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Unformatted text preview: Solutions for Hand-In Set #9 PHYS 212 Spring 2007 A62 Uncertainty The uncertainty in the x position is given in the prob- lem statement: x = 10- 12 m. This implies a minimum uncertainty in the x component of the momentum given by the Heisenberg Uncertainty Principle: x p h 2 p h 2 x . The mass is certain, so this can be rewritten as m v h 2 x , or v h 2 m x 1 . 06 10- 34 J s 2 . 04 kg 10- 12 m 1 . 3 10- 21 m/s A65 Schr odinger equation for classically forbidden situation (a) Rearranging the Schr odinger equation gives d 2 ( x ) dx 2 =- 2 m h 2 ( E- U ) ( x ) . (b) For the first trial solution we have 1 ( x ) = Ax 2 d 1 dx = 2 Ax d 2 1 dx 2 = 2 A. Plugging these into the rearranged Schr odinger equation gives 2 A ? =- 2 m h 2 ( E- U ) Ax 2 . where I have used the symbol ? = to emphasize that were are tying to see if the function 1 works, i.e., makes the equality true for all values of x . In this case 1 is not a soluton. The lefthand side is a positive constant, and the righthand side is quadratic in x ; these clearly cant be equal for all x . The function 1 ( x ) is not a solution. (c) For the second trial solution we have 2 ( x ) = B sin kx d 2 dx = Bk cos kx d 2 1 dx 2 =- Bk 2 sin kx....
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