Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Rotating a Coordinate System

 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, a1', a2', in terms of the old coordinates, a1, a2.

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|ej> = <ej|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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