Physics 410 Midterm Practice Exam Fall 2007, Frey Closed book ...

Physics 410 Midterm Practice Exam
Fall 2007, Frey
Closed book exam. No calculators. There are 6 practice problems. The actual exam will have
no more than 5.
1. (15 points)
Using the definition of matrix multiplication, show that (AB) † = B † A † .

2. (25 points)
Consider the eigenvalue equation A |v〉 = λ |v〉 , where A =
(a) Find the eigenvalues.
⎛
⎜
⎝
1 0 i
0 1 0
−i 0 1
⎞
⎟
⎠
(b) Find the (normalized) eigenvectors.
(c) Let A ′ = SAS −1 be the diagonalized matrix. Write down S.

3. (15 points) Evaluate the following:
(a)
∫ ∞
0
δ(x + 1) [ 3x 2 + 2x − 1 ] dx
(b)
∫ ∞
0
(1 + e x )δ(x + 1) dx
(c)
∫ ∞
0
(5x + 2) δ(2x − 2) dx

4. (20 points)
Let H be a Hermitian matrix. Show that (a) its eigenvalues are real, and (b) its eigenvectors
are orthogonal for non-degenerate eigenvalues.

5. (15 points)
( )
( )
1 0
Two basis vectors, |v 1 〉 = and |v
0
2 〉 = are transformed by a matrix
1
( )
0 1
S =
to a rotated basis S |v
−1 1
1 〉 = |v 1〉 ′ and S |v 2 〉 = |v 2〉 ′ .
(a) Determine |v ′ 1〉 and |v ′ 2〉 .
(b) If an operator A is represented in the original basis by the matrix A =
what is its representation A ′ in the rotated basis?
(
−1 2
2 0
)

6. (10 points) An anti-Hermitian matrix is one for which A † = −A. Show that this property
is invariant under unitary similarity transformations.