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oldexam(3)

# oldexam(3) - EE 439x Exam 1 Oct.18 2006 Name 1...

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EE 439x Name_______________________________________ Exam 1 – Oct.18, 2006 1. Two-dimensional quantum well A two-dimensional, infinitely deep quantum well is defined by the potential U for x < 0, x > L x , y < 0, and y > L y . U = 0 for 0 < x < L x and 0 < y < L y . In 2-D, the Schroedinger equation takes the form " h 2 2 m # 2 \$ x , y ( ) # x 2 " h 2 2 m # 2 \$ x , y ( ) # y 2 + U x , y ( ) \$ x , y ( ) = E \$ x , y ( ) . Find the 2-D wave functions and corresponding energies for electrons trapped in this well.

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EE 439x Name_______________________________________ Exam 1 – Oct.18, 2006 Half of a harmonic oscillator Consider a “half-harmonic-oscillator” potential, defined by U(x) = ½ kx 2 = ½ m ω 2 for x > 0 U(x) for x < 0 Thus, the potential has an infinite barrier at x = 0. Sketch the wave functions and determine the energies (in terms of ω ) for the two lowest states of this potential. V(x) x For reference the wave functions and energies for the 4 lowest states of the full harmonic oscillator (i.e. the one we did in class) are given below. -4 -3 -2 -1 0 1 2 3 4 ! 1 x 0 -4 -3 -2 -1 0 1 2 3 4 !
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