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midsol08w - PHYS 115A Midterm Solutions Winter 2008 Problem...

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PHYS 115A Midterm Solutions Winter 2008 Problem 1 Recall that the normalized energy eigenstates and energies for an infinite well of width a extending from x = - a/ 2 to x = + a/ 2 are: E n = n 2 π 2 ~ 2 2 ma 2 ψ n ( x ) = r 2 a cos nπx a · for odd n = 1 , 3 , 5 , 7 , ... ψ n ( x ) = r 2 a sin nπx a · for even n = 2 , 4 , 6 , 8 , ... and ψ n ( x ) = 0 for | x | > a/ 2. So we recognize that the given wavefunction is just: Ψ( x, 0) = A r a 2 " r 2 a sin 2 πx a + r 2 a cos 3 πx a # = A r a 2 [ ψ 2 ( x ) + ψ 3 ( x )] (a) [5 points] Let’s assume that the normalization factor A is real: 1 = Z -∞ Ψ * ( x, 0) Ψ( x, 0) dx = A 2 a 2 Z -∞ [ ψ 2 + ψ 3 ] * · [ ψ 2 + ψ 3 ] dx = A 2 a 2 Z -∞ [ ψ * 2 ψ 2 + ψ * 3 ψ 3 + ψ * 2 ψ 3 + ψ * 3 ψ 2 ] dx = A 2 a 2 [1 + 1 + 0 + 0] since the wavefunctions ψ n ( x ) are orthonormal A = 1 a (b) [3 points] for either of these correct expressions for Ψ( x, t ) : Ψ( x, t ) = 1 a sin 2 πx a e - iE 2 t/ ~ + cos 3 πx a e - iE 3 t/ ~ = r 1 2 h ψ 2 ( x ) e - iE 2 t/ ~ + ψ 3 ( x ) e - iE 3 t/ ~ i [2 points] for specifying the correct energies: E 2 = 4 π 2 ~ 2 2 ma 2 E 3 = 9 π 2 ~ 2 2 ma 2 1
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