Chapter 7 Problem Solutions

44b m w 14 y0 hxl i p m m w x2 2 the energy of this

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Unformatted text preview: t + dL cn Therefore Xx\ = A cosHw t + dL X px \ = -m w A sinHw t + dL A = 2 — Hn + 1L ê H2 m wL d = Arg Hcn L - Arg Hcn + 1 L where cn cn + 1 and so „ „t „ „t X px \ Xx\ = m X px \ = -m w2 Xx\ which are the Ehrenfest relations. Problem 7.13 The position-space wavefunction of the harmonic oscillator ground state is given by (7.44b) m w 1ê4 y0 HxL = I p — M ‰ - m w x2 2— The energy of this state is E0 = — w ê 2, so the classically disallowed region is that for which V HxL > E0 1 2 m w2 x2 > x > —w 2 — ê Hm wL The probability that a particle in this state is measured to be in the classically disallowed (c.d.) region is thus - —êHm wL pc.d. = KŸ -¶ = = = m w 1ê 2 I p— M 1 p 2 p +Ÿ ¶ —êHm wL - —êHm wL KŸ -¶ -1 ¶ O„x +Ÿ ¶ —êHm wL JŸ -¶ + Ÿ 1 N „ u ‰-u ¶ -u Ÿ1 „ u ‰ 2 J p í [email protected] - erfH1LD pc.d. = 1 - erf H1L y0 HxL 2 2 O„x‰ - u= m w x2 — ¬༼ Let „u...
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