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# Discussion4 - normalized eigenstates of the infinite square...

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Physics 486 Fall 2007 Discussion Problems Week 4 1). This is adapted from Griffiths, problem 1.5 (page 14-15) Consider the wave function: Ψ ( x , t ) = Ae - λ | x | e - i ω t , where A , λ , and ω are real positive constants. (You saw in last week’s homework that this wavefunction is the solution to the δ -function potential). (a). Normalize Ψ . (b). Determine the expectation values <x>, <x 2 >, and σ x = Δ x = (<x 2 > - <x> 2 ) 1/2 ( σ x is called the standard deviation). (c). Calculate the probability that | x | > σ x . Note: 1 0 ! + - = n ax n a n dx e x

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Physics 486 Fall 2007 2). This is similar to part of problem 2.5 in Griffiths. Suppose that at t = 0, the wave function is a superposition of the first two
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Unformatted text preview: normalized eigenstates of the infinite square well ( V = 0 for 0 ≤ x ≤ L , = ∞ otherwise.): Ψ ( x ,0) = A [ ψ 1 (x) + 2 ( x )] (a). Normalize ( x ,0) (i.e., calculate A ). (b). Calculate ( x , t ) and | ( x , t )| 2 . Write | ( x , t )| 2 explicitly as a function of and . Describe the time-dependence of | Ψ (x,t)| 2 . (c). Calculate < > (hint: the result depends on time). Describe your result. NOTE: sin a sin b = 1 2 [cos( a-b )-cos( a + b )] x cos ax dx = ∫ 1 a 2 cos ax + x a sin ax...
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