Chapter 7 Problem Solutions

Chapter 7 Problem Solutions - Intermediate QMI Ch 7 Problem...

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Intermediate QMI Ch 7 Problem Solutions Problem 7.4 We have a ` 0 ] = 0 and thus Y p a ` 0 ] = 0 ¬ a ` = m w 2 I x ` + Â m w p ` x M m w 2 X p I x ` + Â m w p ` x M 0 \ = 0 where Y p x ` 0 ] = Â p X p 0 \ Y p p ` x 0 ] = p X p 0 \ This gives the equation B Â p + Â m w p F X p 0 \ = 0 ¬ Let X p 0 \ = y è 0 H p L „y è 0 p + p m w y è 0 = 0 ¬ Multiply by the integrating factor 1 m w Ÿ p p = ‰ p 2 2 m w p 2 2 m w „y è 0 p + p m w p 2 2 m w y è 0 = 0 p B p 2 2 m w y è 0 F = 0 therefore y è 0 H p L = A - p 2 2 m w We calculate A by imposing normalization Ÿ p y è 0 H p L 2 = 1 Ÿ p A 2 - p 2 m w = 1 ¬ Ÿ p - p 2 a = p a A 2 m p w = 1 ¬ Choose A real; therefore A = H m p w L - 1 ê 4 and we have X p 0 \ = H m p w L - 1 ê 4 - p 2 2 m w Printed by Mathematica for Students
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Problem 7.9 Given the superposition of adjacent energy states (7.63) y H t L\ = ‰ H n + 1 ê 2 L w t B c n n \ + c n + 1 -Â w t n + 1 \F Using x ` = êH 2 m w L J a ` + a ` N p ` x = -Â m w ê 2 J a ` - a ` N
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