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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
i1=2 + h
j + h
i + j
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
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