soll - Physics 741 Graduate Quantum Mechanics 1 Solution...

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Physics 741 – Graduate Quantum Mechanics 1 Solution Set L 1. [10] In class we found the even parity bound states for () 2 , 2 P HV Q m =+ where 0 0i f if Qa VQ a < = > Repeat the analysis for the odd parity bound states. We know from class notes eq. (7.60) that in the three regions, the solution takes the form ( ) () ( ) ( ) cos sin x I II x III xA e x Bk x Ck x xD e β ψ = = We are interested in the odd parity bound states. Since parity relates regions I and III to each other, and region II to itself, this implies ( ) ( ) II I I x x = and () ( ) II II x x =− . We therefore have A = - D and B = 0. We now wish to match boundary conditions. We will choose to do so at x = a, where we follow exactly the notation of (7.62) in the notes to yield () () ( ) ( ) sin exp cos exp II III II III aa C k a Da aak C k a ψβ =⇒ = ′′ = Dividing the second equation by the first, we find ( ) cot kk a = − Using equations (7.57) and (7.59), it is easy to see that 22 2 2 2 00 km V m V k ββ −− += = == Plugging this in and dividing by - k , we find 0 2 cot 1 mV ka k = This is equation (7.66). Solutions of this equation can then be plugged into (7.65) to get
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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soll - Physics 741 Graduate Quantum Mechanics 1 Solution...

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