soll - Physics 741 Graduate Quantum Mechanics 1 Solution Set L 1[10 In class we found the even parity bound states for H= P2 V(Q 2m where 0 V(Q = V0 if

# 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 H V Q m = + where ( ) 0 0 if if Q a V Q V Q 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 x Ae x B kx C kx x De β β ψ ψ 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 ( ) ( ) I III x x ψ ψ = and ( ) ( ) II II x x ψ ψ = . 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 a a C ka D a a a kC ka D a ψ ψ β ψ ψ β β = = = = − Dividing the second equation by the first, we find ( ) cot k ka β = − Using equations (7.57) and (7.59), it is easy to see that 2 2 2 2 2 0 0 2 2 k mV mV k β β + = = = = Plugging this in and dividing by - k , we find ( ) 0 2 2 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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• Spring '04
• Unknow
• Physics, mechanics, Eigenvalue, eigenvector and eigenspace, Hilbert space, odd parity

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