fa02mod_sol

fa02mod_sol - PHY6938 Proficiency Exam Fall 2002 Modern...

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Unformatted text preview: PHY6938 Proficiency Exam Fall 2002 September 13, 2002 Modern Physics and Quantum Mechanics 1. Consider a one-dimensional step potential of the form V ( x ) = ( x < V x ≥ where V > . A particle with total energy E > V and mass m is incident on the step potential “from the left” (in other words: the particle starts at negative values of x and travels toward positive values of x ). (a) Use the time-independent Schr¨ odinger equation to determine the form of the particle’s wave function in the two regions x < and x ≥ . In the x < 0 region (call this region 1) the potential is zero so the Schr¨ odinger equation has the form- ¯ h 2 2 m d 2 Ψ 1 dx 2 = E Ψ 1 , (1) and Ψ 1 has the form of a plane wave moving to the right (the incident wave) and another moving to the left (the reflected wave), so that Ψ 1 = Ae ik 1 x + Be- ik 1 x . (2) Substituting this into the Schr¨ odinger equation we see that it is a solution if- ¯ h 2 2 m (- k 2 1 ) = E k 1 = s 2 mE ¯ h 2 = √ 2 mE ¯ h . In the x > 0 region (call this region 2) we have- ¯ h 2 2 m d 2 Ψ 2 dx 2 + V Ψ 2 = E Ψ 2 .- ¯ h 2 2 m d 2 Ψ 2 dx 2 = ( E- V )Ψ 2 . and Ψ 2 has the form of a plane wave moving to the right (the transmitted wave), so that Ψ 2 = Ce ik 2 x , (3) with k 2 = q 2 m ( E- V ) ¯ h . (4) (b) Derive expressions for the probabilities that the particle is reflected ( R ) and transmitted ( T ) . Hint: Recall that the probability density current is given by j ( x ) = Re Ψ * ¯ h im ∂ Ψ ∂x ! , and that R and T are ratios of probability density currents. We have to derive the values of A , B , and C using the boundary conditions, which are that the wavefunction and its derivative must be continuous at the boundary ( x = 0). Continuity of the wavefunction gives that A + B = C, (5) while that of the derivative at x = 0 gives that k 1 A- k 1 B = k 2 C, (6) since taking the derivative of the plane wave just brings down a factor of- ik , and we have divided by- i . For that reason the incident probability current is simply j in = Re " Ae- k 1...
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This note was uploaded on 03/11/2012 for the course PHY 3900 taught by Professor Staff during the Fall '1 term at FSU.

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fa02mod_sol - PHY6938 Proficiency Exam Fall 2002 Modern...

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