Question

1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential. Sketch what the wave functions would look like for α = 0 for the ground state and the 1st excited state. Write down a formula for all of the bound state energies for α = 0 (no derivation necessary). a) b) Break up the x axis into regions where the Schrödinger equation is easy to solve. Guess solutions in these regions and plug them into the equation State the boundary conditions on ψ(x) at x=a/2 and x=-a/2. Use the Schrodinger equation and properties of the delta function to obtain further matching conditions at x = 0, Eliminate as many of the unknown coefficients as possible. c) d) How do the answers differ for states with even and odd parity? You can write down the energies of the odd states without any further calculations. Obtain a transcendental equation for the even states. Graph the transcendental equation, indicating the values of k, that correspond to bound state energies E,-がk 2m. e) f) Estimate the bound state energies for α « 1 from the transcendental equation. How many bound states are there? How do you know your answer is correct? g) Estimate the bound state energies for α » 1 .

Answer 1

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