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CHEM 5 (21)
Lecture

# 04 3D plots, Contour plots, Density plots.pdf Premium

21 Pages
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Department
Chemistry
Course
CHEM 5
Professor
Douglas Tobias
Semester
Fall

Description
Three-dimensional plots of functions of two variables Plots of three-dimensional surfaces We have seen that Mathematica is very useful for making plots of functions of one variable. In this lesson we explore some of Mathematica's capabilities for plotting functions of two variables. As an example, we will consider the quantum mechanical problem of a particle in a two-dimensional box, i.e., quan- tum particle confined in two dimensions between "walls" that correspond to regions of infinite potential energy. Solution of the Schrödinger equation for this system gives the quantized energies: h2 En , n = In + n M2 , 1 2 1 2 8 mL where m is the mass of the particle, h is Planck's constant, L is the length of the box (assumed to be the same in the x and y directions, and n and1n are q2antum numbers that are restricted to the positive integers (1, 2, 3,...). The corresponding wavefunctions are: 2 n1px n2py yn1, 2Hx, yL = Lin( L ) sin( L ). According to the Born interpretation of the wavefunction, y 2 is the probability density of the quantum particle, i.e., y dxdy is the probability of finding the particle within dxdy of the point (x,y). In the following, we'll plot a couple of the wavefunctions and the corresponding probability densities with x and y in units of L (i.e., we set L = 1 in the above equation) using the Plot3D command, which is analogous to the Plot command for functions of one variable: In[1]:[email protected]_, y_D := 2 [email protected] Pi xD [email protected] Pi yD; H* plot using replacement rules to set the values of n1 and n2 *L [email protected]@x, yD ê. 8n1 Ø 1, n2 Ø 1
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