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# HW3sol - Physics 731 Assignment#3 Solutions 1 An operator L...

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Physics 731 Assignment #3, Solutions 1. An operator L is self-adjoint if f | Lg w = Lf | g 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 f | Lg = 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 1 - 1 f * ( x )( - (1 - x 2 )) d 2 g dx 2 dx = f * ( x )( - (1 - x 2 )) dg dx 1 - 1 - 1 - 1 df * dx ( - (1 - x 2 )) + 2 xf * 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 = 1 - 1 df * dx (1 - x 2 ) dg dx dx. (4) Integrating again by parts, we have I = df * dx (1 - x 2 ) g 1 - 1 - 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 = 1 - 1 d 2 f * dx 2 ( - (1 - x 2 )) + 2 x df * dx gdx = Lf | g , (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 ) = 1 a sin ( x + a ) 2 a , n = 1 , 2 , 3 , . . . (7) Therefore, x = 1 a a - a x sin 2 ( x + a ) 2 a dx = 0 (8) x 2 = 1 a a - a x 2 sin 2 ( x + a ) 2 a dx = a 2 1 3 - 2 n 2 π 2 , (9) and p = - i ¯ h a a - a sin ( x + a ) 2 a d dx sin ( x + a ) 2 a dx = 0 (10) p 2 = - ¯ h 2 a a - a sin ( x + a ) 2 a d 2 dx 2 sin ( 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 x ) 2 p ) 2 = ¯ h 2 4 π 2 n 2 3 - 2 1 4 | [ X, P ] | 2 = ¯ h 2 4 . (12) For n
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HW3sol - Physics 731 Assignment#3 Solutions 1 An operator L...

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