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Angelina Cole
5 years ago
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1 In[0]:= Out[0]= 82009, 0, 9, 2, 35, < ü Bo 0 < < -> 2 Consider the case when we have a particle in the ground state of PIB of length. In[3]:= Y@_, _D := Definitions: 2 SinB p F In[4]:= F@_, _D := If@0 < <, Y@, D, 0D In[5]:= f@m_d@_, _D := Now we have two normalized functions SinB m p 2 F In[6]:= 0 Y@, D Y@, D Out[6]= 2 In[7]:= 0 f@md@, D f@md@, D Out[7]= Sin@2 m pd - 2 m p For completeness we plot the functions and the square of the functions, assuming that =. Plot@8F@, D, Hf@D@, D<, 8, 0, 2<, PlotStyle Ø 88Thick, Red<, 8Thick, Blue<<, BaseStyle Ø 8Bold, 4<, Frame Ø True, Frameabel Ø 8, y<, Rotateabel Ø False, ImageSize Ø 400D y

2 2 DoublePiB.nb D^2, D^2<, 8, 0, 2<, PlotStyle Ø 88Thick, Red<, 8Thick, Blue<<, BaseStyle Ø 8Bold, 4<, Frame Ø True, Frameabel Ø 9, "»y 2 "=, Rotateabel Ø False, ImageSize Ø 400E 2.0.5»y To find the epansion coefficients and the probabilities, we first need to define the coefficients in the new basis set of the bigger bo: In[8]:= c@m_, _D := SimplifyB f@md@, D Y@, D, Assumptions Ø 8m œ Integers<F 0 In general the coefficients are given by the following epression: In[9]:= Out[9]= c@m, D SinA m p 4 p - m 2 p Notice that n=2 may give us a problem. Now we calculate the probabilities as the square of the coefficients prob = Table@8n, c@n, D^2< êê N, 8n,, 0<D; and show them in a tabular form TableForm@prob, TableHeadings Ø 88"n =", "n =", "n =", "n =", "n =", "n =", "n =", "n =", "n =", "n ="<, 8n, Prob<<D n Prob n = n = n = n = n = n = n = n = n = n = 0. 0.

3 DoublePiB.nb 3 From the table we notice that the probability of finding the particel in the grouns state of the bigger bo is only 36% compared to 50% in the first eicted state. Furthermore we can calculate the sum of the square of the coefficients and include up to 50 terms We can also calcuate the average energy 50 probt = c@m, D^2 êê N 50 m= avere = NB c@m, D^2 m ^2 ê 4F m= Finally we construct the approimate wave function thet include the contribution of the first 0 eigenfunctions of the bogger bo 0 aprof@_, _D = c@m, D f@md@, D m= 3 p SinA p 2 E SinA p E 2 5 p SinA 3 p 2 E - 2 p SinA 5 p 2 E 45 p SinA 7 p 2 E - 77 p SinA 9 p 2 E et us plot the approimate wave function PlotAaproF@, D^2, 8, 0, 2<, PlotStyle Ø 8Green, Thickness@5D<, BaseStyle Ø 8Bold, 4<, Frame Ø True, Frameabel Ø 9, "»y 2 "=, Rotateabel Ø False, ImageSize Ø 400E 2.0.5»y If we measure the energy in units of ue

4 4 DoublePiB.nb ue = h 2 8 m 2 and time in units of we can construct the approimate time dependent wave function ut = 8 m 2 h 0 apro@_,, t_d = c@m, D f@md@, D Ep@ -Â m ^2 m= 2 p t ê 4D - 2 Â p t SinA p 3 p Â p t SinA 5 p 2 p -2 Â p t Sin@p D Â p t SinA 7 p 45 p But we really want to visualize the probabilitydensity of finding the particel at Â p t SinA 3 p - 5 p Â p t SinA 9 p 77 p prob@_,, t_d = Simplify@Epand@apro@,, td * apro@,, -tddd Now we consider the middle of the bo and consider the probability as a function of time Plot@Evaluate@Re@prob@,, tddd, 8t, 0, 2<D movie = TableAPlotAEvaluate@Re@prob@,, nddd, 8, 0, 2<, PlotRange -> 80, 2.25<, Frame Ø True, ImageSize Ø 400, Plotabel Ø N@nD, Frameabel Ø 9 ê, "»y 2 "=, PlotPoints Ø 00, Rotateabel Ø False, PlotStyle Ø 8Black, Thickness@D<, BaseStyle Ø 8Bold, 4<E, 8n, 0, 4, ê 00<E; Movie: Using printed plots from movie, selecting the cell and saving selection from File->Save Selection As menu as Quick- Movie, or select all the figures and use Cell -> ConverTo.-> QuickTime. Also we can animate the cell using Graphics- >Rendering->Animate, but istmanipulate is better Do@Print@movie@@iDDD, 8i,, ength@movied<dh*flush the figures to the left*

5 DoublePiB.nb »y istanimate@movied »y ü Bo -/2 < < /2 -> - < <

ME201/MTH281ME400/CHE400 Convergence of Bessel Expansions

ME201/MTH281ME400/CHE400 Convergence of Bessel Expansions 1. Introduction This notebook provides a general framework for the construction and visualization of partial sums of Fourier-Bessel series. The