as shown in the following equation:
()
()
()
()
()
()
These are the full forms of the direct products of equations (9)–(11). By using them, equation (18) may be written as
()
The two-spin density matrix is thus composed of the sum of all fifteen operators and the identity matrix E [6]. As with the one-spin density matrix, for states with iα, sβ = ±1 the density matrix of equation (25) simplifies considerably. The I and S operators commute and trace calculations using equation (25) show that a product operator expectation value equals the product of the individual operator expectation values:
()

The spins act independently and are therefore uncorrelated.

2.4. Two-Spin Density Matrix: Mixed States

Mixed states of the two-spin system are constructed from linear combinations (sums or differences) of two pure eigenstates. We first consider the case of indistinguishable spins [6].

2.4.1. Indistinguishable Spins: Singlet and Triplet Mixed States

The eigenvector matrix Uz associated with the operators Iz, Sz and IzSz is
()
corresponding to the IS |αα〉, |αβ〉·|βα〉, and |ββ〉 product eigenvectors, with α = spin “up” and β = spin “down” [5]. Indistinguishability requires that new states be defined as linear combinations of product eigenvectors. These give a (normalized) singlet mixed state and a set of three mixed states comprising a triplet (so-called Bell states [6]):
()
()
()
()
where the superscript T denotes a vector transpose. The same set of four mixed states may be constructed from suitable combinations of the eigenvectors of Ix, Sx, IxSx and of Iy, Sy, IySy.

2.4.2. Singlet State Density Matrix

The singlet state density matrix is given by
()
using the operators of equations (19)–(24).
It is informative to calculate various expectation values for σS, equation (29). We find from trace relations that
()
and
()

The individual operator expectation values of zero indicate that neither spin is “up” or “down” but rather averaged over both states [6]. The singlet state vector equation (28a) is not an eigenvector of any individual operator (equations (19)–(24)). It is an eigenvector of the coaxial product operators, which return an eigenvalue of −1 for all three.

The trace calculation for IzSz applied to σS (equation (29)) is shown in equation (32) using operators.
()
where IαIβ = iIγ, IβIα = −iIγ, and similarly for the S operators.

2.4.3. Triplet State Density Matrices

The density matrices of the triplet states of equations (28a)–(28d) are
()
()
()

Expectation values for orthogonal product operators are zero for the singlet and triplet states, Equations (28a)–(28d), as are the expectation values for the individual operators. Eigenvalues for coaxial product operators of the singlet and triplet eigenvectors are collected in Table 1.

Table 1. Eigenvalues of product operators for mixed eigenvectors.
|ψ〉 〈IxSx〉 〈IySy〉 〈IzSz〉
|S〉 −1 −1 −1
|T1〉 1 1 −1
|T2〉 1 −1 1
|T3〉 −1 1 1

2.5. Properties of the Singlet and Triplet Density Matrices

2.5.1. Idempotence

The singlet and triplet density matrices of equations (29) and (33a)–(33c) are idempotent. For example, , or as verified by matrix multiplication.

2.5.2. Factoring

The singlet density matrix equation (29) may be factored into sums of diagonal and off-diagonal matrices as shown in the following equation:
()
whereas the full singlet density matrix is idempotent, the submatrices are not. The idempotent triplet density matrices may be similarly factored into sums of nonidempotent submatrices.

2.6. Partially Mixed States and Density Matrices

2.6.1. Hamiltonian Operator and Eigenvectors

Partially mixed eigenvectors are exemplified by bilinear coupling between spins I and S with coupling constant J and different Larmor frequencies. The Hamiltonian operator for this system is [1]
()
with I·S = IxSx + IySy + IzSz and Δ = ωI − ωS, ωS = −ωI
The unitary eigenvector matrix U that diagonalizes the Hamiltonian is
()
with tan 2θ = (J/Δ),  cos 2θ = (Δ/Ω),  sin 2θ = (J/Ω) and .
The diagonal matrix of eigenvalues of the Hamiltonian is
()

Eigenvectors |1〉 and |4〉 of equation (36) are the |αα〉 and |ββ〉 product spin states [5].

The product eigenstates |αβ〉 and |βα〉 are given by
()
Off-diagonal J terms in the Hamiltonian couple these and result in the mixed eigenvectors
()
()

In the limit of equal resonance frequencies, the spins become indistinguishable so that |2〉⟶|T1〉 (equation (28b)) and |3〉⟶|S〉 (equation (28a)). For J values small compared to the resonance frequency difference, the off-diagonal terms in the Hamiltonian may be omitted and the eigenvector matrix equation (36) becomes equation (27).

2.6.2. Difference Density Matrix

The density matrix constructed from the difference combination equation (40) is
()
Equation (41) factors into a sum of diagonal and off-diagonal submatrices
()

Although the full density matrix equation (41) is idempotent, the submatrices are not.

2.6.3. Sum Density Matrix

The density matrix constructed from the sum combination equation (39) is
()
It factors into a sum of diagonal and off-diagonal submatrices
()

Although σsum is idempotent, the individual submatrices are not.

Expectation values for various operators of the difference and sum density matrices are collected in Table 2.

Table 2. Expectation values for σdiff and σsum.
σ 〈Iz〉 〈Sz〉 〈IxSx〉 〈IySy〉 〈IzSz〉
σdiff −cos 2θ cos 2θ −sin 2θ −sin 2θ −1
σsum cos 2θ −cos 2θ sin 2θ sin 2θ −1

Expectation values for orthogonal product operators are zero. For the z and zz operators equation (26) no longer holds and the spins exhibit partial correlation.

2.6.4. Calculation of the NMR Spectrum

The NMR spectrum consists of four lines, with transition frequencies given by differences of eigenvalues of equation (40) and intensities (transition probabilities) given by [1].
()
For example, using equations (19), (20), and (40), we find that
()

Transition frequencies and intensities are collected in Table 3. This is the well-known quartet.

Table 3. Frequencies and intensities of the two-spin NMR spectrum.
Spin Transition Frequency 2I
I |1〉⟶|3〉 (1/2)(Ω + J) 1 − (J/Ω)
I |2〉⟶|4〉 (1/2)(Ω − J) 1 + (J/Ω)
S |1〉⟶|2〉 (1/2)(−Ω + J) 1 + (J/Ω)
S |3〉⟶|4〉 (1/2)(−Ω − J) 1 − (J/Ω)

Table 3 shows that for indistinguishable spins (Δ = 0), the two singlet-triplet transitions become strictly spin-forbidden whereas the two transitions within the triplet manifold each appear at the center frequency with unit intensity.

3. Discussion

3.1. Pure Density Matrix: Distinguishable Spins

Equation (16) is the idempotent two-spin density matrix σIS for the product states of I and S. It is valid for any values of iα, sβ α, β = x, y, z with . It is constructed from the eigenvector outer product of the general operator O or equivalently as the direct product of the individual density matrices. The expectation value of a product operator is the product of the individual operator expectation values and the spins act independently.

3.2. Mixed Density Matrix

3.2.1. Indistinguishable Spins

Linear combinations of product eigenvectors equation (27) produce a singlet state density matrix of equation (29) and three density matrices comprising the triplet of equations (33a)–(33c). The singlet and triplet state vectors in equations (28a)–(28d) are not eigenvectors of the individual operators I and S, and the resulting null expectation values reflect spin state averaging. They are eigenvectors of the coaxial product operators, and the +1 or −1 zz eigenvalue (Table 1) indicates that the spins in each state are respectively parallel or antiparallel [6]. The mixed density matrices in equations (29) and (33a)–(33c) of the singlet and triplet states are idempotent and factor into sums of nonidempotent submatrices.

3.2.1.1. Parahydrogen

The nuclear spin isomer parahydrogen [1] is an example of a singlet state, with a density matrix given by equation (29). The singlet state vector in equation (28a) is antisymmetric with respect to exchange of the two nuclei whereas the triplet state vectors (which describe orthohydrogen) are symmetric. The antisymmetry of the singlet state provides a means (owing to required overall antisymmetry of the molecular hydrogen wave function [1, 2, 7]) of converting a mixture of ortho- and parahydrogen entirely to the para spin isomer by cooling to low temperature, where the system is placed in the lowest (symmetric) rotational state of the electronic-vibrational ground state [7] and consequently the singlet spin state.

In the singlet state, only the intermediate spin energy levels (diagonal elements of equation (29)) of H2 are populated. Concerted addition of parahydrogen to an alkene transfers this density matrix to the (inequivalent) hydrogen atom pair of the product alkane (from which rapidly oscillating off-diagonal zero-quantum terms may be deleted) so that the isotropic density matrix in equation (29) becomes σS = (1/4)(E − IzSz). This non-Boltzmann population distribution results in dramatically polarized NMR signals for the added hydrogens [7].

3.2.1.2. Orthohydrogen

The density matrix for triplet orthohydrogen [1] is given by the average of the sum of equations (33a)–(33c):
()
with expectation values
()
()

It is not idempotent, and the null expectation values of equation (48) again reflect spin averaging. The diagonal elements of equation (47) show that all four spin energy levels are (unequally) populated. The density matrix of a mixture of parahydrogen (with fractional population fP) and orthohydrogen (with fractional population 1 − fP) is calculated with equations (29) and (33a)–(33c) to be σmix = (1/4)E − (1/12)(4fP − 1)I·S. All four spin states are then equally populated at equilibrium, where fP = 1/4.

3.2.2. Partially Distinguishable Spins

Density matrices of partially mixed states in equations (41) and (43) are constructed from the corresponding mixed eigenvector linear combinations (equations (39) and (40)). The Iz and Sz operator expectation values of Table 2 are a measure of the extent of mixing [6]. The sum and difference mixed states |2〉 and |3〉 (equations (39) and (40)) are eigenvectors of the two-spin product operator IzSz, and the 〈IzSz〉 eigenvalue of −1 in Table 2 indicates that the spins are of opposite orientation regardless of the extent of mixing. The general mixed density matrices are idempotent and factor into sums of nonidempotent submatrices.

4. Conclusion

4.1. Distinguishable Spins

The pure product density matrix (equation (16)) is constructed from the eigenvector outer product of the two-spin linear operator O or equivalently as the direct product of the one-spin density matrices (equation (17)). The expectation value of a product operator equals the product of the expectation values of the individual operators (equation (26)), and the spins are therefore uncorrelated.

4.2. Indistinguishable Spins

Indistinguishable spins must be described by linear combinations of product eigenvectors to give mixed singlet and triplet state vectors (equations (28a)–(28d)). Individual operator expectation values are zero, indicating spin averaging. The zz product operator returns an eigenvalue of +1 or −1 for these eigenvectors (Table 1), and the spins are respectively parallel or antiparallel (fully correlated or anticorrelated). The corresponding singlet and triplet density matrices are idempotent and can be factored.

4.3. Partial Mixing

Partial mixing of product eigenvectors results in the state vectors in equations (42) and (43) with corresponding density matrices which are idempotent and factorable. Expectation values for individual spin operators (Table 2) are a measure of the extent of mixing. The zz product operator eigenvalue of −1 nevertheless indicates that the spins are of opposite orientation regardless of the extent of mixing.

4.4. NMR Spectrum

The NMR spectrum (Table 3) displays the degree of eigenvector mixing in the form of unequal transition intensities and perturbed line positions. In the limit of spin equivalence, the two transitions which would reveal the J coupling vanish, and a pair of transitions, each of unit intensity, appears at the center frequency.

4.5. Spin Dynamics

The density matrix is central to the calculation of spin dynamics as it represents the initial state of the spin system in the integrated solution of equation (6) of the differential Liouville–von Neumann equation of motion of the density matrix [3, 4]. Expectation values are calculated from the transformed density matrix with the trace operation of equation (7).

The present work demonstrates that for pure product states, the two-spin density matrix is simply the direct product of the individual one-spin density matrices and the spins behave independently (are not “entangled”). A remarkable feature of the mixed singlet state equation (28a) is that the two particles which constitute it remain anticorrelated (“entangled”) after physical separation and the system must be viewed as a single entity even when the particles are widely separated [6]. The relaxation behavior and correlation characteristics of the singlet state have recently been theoretically investigated [8].

Conflicts of Interest

The author declares no conflicts of interest.

Funding

No funding was received for this manuscript.

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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