# solm - Physics 741 Graduate Quantum Mechanics 1 Solution...

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Physics 741 – Graduate Quantum Mechanics 1 Solution Set M 1. [20] A particle of mass m and energy E in two dimensions is incident on a plane step function given by () 0 0 if 0, , if 0. x xy x Q VQQ VQ < = > The incoming wave has wave function ( ) in , ikx ky xy e ψ + = for x < 0. (a) [7] Write the Hamiltonian. Determine the energy E for the incident wave. Convince yourself that the Hamiltonian has a translation symmetry, and therefore that the transmitted and reflected wave will share something in common with the incident wave (they are all eigenstates of what operator?). The Hamiltonian, of course, is just 22 2 ,, pp HV x y V x y mm x y + ⎛⎞ ∂∂ =+ = + + ⎜⎟ ⎝⎠ = The incident wave will therefore have an energy ()() ( ) ( ) 22 2 2 2 2 , kk EH i k i k V x y ψψ + == + + = = = , so that 2 Ek k m = . The potential depends on x , but is independent of y . Therefore, they are eigenstates of the translation operator in the y -direction, ( ) ˆ Ta y , for any a . Since this is a continuous symmetry, we can consider small translations, and we define the momentum operator in the y -direction P y in the usual way, and our state must be an eigenstate of this operator as well. Our wave must therefore take the form ( ) ( ) ,. y ik y x ye X x = It remains only to find the function X ( x ).

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(b) [7] Write the general form of the reflected and transmitted wave. Use Schrödinger's equation to solve for the values of the unknown parts of the momentum for each of these waves (assume 22 0 2 x km V > = ). If we plug our general solution into Schrödinger’s equation, we have () ( ) () ( ) () 2 22 2 2 ,, 2 ,. 2 yy y y ik y ik y ik y xy ik y ik y ik y y Ee X x e X x V x y e X x mx y kk k e Xx e Xx Vxye Xx mm m x ⎛⎞ ∂∂ =− + + ⎜⎟ ⎝⎠ + + = = = = The exponentials cancel everywhere, and some factors can be cancelled.
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solm - Physics 741 Graduate Quantum Mechanics 1 Solution...

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