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

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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

Ψ_{n1x n2x}(x,t) = 2^-1/2 [Ψ_n1x(x,t) + Ψ_n2x(x,t)],                     (13.13)

 and

Ψ_{n1y n2y}(y,t) = 2^-1/2 [Ψ_n1y(y,t) + Ψ_n2y(y,t)],                     (13.14)

where Ψ(x,y,t) = Ψ_{n1x n2x}(x,t)Ψ_{n1y n2y}(y,t), and Ψ_n1x(x,t), Ψ_n2x(x,t), Ψ_n1y(y,t), and Ψ_n2y(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 2π ħ/E₁. You can change n_1x, n_2x, n_1y, and n_2y. The default values, n_1x = n_1y = 1 and n_2x = n_2y = 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 n_1x, n_2x, n_1y, and n_2y. In particular:

1. For n_1x = 1, n_2x= 2, n_1y= 1, and n_2y = 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> e_x + <y> e_y.
2. For n_1x = 1, n_2x= 2, n_1y = 1, and n_2y = 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 n_1x = 1, n_2x= 3, n_1y = 1, and n_2y = 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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