HW3 solutions - Physics 4120 Spring 2008 Homework #3 1)...

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Unformatted text preview: Physics 4120 Spring 2008 Homework #3 1) Reif 2.1 ( 29 E E m p p mE p δ δ + = + = 2 2 Shaded region is accessible to our particle. 2) Reif 2.2 Slice of phase space through position is simple. Slice of phase space through momentum forms a ring, with inner radius p and outer radius p+ δ p, where ( 29 E E m p p mE p δ δ + = + = 2 2 3) Reif 2.3 a) Total probability of being within the darkly shaded region is given by the probability density integrated over that interval. We can use either P(x) or w ( ϕ ) to do this, so the integrals must be equal. Note that we add a factor of 2 since when we integrate over dx , we’re automatically taking both the top and bottom half of the circle: ( 29 ( 29 ( 29 2 2 2 2 2 cos 1 sin cos 2 1 2 ) ( 2 ) ( 2 1 2 1 2 1 x A dx d d x A d t A d t A dx t A x d d w dx x P x x-- =-- = +-- = +- = + = = = ∫ ∫ ∫ ϕ ϕ ϕ ϕ ϖ ϕ ϕ ϖ ϕ ϖ ϕ π ϕ ϕ ϕ ϕ ϕ ϕ Note the minus sign: we need that because we have to switch the direction we’re integrating. ∫ ∫ ∫ ∫ ∫ ∫- =- =-- = = 2 1 2 1 1 2 2 1 2 1 2 1 2 2 2 2 2 2 ) ( 1 1 ) ( x x x x x x x A dx dx x P x A dx x A dx d dx x P π π π ϕ π ϕ ϕ ϕ ϕ ϕ ϕ Note the switch in integration order gets rid of our minus sign and aligns the integration...
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This note was uploaded on 04/17/2008 for the course PHYS 4120 taught by Professor Vajk during the Spring '08 term at Missouri (Mizzou).

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HW3 solutions - Physics 4120 Spring 2008 Homework #3 1)...

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