Sequential Stern Gerlach Experiment

We will show how the presence of an observer affects measurement in nonrelativistic quantum mechanics, using the familiar Stern Gerlach Experiment. This observer-measurement relationship is the heart of quantum mechanics and is mathematically captured by the uncertainty principle.

Consider three consecutive Stern Gerlach apparatuses, A, B and C, which we shall refer to as boxes for linguistic convenience. Box A allows a particular ket $|a'\rangle$ to pass through and suppresses all other states. Similarly, box B allows a particular $|b'\rangle$ to pass through, suppressing all others and box C allows only a particular $|c'\rangle$ to pass through suppressing all other states.

The probability of obtaining $|c'\rangle$ when the kets out of box A are normalized is given by

(1)
\begin{align} |\langle c'|b'\rangle|^{2}|\langle b'|a'\rangle|^{2} \end{align}

But we need to sum over all possible $|b'\rangle$ paths. Operationally, this amounts to keeping all but one $|b'\rangle$ blocked, and recording the value of the above probability term, and repeating the experiment with all but the second $|b'\rangle$ blocked and continuing until all the $|b'\rangle$ paths have been exhausted. Thus, the total probability is given by summing over all $|b'\rangle$ paths

(2)
\begin{align} \sum_{b'}|\langle c'|b'\rangle|^{2}|\langle b'|a'\rangle|^{2} = \sum_{b'}\langle c'|b'\rangle \langle b'|c'\rangle \langle b'|a'\rangle \langle a'|b'\rangle \end{align}

Next, consider the experimental setup when box B is either absent or non-operative. In this case the probability is simply

(3)
\begin{align} |\langle c'|a'\rangle|^{2} \end{align}

Now, since the ket $|a'\rangle$ can be expanded in terms of the complete set of states $|b'\rangle$ as

(4)
\begin{align} |a'\rangle = \sum_{b'}|b'\rangle \langle b'|a'\rangle \end{align}

the probability term $|\langle c'|a'\rangle|^{2}$ can be rewritten as

(5)
\begin{align} |\sum_{b'}\langle c'|b'\rangle \langle b'|a'\rangle|^{2} = \sum_{b'}\sum_{b''}\langle c'|b'\rangle \langle b'|a'\rangle \langle b''|c' \rangle \langle a'|b''\rangle \end{align}

Clearly, the expressions for the probability are not identical!

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