These pages make extensive use of Greek symbols. The font is SPIONIC.TTF and is available for free download here.

In order to emphasize various things I have coloured operators, eigenfunctions etc.:

eigenfunctions

operators

eigenvalues

Density matrices and operators are an important stepping stone on the road to product operators.

The wave function for the spin system, Y contains all of the information about the system it describes that can be known. In other words, it is as complete a description of the state of the system as we can ever hope to have. For a spin-1/2 system (which is most often encountered in nmr spectroscopy) there are two pure spin states corresponding to the two possible eigenvalues of I_z with Y as the eigenfunction. See angular momentum for an explanation of this. The convention is to name the state with eigenvalue +1/2 "a" and the state with eigenvalue -1/2 "b"

Often, it will be erroneously noted that a spin must be in either of these two pure states. However, there is no justification for this assumption. Nowhere in quantum mechanics is this required. Rather, a spin may be in a mixed or superposition state which is a mixture of the two pure states:

We may represent the bra and ket functions as a row and column matrix:

The expectation value for an operator operating on y is:

or, in terms of the superposition state in matrix-vector notation:

(This is my html version of matrices .. a row matrix then a 4x4 square matrix then a column matrix)

Multiplication of these matrices gives:

Now, bear with me because this is cute. If you look at the preceding equation you see several terms involving "c" which are generated from the row and column matrix. You can generate each of these terms by multiplying the column and row matrices together:

which gives a square matrix:

containing each of the coefficient terms.

In terms of the wavefunction, Y this could be written:

Now, it we multiply this square matrix of coefficients into the operator matrix we get:

and the resulting 4x4 square matrix from this operation is:

The trace of this matrix is:

which is identical to what we got above for the expectation value! Nice.

|Y><Y| is called the density operator and given the symbol 'r' in general quantum mechanics and 's' in spin quantum mechanics (I don't know why they are different .. they mean essentially the same thing).

Using this definition and what has just been shown we can say:

The matrix form of r is, of course, called the density matrix.

The diagonal terms, diagonal being from upper left to lower right, are called populations and in this case correspond to populations in the a and b states. The off-diagonals are referred to as coherences.

As nmr spectroscopists, our interest lies in the time evolution of the density matrix. We do this generally by comparing states as described by r at different times. Thus, we might be interested in the changes that take place as the result of a rf pulse. This is done using rotation operators:

where q is the angle of rotation, r₀ is the density matrix at time zero and r₁ is the density matrix after the rotation time period.