What is wrong with the following argument We know that the functions n (x) 2 a sin(n x a) are eigenfunctions of the total energy operator for the particle in the infinitely deep box We also know that in the box, E P 2 x 2m V(x) P 2 x 2m Therefore, the operator for E total is proportional to the operator for p 2 x Because the operators for p 2 x and p x commute as you can easily demonstrate, the functions n (x) 2 a sin (n x a) are eigenfunctions of both the total energy and momentum operators

Question: What is wrong with the following argument? We know that the functions n (x) = 2/a sin(n x/a) are eigenfunctions of the total energy

What is wrong with the following argument? We know that the functions ψ _n (x) = √2/a sin(nπ x/a) are eigenfunctions of the total energy operator for the particle in the infinitely deep box. We also know that in the box, E = P²_x/2m + V(x) = P²_x/2m. Therefore, the operator for E _total is proportional to the operator for p²_x. Because the operators for p²_x and p_x commute as you can easily demonstrate, the functions ψ _n (x) = √2/a sin (nπ x/a) are eigenfunctions of both the total energy and momentum operators.

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