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p827aps9

p827aps9 - 3 Given an attractive central potential of the...

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Phys. 827: Problem Set 9 Due Wednesday, November 28, 2007 at 11:59 P. M. 1. Shankar, problem 12.5.3 (p. 329). 2. Consider a particle mass m moving in a spherically symmetric square well with potential V = - V 0 for r < a; V = 0 for r > a ( V 0 > 0). (a). Obtain the radial equation. (b). Write down the solution for r < a and for r > a in the case = 0. Hint: make the substitution R(r) = u(r)/r and Fnd u(r). (c). By matching boundary conditions, obtain a transcendental equa- tion for the energy E. (“Transcendental” means an equation involving trigonometric or exponential functions.) (d). What is the minimum potential depth in order for there to be an = 0 bound state? (e). At what depth does the second bound level appear?
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Unformatted text preview: 3. Given an attractive central potential of the form V =-V exp(-r/a ), solve the Schr¨odinger equation for the s-states by making the substitu-tion ξ = exp[-r/ (2 a )]. Obtain an equation for the eigenvalues. 4. Shankar, problem 12.6.11, parts (1) and (2) only. By “parity” is meant the following: Each solution will satisfy ψ (-x ) = ± ψ ( x ), where the + sign means that the state has even parity, and the-sign means that the state has odd parity. In spherical coordinates, if we write x = r, θ, φ , then-x = r, π-θ, π + φ . 1...
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