Quantum Engi Q and A_Part_4

# Quantum Engi Q and A_Part_4 - 10 CHAPTER 2 MATHEMATICAL...

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Unformatted text preview: 10 CHAPTER 2. MATHEMATICAL PREREQUISITES the following multiples of sin( x ) are all normalized: sin( x ) / √ π , (for α = 0), − sin( x ) / √ π , (for α = π ), and i sin( x ) / √ π , (for α = π/ 2). Answer: A multiple of sin( x ) means c sin( x ), where c is some complex constant, so the magnitude is || c sin( x ) || = radicalBig ( c sin( x ) | c sin( x ) ) = radicalBigg integraldisplay 2 π ( c sin( x )) ∗ ( c sin( x )) d x You can always write c as | c | e i α where α is some real angle, and then you get for the norm: || c sin( x ) || = radicalBigg integraldisplay 2 π ( | c | e − i α sin( x )) ( | c | e i α sin( x )) d x = radicalBigg integraldisplay 2 π | c | 2 sin 2 ( x ) d x = | c | √ π So for the multiple to be normalized, the magnitude of c must be | c | = 1 / √ π , but the angle α can be arbitrary. 2.3.7 Solution dot-g Question: Show that the functions e 4i πx and e 6i πx are an orthonormal set on the interval ≤ x ≤ 1. Answer: You need to show that both functions are normalized, vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle e 4i πx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 1 and vextendsingle vextendsingle...
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