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Energy_and_Sketching_Wave_Functions

Energy_and_Sketching_Wave_Functions - Energy and the Art of...

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Energy and the Art of Sketching Wave Functions 1 I: Sketching wave functions A. Review: The figure to the right shows an infinite square well potential ( V = 0 from - L/ 2 to L/ 2 and is infinite everywhere else). 1. Write down the formula for the energies of the first two energy eigenstates, E a 1 and E a 2 . 2. On the graphs below, sketch the ground state ( u 1 , the lowest energy eigenstate) and first excited state ( u 2 ) for a particle of mass m in this potential. B. Energetic curves: The figure below shows normalized energy eigenstate wave functions for three identical particles in (possibly different) infinite square wells. The drawings are to scale with each other. 1. Rank the functions from lowest energy (1) to highest energy (2, 3). If you think that two of the functions have the same energy, be sure to indicate the tie. 2. What feature(s) are important in making this determination? PHYS 3220 Tutorials 2008 c S. Goldhaber, S. Pollock, and the Physics Education Group University of Colorado, Boulder
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Energy and the Art of Sketching Wave Functions 2 C. Stepped Potential: Now, consider a slightly more complicated potential, shown on the right. Let’s assume that V c E a 2 , i.e. , that “step up” in the middle is a big step. 1. On the graphs below, sketch the ground state and first excited state for the same particle in this potential. Make sure you use the Schr¨ odinger equation to guide your work. What is the effect of the potential step?
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