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68111127-3-2-quantumwell

# 68111127-3-2-quantumwell - Quantumwell problems This is a...

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Quantumwell problems This is a Mathematica program, presented for quantum mechanical calculus. This first section include analysis of potential well and barrier problems. Quantum well systems are particularly play important role in optoelectronics, and are used in devices such as the quantum well laser etc. Infinite potential well In quantum mechanics, the particle in a box model is the simplest example of a confined particle. The model is used to explain the differences between classical and quantum systems. ª Definition of the potential In this section we ar dealing with constant or zero potentials. For our purpose let us define a potential V[L_, V1_,V2_,V3_][x_] whose parameters are given by L is length of the well. Vi is potential values of the region i. We laso define the parameters , m and Ε are Planck’s constant, mass of the particle and energy of the particle respectively. Infinite well potential can be defined as follows: V @ L_, V1_, V2_, V3_ [email protected] x_ D : = V1 ; x < 0 V @ L_, V1_, V2_, V3_ [email protected] x_ D : = V2 ; x > 0 && x < L V @ L_, V1_, V2_, V3_ [email protected] x_ D : = V3 ; x > L In the case of infinite well potential V1 = V3 fi ¥ and V2 = 0. Plot of potential for some specific values o f the parameters as follows: Plot @ V @ 6, 3, 0, 3 [email protected] x D , 8 x, - 0.01, 6.01 < , PlotStyle -> 8 Thickness @ 0.01 D< , Frame fi 8 True, True, False, True < , Axes -> False, PlotLabel -> "Potential Well", FrameLabel -> 8 "x", " Ε " < , RotateLabel -> False, PlotRange fi All D 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x Ε Potential Well Schr dinger equation is - 2 2 m Ψ ¢¢ @ x D + V Ψ @ x D = ΕΨ @ x D OR we can write Ψ ¢¢ @ x D + k 2 Ψ @ x D = 0 where k 2 = 2 m H Ε- V L 2 . Solution of the equation can be obtained: Ψ in @ x D = DSolve B Ψ ¢¢ @ x D + k 2 Ψ @ x D == 0, Ψ @ x D , x 1, 1, 2 DD C @ 1 D Cos @ k x D + C @ 2 D Sin @ k x D Let us turn our attention to the infinite well potential: outside the well Ψ @ x D = 0 and inside the well V = 0. Then k 2 = 2 m Ε 2 . Using the boundary conditions Ψ @ 0 D = Ψ @ L D = 0

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sol1 = Solve @8 Ψ in @ x D 0 . x fi 0, Ψ in @ x D == 0 . x fi L <D Solve::svars: Equations may not give solutions for all "solve" variables. 88 C @ 2 D fi 0, C @ 1 D fi 0 < , 8 Sin @ k L D fi 0, C @ 1 D fi 0 << We can easily deduce that kL = n Π or Ε = n 2 Π 2 2 2 L 2 m . The integral constant can be obtained by the normalization of the wave function: Ψ in @ x D = Ψ in @ x D . 8 C @ 1 D fi 0, k fi n Pi L < C @ 2 D Sin B n Π x L F Ε = n 2 Π 2 2 2 L 2 m sol1 = Solve @ Assuming @ Element @ n, Integers D , Integrate @ Ψ in @ x D in @ x D , 8 x, 0, L <DD 1, C @ 2 DD :: C @ 2 D fi - 2 L > , : C @ 2 D fi 2 L >> Ψ in @ x D = Ψ in @ x D . sol1 @@ 2 DD 2 Sin B n Π x L F L We define the parameters and then plot the graph of the wavefunction and energy. parameters1 = 8 L fi 4, m fi 1, fi 1 < ; Do @ pe @ i D = Plot @ Evaluate @ Ε . parameters1 . n fi i D , 8 x, 0, L . parameters1 < , PlotStyle -> 8 Dashed, Red <D ; pw @ i D = Plot @ Evaluate @ Ε + Ψ in @ x D . parameters1 . n fi i D , 8 x, 0, L . parameters1 <D , 8 i, 1, 5 <D Show @ pe @ 1 D , pw @ 1 D , pe @ 2 D , pw @ 2 D , pe @ 3 D , pw @ 3 D , pe @ 4 D , pw @ 4 D , pe @ 5 D , pw @ 5 D , Plot @ V @ L . parameters1, 9, 0, 9 [email protected] x D , 8 x, - 0.01, L + .01 . parameters1 < , PlotStyle -> 8 Thickness @ 0.01 D<D , Frame -> 8 True, True, False, True < , Axes -> False, PlotLabel -> "Infinite Potential Well", FrameLabel -> 8 "x", " Ε " < , PlotRange fi All D 0 1 2 3 4 0 2 4 6 8 x Ε Infinite Potential Well 2 quantumwell.nb
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