Posted: May 3rd, 2015

Quantum Problem Paper

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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)