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
€
|
α
+>= 1
2|u> + 3
2|d>
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?