HW6 - 0(b[10pts Show that this equation can be written as...

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PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 6: Motion in 1D, Probability Current: 1. [10pts] Let P ( t | a, b ) be the probability that a particle is in the range a < x < b . Show that d dt P ( t | a, b ) = j ( a, t ) - j ( b, t ) where j ( x, t ) is the probability current at position x at time t . 2. [10pts] For the infnite square-well potential, calculate the momentum uncertainty Δ P ±or each energy eigenstate. You should write the eigenstate wave±unctions as φ n ( x ) = r 2 L sin p nπx L P u ( x ) u ( L - x ) where u ( x ) is the unit step ±unction. 3. Finite square well Bound State: Consider a particle o± mass m which ±eels the potential V ( x ) = 0 , x < - a - V 0 , - a < x < a 0 , x > a (a) [10pts] Start by writing a general ±orm (ansatz) ±or the solutions in the regime - V 0 < E < 0. By applying the required boundary conditions, fnd an equation which determines the energies o± the bound states (those states with energy satis±ying - V 0 < E <
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Unformatted text preview: 0). (b) [10pts] Show that this equation can be written as tan z = R ( z /z ) 2-1 where z = a R 2 m ( E + V ) / p and z = a √ 2 mV / p . This equation holds ±or solutions which are even ±unctions o± x. What about solutions which are odd ±unctions? (c) [10pts] Make a plot o± tan z versus z and on the same plot show R ( z /z ) 2-1 versus z . Indicate on the plot where the curves intersect. Plot must be computer generated. (d) [10pts] Write E i as a ±unction o± z i , where z i would be the z-value o± the i th intersection point and E i would be the corresponding energy eigenvalue. You do not need to fnd expressions or values ±or the z i ’s. In principle, you could use a computer to fnd the z i ’s and that would then give you the energy eigenvalues....
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