# Quantum Problem Paper

**Paper, Order, or Assignment Requirements**

PHYS 3711 Quantum Physics

Problem Set 1

1. Prove the Schwartz inequality:

h j ih‑j‑i jh j‑ij2

Note that this is equivalent to 1 cos2 for usual three-dimensional vectors

Hint: Consider the function f(; ) hji 0 where ji j i + j‑i and calculate

the value of the (complex) that minimizes f(; ). Note: a complex number and its

complex conjugate are independent variables.

2. Prove that, for any two state vectors j

i and j

i

[h

j

i1=2 + h

j

i1=2]2 (h

j + h

j)(j

i + j

i)

Hint: Use the Schwarz inequality.

3. Show that

(a) (A + B)y = Ay + By

(b) (cA)y = c Ay

(c) (AB)y = ByAy

(d) (Ay)y = A

(e) (Ay)

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