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lecture4

# lecture4 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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2.57 Fall 2004 – Lecture 4 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 4 Quick review of Lecture 3 Photon: E h ν = , / p h λ = . Assuming Ψ t ( r ,t)= Ψ ( r )Y(t), we use separation of variables to solve the Schrödinger equation t i U m t t t Ψ = Ψ + Ψ = = 2 2 2 . The solutions are [ ] 1 1 exp exp E Y C i t C i t ω = = = , and ( ) 2 2 0 2 U E m ∇ Ψ + Ψ = = , where the eigen value E represents the total energy of the system. Heisenberg uncertainty principle states ; 2 2 p x t E < ∆ >< ∆ >≥ < ∆ >< ∆ >≥ = = . 2.3 Example solutions: Here we determine ( ) u r G by the boundary conditions and will not consider the ( , ) u r t G case. 2.3.1 Free particles in 1D In this case, there are no constraints for the particles. The potential energy u=0 so that 2 2 2 0 2 d E m dx Ψ Ψ = = . This gives ikx ikx Ae Be ψ = + , where 2 / / k mE p = = = = (note 2 / 2 E p m = ). The final solution is ( ) ( ) ( , ) i t kx i t kx t x t Ae Be ω ω + Ψ = + , in which the first term corresponds to negative-direction propagation, the second term is positive-direction wave. Please also recall problem 2.5 in assignment 2. 2.3.2 Quantum well Consider the general case of a particle in a one-dimensional potential well, which can be, for example, an electron subject to an electric potential field as shown in the figure. This is actually the model for thin films. The steady-state Schrödinger equation for the particle in such a potential profile is 2 2 2 0 2 d E m dx Ψ Ψ = = (0<x<D); 0 Ψ = (x<0 or x>D).

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2.57 Fall 2004 – Lecture 4 2 Note: for u → ∞ (x<0 or x>D), only 0 Ψ = can satisfy the Schrödinger equation. Same as the free particles, the solution for first equation is still ikx ikx Ae Be Ψ = + , where 2 2 2 mE mE p k = = = = = = . The general boundary conditions are the continuity of the wave functions and their first derivatives at the boundaries. The latter derives from the continuity of particle flux at the boundary. For the current problem, the continuity of the first derivatives is not required because the wavefunction at the boundaries are already known to be zero. With the continuity of the wave function at x=0 and x=D, we have x=0 A+B=0 [ ] [ ] x=D exp exp 0 A ikD B ikD + = Above equations yield ( ) 0 ikD ikD A e e = .
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