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Showing two 1D box potentials merge into a 2D

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    So I showed

    U_1(x) = \begin{Bmatrix} 0, \ 0<x<a \\ 0, \ ow

    is represented by the eigenfunction

    \phi_n(x) = \sqrt{\dfrac{2}{a}} sin \dfrac{n \pi x}{a}, \ n=1,2,3...

    with eigenenergies

    E_n = \dfrac{\hbar^2 n^2 \pi ^2}{2ma^2}

    Then to show that the 2D box

    U_2(x) = \begin{Bmatrix} 0, \ 0<x<a, \ 0<y<a \\ 0, \ ow

    is represented by the eigenfunction

    \psi_{{n_x},{n_y}}(x,y) = \phi_{n_x}(x) \phi_{n_y}(y) = \dfrac{2}{a}sin \dfrac{n_x \pi x}{a} sin \dfrac{n_y \pi y}{a}

    with eigenenergies

    E_{{n_x},{n_y}} = E_{n_x}+E_{n_y} = \dfrac{\hbar^2 \pi^2}{2ma^2}(n_x^2 + n_y^2)

    Is it sufficient to simply state that by separating the variables, and letting \psi_{{n_x},{n_y}} = f(x)g(y), one can separate the 2D box into two independent 1D boxes?

    Just to clarify here, I'm not explicitly asked to separate the 2D box, but to show that two 1D boxes will combine into a 2D box... so is it okay to just say that, "since we can seperate the 2D box eigenfunction into two 1D box eigenfunctions, it is clear that the 2D eigenfunction is a combination of two 1D eigenfunctions."
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    Yeah, just assume you can always separate variables.

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