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Section 13.3: Exploring Superpositions in the Two-dimensional Infinite Well


n1x = | n2x = | n1y = | n2y =

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One of the simplest examples of non-trivial time-dependent states is that of an equal-mix, two-state superposition in the infinite square well. Here we explore what these superpositions look like in two dimensions for a symmetric infinite square well. The individual position-space wave functions are

Ψn1x n2x(x,t) = 2-1/2n1x(x,t) + ψn2x(x,t)],

and

Ψn1y n2y(y,t) = 2-1/2n1y(y,t) + ψn2y(y,t)],

where Ψ(x,y,t) = Ψn1x n2x(x,tn1y n2y(y,t), and ψn1x(x,t), ψn2x(x,t), ψn1y(y,t), and ψn2y(y,t) are the individual one-dimensional solutions.

The animation depicts the time dependence of an arbitrary equal-mix two-state superposition by showing the probability density as a three-dimensional plot and also as a contour plot. The time is given in terms of the time it takes the ground-state wave function to return to its original phase, i.e., Δt = 1 corresponds to an elapsed time of 2πħ/E1. You can change n1x, n2x, n1y, and n2y. The default values, n1x = n1y = 1 and n2x = n2y = 2, are the two-dimensional extension of the standard one-dimensional case treated in almost every textbook, and treated here in Section 10.6.

Explore the time-dependent form of the position-space and momentum-space wave functions for other n1x, n2x, n1y, and n2y. In particular:

  1. For n1x = 1, n2x= 2, n1y= 1, and n2y = 2, describe the time dependence of the wave function. Describe the trajectory of the expectation value of position as a function of time: <r> = <x> ex + <y> ey.
  2. For n1x = 1, n2x= 2, n1y = 1, and n2y = 3, describe the time dependence of the wave function. Describe the trajectory of the expectation value of position <r> as a function of time.
  3. For n1x = 1, n2x= 3, n1y = 1, and n2y = 3, describe the time dependence of the wave function. Describe the trajectory of the expectation value of position <r> as a function of time.
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