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Discussion5 - α> 0 Remember that x δ = 0 x x ≠ ∞ =...

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Physics 486 Fall 2007 Discussion Problems 5 1). Quantum “pressure”: Consider a quantum particle of mass m in the n th energy state of a one-dimensional infinite square potential of width L . (a). Calculate the opposing force, F = - E / L , encountered when the walls are slowly pushed in, assuming the particle remains in the n th state of the well as L changes. (b). Now, consider a classical particle moving with kinetic energy E in a well of width L. By determining the frequency of its collisions with a given wall, and its momentum transfer per collision, calculate the average force the particle exerts on the wall. How does this result compare with (a)?
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Physics 486 Fall 2007 2) In this problem, you will revisit a problem from Homework Set 2 and evaluate the bound state solution for the delta function potential, V(x) = - αδ
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Unformatted text preview: α > 0. Remember that, ( ) x δ = 0, , x x ≠ ∞ = ; and ( ) 1 x dx ∞-∞ = ∫ ; and ( ) ( ) ( ) f x x a dx f a ∞-∞-= ∫ (a). By integrating the SEQ from – ε to + ε , then taking the limit as ε → 0, show that the condition on the derivative of the wavefunction, d ψ /d x , for the case of the delta function potential is given by: ( ) 2 2 d d m dx dx ε α +--= -ℏ (b). Qualitatively sketch the wavefunction for this potential (it should look familiar!). (c). What are the solutions (x) to the SEQ in regions I and II (see figure above)? (d). By applying the continuity condition on at x=0, as well as the condition derived in (a) above, derive bound state energy E associated with the wavefunction you sketched in (b). x V(x) -αδ (x) V=0 E I II +0 -0...
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