{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

midterm solutions 2009

# midterm solutions 2009 - Physics 341 Midterm Solutions...

This preview shows pages 1–2. Sign up to view the full content.

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 ) = ( 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 (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 ψ = A 1 e k 1 x which is normalizable for k 1 > 0. For x > 0, similarly ψ = 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 ψ 00 + (0) = Z - ( x ) . (1) The left hand side integral becomes - ~ 2 2 m ( ψ 0 ( ) - ψ 0 ( - )) = ~ 2 k 2 m ( A 1 + A 2 ) . So ψ 0 is a step function just like your homework problem. That means ψ 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 . This makes sense since k > 0 for normalizability and g < 0. If g had the opposite sign, there would be no normalizable solution. The energy is therefore E = - g 2 m 2 ~ 2 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

midterm solutions 2009 - Physics 341 Midterm Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online