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

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

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

(5)Clearly, the expressions for the probability are **not** identical!