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A problem from a university-level quantum mechanics course that involves manipulating bra-ket notation. Students are required to normalize and find the probability of finding a particle in the 'up' or 'down' state, find a ket orthogonal to a given ket, express the 'up' and 'down' kets in terms of the new kets, and calculate the average rate of clicks from a measurement device based on the initial state and the measurement outcomes.
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Chem 212 Problem 1. (Due Thursday, September 4, 2008) This problem is meant to give you practice manipulating bras and kets. A two-state system can be represented by the kets |u> and |d> depending whether the particle is in the “up” or “down” state. Assume that the |u> and |d> kets are normalized and orthogonal. (a) If the ket for a prepared state of the particle is known to be €
then show that |α+> is normalized, and find the probability that the particle is (i) up or (ii) down. (b) Find a ket that is orthogonal to |α+> and normalize it. Call it |α−>. (c) Express the kets |u> and |d> in terms of the kets |α+> and |α−>. (d) If a ket is known to be in the |u> state, then what is the probability that it is in the |α+> state? The |α−> state? (e) Suppose that the particle is known to be in |α+> initially, and is passed through a measuring device that measures only |u> or |d>. Particles that are measured as “up” are deflected up, and those that are measured “down” are deflected down. Another measurement device follows which measures whether the state is |α+> or |α−>. The “+” response triggers and audible detector as a “click”. If a beam of 1000 particles per second is sent through the first detector, what is the average rate of clicks?