• This lecture is best performed following the Describing the Sample Oven in the Stern-Gerlach Experiment Mathematically ( Needs link) activity.

• So, density matrices are very useful for finding the probabilities of mixed states. To confirm that this strategy really does work, lets find the probability of finding the oven in the state $\vert+ \rangle_{y}$. Using the same form we would for finding the probability that some general state is in $\vert+ \rangle_{y}$, we would have

$$\vert+ \rangle_{y}{_{y}\langle +\vert}\psi \rangle\langle \psi\vert \; \; .$$

Now, we insert the oven mixed state into where the density matrix for $\vert\psi \rangle$ would be to get

$$P_{+_{y}} \,=\, \vert+ \rangle_{y}{_{y}\langle +\vert}\left(\frac{1}{2}\vert+ \rangle\langle +\vert \: + \: \frac{1}{2}\vert- \rangle\langle -\vert\right) \; \; .$$

Matrix notation will be necessary to move from this point. Rewriting the above statement, we have

$$\frac{1}{2}\left(\begin{array}{c} 1\\ i\\ \end{array}\right) \left(\begin{array}{cc} 1&-i\\ \end{array}\right) \frac{1}{2}\left[\left(\begin{array}{c} 1\\ 0\\ \end{array}\right) \left(\begin{array}{cc} 1&0\\ \end{array}\right) \; + \; \left(\begin{array}{c} 0\\ 1\\ \end{array}\right) \left(\begin{array}{cc} 0&1\\ \end{array}\right)\right] \; \; . $$

Computing the outer products, factoring, and adding the oven density operator terms will give

$$\frac{1}{4}\left(\begin{array}{cc} 1&i\\ -i&1\\ \end{array}\right) \left(\begin{array}{cc} 1&0\\ 0&1\\ \end{array}\right) \; \; . $$

The right matrix is just the identity matrix, so the trace of the left matrix multiplied by $\frac{1}{4}$ will give

$$P_{+_{y}}=\frac{1}{2} \; \; . $$ This is the probability we receive from the experiments previously performed.

• The same concept of using density matrices can also be applied to expectation values. For example, the expectation value for the spin operator $\hat{S}_{z}$ for an arbitrary state as such:

$$\langle \hat{S}_{z} \rangle \, = \, \langle \psi \vert \hat{S}_{z} \vert \psi \rangle \, = \, \hat{S}_{z} \vert \psi \rangle \langle \psi \vert \; \; . $$

Now, we can insert the density matrix representation of the oven into the outer product of the arbitrary $\vert \psi \rangle $ to get

$$\langle \hat{S}_{z} \rangle \, = \, \hat{S}_{z} \frac{1}{2} \left(\vert + \rangle \langle + \vert \, + \, \vert - \rangle \langle - \vert \right) \; \; . $$

If you wish, have one-third of the class each perform this calculation for the spin operators in the x, y, and z orientation.

To carry this concept further, the uncertainty of an operator is just composed of two expectation values. If the uncertainty of the operator $\hat{S}_{z}$ is defined as

$$\sqrt{\langle \hat{S}_{z}^{2} \rangle - \langle \hat{S}_{z} \rangle ^{2} } \; \; , $$

then the uncertainty of a measurement out of the oven could also be calculated using density matrices.

Again, time permitting, have one-third of the class each perform this calculation for the spin operators in the x, y, and z orientation.