midterm solutions 2009 - Physics 341 Midterm Solutions...

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Unformatted text preview: Physics 341 Midterm Solutions November 10 Do not use calculators, cell phones, or cheat sheets. All you need is a pen or pencil. Please show your work on these sheets. 1 . Consider an electron of mass m in one dimension subject to a potential V ( x ) = g ( x ) where g is a constant and ( 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 (not a scattering state)? (ii) Please find a relation between the bound state energy E and g . Solution: (i) Think of the potential as a limit of Gaussians. The attractive potential is the upside down Gaussian so you want g < 0 for a bound state. (ii) Away from x = 0, the potential is vanishing so the wavefunction is free particle. For x < 0, we can take = A 1 e k 1 x which is normalizable for k 1 > 0. For x > 0, similarly = A 2 e- k 2 x . This should be an energy eigenstate and the energy is E =- ~ 2 2 m ( k 1 ) 2 =- ~ 2 2 m ( k 2 ) 2 so k 1 = k 2 = k with k > 0. Lastly, we need to patch the wavefunctions at x = 0. By definition of the delta-function if we integrate around x = 0, we find: Z-- ~ 2 2 m 00 + g (0) = Z- E ( x ) . (1) The left hand side integral becomes- ~ 2 2 m ( ( )- (- )) = ~ 2 k 2 m ( A 1 + A 2 ) . So is a step function just like your homework problem. That means is continuous at x = 0 so A 1 = A 2 (integrate again around the junction to check this). The right hand side of (1) therefore vanishes. This gives the relation: ~ 2 k m + g = 0 ....
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This note was uploaded on 02/27/2010 for the course PHYSICS 341 taught by Professor Sghy during the Spring '10 term at King's College London.

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midterm solutions 2009 - Physics 341 Midterm Solutions...

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