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

Animation 1: Probability Density | Animation 2: Probability Density Contours

Please wait for the animation to completely load.

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*)],

and

Ψ_{n1y n2y}(*y*,*t*) = 2^{-1/2} [ψ_{n1y}(*y*,*t*) + ψ_{n2y}(*y*,*t*)],

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*., Δ*t* = 1 corresponds to an elapsed time of 2π*ħ/E*_{1}. 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:

- 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}. - 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 <> as a function of time.**r** - 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 <> as a function of time.**r**

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