Download 763313A QUANTUM MECHANICS II Exercise 1 1. Let A and B be

763313A QUANTUM MECHANICS II
Exercise 1
spring 2017
1. Let A and B be two matrices.
a) Assume that A and B are such matrices that the product AB is
well-defined. Show that (AB)† = B † A† .
b) Show that for 2 × 2 matrices det(AB) = det(A) det(B). Does this
result hold for all square matrices?
2. Calculate the eigenvalues and eigenvectors of the Pauli spin matrices
σ1 =
0 1
1 0
,
σ2 =
0 −i
i 0
,
σ3 =
1 0
.
0 −1
Show that the Pauli matrices are hermitian.
3. Calculate the expectation values hα|σi |αi, i = 1, 2, for Pauli spin matrices σ1 and σ2 with respect to an arbitrary state |αi = α1 |1i + α2 |2i.
Here |1i and |2i are the eigenvectors of σ3 .
4. Consider a three-dimensional vector space spanned by an orthonormal
basis |1i, |2i, |3i. Kets |αi and |βi are given by
|αi = i|1i − 2|2i − i|3i, |βi = i|1i + 2|3i.
Find all nine matrix elements of the operator  = |αihβ|, in this basis,
and construct the matrix representing Â. Is it hermitian?
5. The Hamiltonian for a certain two-level system is
Ĥ = (|1ih1| − |2ih2| + |1ih2| + |2ih1|),
where |1i, |2i is an orthonormal basis and is a number with the dimensions of energy. Find its eigenvalues and eigenvectors (as linear
combinations of |1i and |2i). What is the matrix representing Ĥ with
respect to this basis?