Problems for Edward F. Redish |
A two mass system where each mass can be displaced in one dimension is described as a linear space spanned by the basis vectors
.
Suppose the normal modes of the system are found to be described by the orthonormal set of vectors
where α, β, γ, and δ could be complex.
(a) If we choose to describe these 4 vectors as Dirac states, |f_{i}>, i = 1,2 and |e_{i}>, i = 1,2, find the 8 inner products <e_{i}|f_{i}> and <f_{i}|e_{i}>.
(b) What is the matrix (in the f-basis) that represents the sum ?
(c) A vector is represented in the f-basis by the sum . This same vector is represented in the e-bases by . Find the A'_{i} in terms of the A_{j}.
Solution
(a) The dot products are easily obtained if one recalls that
that is, when one takes the transpose of a complex vector from a ket to a bra, it corresponds to taking the complex conjugate.
The results are
.
[Grading: +2 for each of the answers. -2 if you forgot the *]
(b) The result is the identity matrix in any orthonormal basis. One was not required to prove this, but if you construct the outer product the result is:
.
To see that this is in fact the identity matrix, you have to use orthonormality of the basis vectors:
A little manipulation shown that the matrix is the identity.
[Grading: 5 pts for the identity matrix; 4 is the correct outer product is given but not recognized as I.]
(c) We equate the two sums since they represent the same vector. Dotting it with <e_{1}| picks out A_{1}' and gives an equation for it. Similarly, dotting with <e_{2}| picks out A_{2}' and gives an equation for it. The results are:
[Grading: +3 for knowing how to pick out the A' terms, +4 for getting the results. If you forgot the * in (a) no points were deducted for not putting it here.]
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Last revision 12. December, 2005.