HW3sol - Physics 731 Assignment #3, Solutions 1. An...

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Physics 731 Assignment #3, Solutions 1. An operator L is self-adjoint if h f | Lg i w = h Lf | g i w (1) for all f, g on the interval [ a, b ] with weight function w ( x ) . For the Legendre operator on the interval [ - 1 , 1] with w ( x ) = 1 and functions that are finite at the endpoints, the LHS of Eq. ( 1 ) is I ≡ h f | Lg i = Z 1 - 1 f * ( x ) " - (1 - x 2 ) d 2 dx 2 + 2 x d dx # g ( x ) dx. (2) Integrating the first term by parts, we have Z 1 - 1 f * ( x )( - (1 - x 2 )) d 2 g dx 2 dx = ± f * ( x )( - (1 - x 2 )) dg dx ² 1 - 1 - Z 1 - 1 ± df * dx ( - (1 - x 2 )) + 2 xf * ² dg dx dx. (3) The boundary term is zero, since f and g are finite at x = ± 1 . In the remaining integral, note that the second term is equal in magnitude and opposite in sign to the second term in Eq. ( 2 ). Therefore, I = Z 1 - 1 df * dx (1 - x 2 ) dg dx dx. (4) Integrating again by parts, we have I = ± df * dx (1 - x 2 ) g ² 1 - 1 - Z 1 - 1 " d 2 f * dx 2 (1 - x 2 ) g - 2 x df * dx g # dx. (5) Again, the boundary term vanishes. I is then given by I = Z 1 - 1 " d 2 f * dx 2 ( - (1 - x 2 )) + 2 x df * dx # gdx = h Lf | g i , (6) which is what we wanted to prove. 2. The even and odd parity eigenfunctions of the infinite square well can be summarized as follows: ψ n ( x ) = r 1 a sin ³ ( x + a ) 2 a ´ , n = 1 , 2 , 3 , . . . (7) Therefore, h x i = 1 a Z a - a x sin 2 ³ ( x + a ) 2 a ´ dx = 0 (8) h x 2 i = 1 a Z a - a x 2 sin 2 ³ ( x + a ) 2 a ´ dx = a 2 ± 1 3 - 2 n 2 π 2 ² , (9) and h p i = - i ¯ h a Z a - a sin ³ ( x + a ) 2 a ´ d dx sin ³ ( x + a ) 2 a ´ dx = 0 (10) h p 2 i = - ¯ h 2 a Z a - a sin ³ ( x + a ) 2 a ´ d 2 dx 2 sin ³ ( x + a ) 2 a ´ dx = ¯ h 2 n 2 π 2 4 a 2 . (11) 1
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Thus, the x - p uncertainty product and the generalized uncertainty principle take the form h x ) 2 ih p
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This note was uploaded on 12/14/2009 for the course QUANTUM I 731 taught by Professor Everett during the Fall '09 term at University of Wisconsin Colleges Online.

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HW3sol - Physics 731 Assignment #3, Solutions 1. An...

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