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(September 9, 2003)
![Cell[TextData[The Infinite Potential Well]]](HTMLFiles3/index_1.gif){kind=small}
![1 (+) (-) 2 2 2 2 2 (n - -) πx 2 2 2 2 2 2 -iE t/h -iE t/h ^ -h ^2 ^ ^ ^ h d h d ∞ (+) (+) (-) (-) (+) (+ /-) (+) 1 2 (-) 1 nπx (+) h (2 n - 1) π (-) h n π .. d ^ ∞ (+) (+) n (-) (-) n Cell[TextData[{In one dimension, the general form for the Hamiltonian of a particle moving in a potential is given by <br /><br />, Cell[BoxData[H = --- p + V(x)], Background -> RGBColor[0, 1, 1]], <br /><br />the energy eigenvalues of a particle in a state |ψ> are given by the following operator equation:<br /><br />, Cell[BoxData[H | ψ > = E | ψ >], Background -> RGBColor[0, 1, 1]], <br /><br />or in the position representation, the equation can be written in the more familliar form:<br /><br />, Cell[BoxData[< x | H | ψ > = E < x | ψ > --> ---- --- ψ(x) + V(x) ψ(x) = E ψ(x) ], Background -> RGBColor[0, 1, 1]], <br /><br />If the potential is an infinite well, of width a, positioned symmetrically about the position x=0, it may be represented as:<br /><br />, Cell[BoxData[V(x) =], Background -> RGBColor[0, 1, 1]], Cell[BoxData[{0 for | x | < a ], Background -> RGBColor[0, 1, 1]], <br /><br />For the infinite potential well, the wave equation becomes inside of the well <br /><br />, Cell[BoxData[---- --- ψ(x) = E ψ(x) for | x | < a ], Background -> RGBColor[0, 1, 1]], <br /><br />and<br /><br />, Cell[BoxData[ψ(x) = 0 for | x | >= a], Background -> RGBColor[0, 1, 1]], <br /><br />After solving the second order differential equation and imposing the boundary conditions, the solutions to the infinite potential well are :<br /><br />, Cell[BoxData[ψ(x) = ∑ (a u (x) + a u (x) )], Background -> RGBColor[0, 1, 1]], <br /><br />Where , Cell[BoxData[a ], Background -> RGBColor[0, 1, 1]], is a c-number and , Cell[BoxData[u (x) ], Background -> RGBColor[0, 1, 1]], are given by the following equations:<br /><br />, Cell[BoxData[u (x) = ------- cos(---------------)], Background -> RGBColor[0, 1, 1]], <br /><br />, Cell[BoxData[u (x) = ------- sin(--------)], Background -> RGBColor[0, 1, 1]], <br /><br />The energy eigenvalues corresponding to these solutions are given by:<br /><br />, Cell[BoxData[E = --------------------], Background -> RGBColor[0, 1, 1]], <br /><br />, Cell[BoxData[E = -------------], Background -> RGBColor[0, 1, 1]], <br /><br />The time evolution of ψ(x) can be determined from the Schr, Cell[BoxData[o ], Background -> RGBColor[0, 1, 1]], dinger equation:<br /><br />, Cell[BoxData[ih -- | ψ > = H | ψ > = E | ψ >], Background -> RGBColor[0, 1, 1]], <br /><br />solving the equation in the position representation yields<br />, Cell[BoxData[ψ(x, t) = ∑ (a u (x) e + a u (x) e )], Background -> RGBColor[0, 1, 1]], <br /><br />The solutions to the infinite potential well are plotted below versus energy. <br />}], Background -> RGBColor[0., 1., 1.]] 2 m 2 m 2 2 m 2 n = 1 n n n n n n n Sqrt[a] a n Sqrt[a] a n 2 n 2 dt n = 1 n n n n dx ∞ for | x | >= a dx 8 ma 2 ma](HTMLFiles3/index_2.gif)
![h = 1.6 * 10^(-34) ; "Planck's Constant J*s" ; a = 10 * 10^-10 ; "Potential Well Width" ; m = 9.109 * 10^(-31) ; (* mass of electron *) E[n_]^+ := (h^2 * (2 * n - 1)^2 * π^2)/(8 * m * a^2) ; (* Even eigenenergies *) E[n_]^- := (h^2 * n^2 * π^2)/(2 * m * a^2) ; (* Odd eigenenergies *) u[n_, x_]^+ := Cos[((n - 1/2) π * x)/a] ; (* Even eigenfunctions *) u[n_, x_]^- := Sin[(n * π * x)/a] ; (* Odd eigenfunctions *) "Plots of even and odd eigenfunctions with their energies" ; ψ _ 1[x_, t_] := 1/2 (u[1, x]^+ * e^(-(I * E[1]^+ * t)/h) + u[1, x]^- * e^(-(I * E[1]^- * t)/h)) Wavfuns = Table[{Plot[u[n, x]^+, {x, -a, a}, DisplayFunction -> Identity, Frame -> True, FrameLabel -> {"position (m)", amplitude, "even " n, o}], Plot[u[n, x]^-, {x, -a, a}, DisplayFunction -> Identity, Frame -> True, FrameLabel -> {"position (m)", amplitude, "odd " n, o}]}, {n, 1, 8}] ; Energy1 = Table[E[n]^+, {n, 1, 10}] ; "Energy levels of even eigenfunctions" ; Energy2 = Table[E[n]^-, {n, 1, 10}] ; "Energy levels of odd eigenfunctions" ; Show[ListPlot[Energy1, DisplayFunction -> Identity, PlotStyle -> RGBColor[1, 0, 0]], ListPlot[Energy2, DisplayFunction -> Identity, PlotStyle -> RGBColor[0, 1, 0]], DisplayFunction -> $DisplayFunction, Frame -> True, FrameLabel -> {"n", "Energy (J)", "Energy of Even(Red) and Odd(Green) Eigenfunctions", o}, Prolog -> AbsolutePointSize[4]] Show[GraphicsArray[Wavfuns], DisplayFunction -> $DisplayFunction] "Two state wavefunction generated by the combination of first excited state and ground state wave functions" ; ψ _ 1[x_, t_] := 1/2 (u[1, x]^+ * e^(-(I * E[1]^+ * t)/h) + u[1, x]^- * e^(-(I * E[1]^- * t)/h)) Plot[ψ _ 1[x, 0 * 1 * 10^-8] * ψ _ 1[x, 0 * 1 * 10^-8], {x, -a, a}, PlotRange -> {{-a, a}, {0, 1}}, Frame -> True, FrameLabel -> {"Position(m)", "Probability", "Two state evolution in infinite potential well", "=time (sec)" N[n * 1 * 10^-8] }, AspectRatio -> 1.2, PlotStyle -> RGBColor[1, 0, 0]]](HTMLFiles3/index_3.gif)
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