p4 - Problem Set 4 Physics 341 Due November 19 Some...

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Problem Set 4 Physics 341 Due November 19 Some abbreviations: S - Shankar. 1 . Understanding the spectrum for simple potentials is very important. So we will continue with a problem from the midterm. Start by considering the case in the exam: an electron of mass m in one dimension subject to a potential V ( x ) = ( x ) where g is a constant and δ ( x ) is the Dirac delta-function. This system approximates a highly localized potential around the point x = 0. (i) For what sign(s) of g is there a quantum mechanical bound state? (ii) Please find a relation between the bound state energy E and g . (iii) Now consider the double well potential V ( x ) = ( x + a ) + ( x - a ) . Again find a relation between E,g and a for any bound states. What happens to the energy E of these states as a 0? 2 . This problem is inspired by a discussion after lecture. For a repulsive square well potential of width a and height V 0 , we saw in lecture that perfect transmission is possible as long as sin(
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p4 - Problem Set 4 Physics 341 Due November 19 Some...

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