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p5 - E for which transmission is perfect(ii Although...

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Problem Set 5 Physics 341 Due December 1 Some abbreviations: S - Shankar. 1 . Let’s consider some additional aspects of coherent states | λ i for the harmonic oscillator. Recall that λ is a complex number. The identity operator can be expressed in terms of coherent states: 1 = Z dxdy f ( x, y ) | λ ih λ | for some function f ( x, y ). Here λ = x + iy . Find the function f ( x, y ). 2 . S 21.1.18 3 . S 21.1.21 4 . At this point, I know you love δ -function potentials. Read section 5.4 and make sure you note the role of the probability current in defining R and T . Solve S 5.4.2. 5 . Consider an attractive square well potential V ( x ) = - V 0 for - a/ 2 < x < a/ 2 and V ( x ) = 0 elsewhere. Here V 0 is a positive constant. Let’s consider the scattering of particles with E > 0 off this potential. (i) Compute the transmission coefficient T ( E ) as a function of energy. Find the values of
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Unformatted text preview: E for which transmission is perfect. (ii) Although physically the energy is real and positive, it is often useful to view T ( E ) as a function of complex energy E . Show that T ( E ) has a pole in the complex E plane at the values of E which correspond to bound state energies of the square well. (iii) Repeat exercise (ii) for the single delta-function potential using the results of the previous question. 6 . Suppose is a simultaneous eigenstate of two anti-commuting Hermitian operators A and B . What can you say about the eigenvalues of A and B for the state ? Illustrate your point using the parity operator (chosen so that = -1 = ) and the momentum operator. 1...
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