Problems for Edward F. Redish |
In a 2-D Cartesian vector space the basis is expressed in Dirac notation as
An arbitrary vector is expressed in this basis as
Suppose we consider a new basis rotated from the first by an angle θ as shown in the figure at the right:. |
(a) Construct the 8 dot products
.
(b) Use the dot products you have constructed to express the coordinates of the vector |A> in the new basis, a_{1}', a_{2}', in terms of the old coordinates, a_{1}, a_{2}.
Solution
(a) The dot products are easily read off the graph. They are either a sine or a cosine and either positive or negative. The result for the first four is:
The result for the transposes of these is the same since <e'_{i}|e_{j}> = <e_{j}|e'_{i}>*. Since they are all real the transposes of the dot products are the same.
(b) Writing |A> in two ways and taking the dot product of both equations with the primed basis states we get the results
Therefore we can conclude
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Last revision 15. November, 2005.