Issued Wed, Oct 16 Due Wed, Oct 23

(Required reading for this week: Finish Gas. Ch. 6 , start Ch. 7

There are HINTS for this homework!

1a) Gas. #6-2

b) part 1 of problem #6-1

2a) Gas 6-3

b) Gas 6-9 (see prob 6-8 for the definition of unitary, if you've forgotten)

3) Gas 6-4

4a) Gas 6-6

b) Gas 6-8

5) Operators in quantum mechanics can always be represented by matrices. Sometimes these matrices are infinite dimensional(!), which is a pain, but in many problems we will work with finite matrices after all.

Suppose the operator O can be written as

(with "a" and "b" real, nonzero constants)

a) Show that O is Hermitian.

b) Find all the possible eigenvalues, , of O, and the corresponding eigenvectors. Please normalize your eigenvectors. (Are the eigenvalues you get real? Should they be?)

c) If this operator represents the Hamiltonian of some system, how many allowed energies does your system have?

Show that the eigenvectors you found in b) are orthogonal.

Show that they also form a complete basis for all 2-D vectors (i.e. all 2 by 1 column vectors)

(F.Y.I, this is the way one would describe e.g. spin 1/2 particles in a magnetic field in quantum mechanics!)

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