Answer
No pair of adjacent levels has an energy difference equal to the energy of the $n=6$ level.
Work Step by Step
We can find an expression for energy of an electron in this potential well when $n = 6$:
$E_n = (\frac{h^2}{8m~L^2})~n^2$
$E_6 = (\frac{h^2}{8m~L^2})~(6)^2$
$E_6 = 36~(\frac{h^2}{8m~L^2})$
Let $n_h$ be the state with the higher quantum number.
Let $n_l$ be the state with the lower quantum number.
We can use the expression for the energy difference to try to find $n_h$ and $n_l$:
$E_{n_h} - E_{n_l} = E_6$
$(\frac{h^2}{8m~L^2})~n_h^2 - (\frac{h^2}{8m~L^2})~n_l^2 = 36~(\frac{h^2}{8m~L^2})$
$n_h^2 - n_l^2 = 36$
$(n_h - n_l)(n_h+n_l) = 36$
Let's suppose that:
$(n_h - n_l)(n_h+n_l) = (1)(36)$
However, there are no two integers $n_h$ and $n_l$ which satisfy this equation.
Let's suppose that:
$(n_h - n_l)(n_h+n_l) = (2)(18)$
However, there are no two integers $n_h$ and $n_l$ which satisfy this equation.
Let's suppose that:
$(n_h - n_l)(n_h+n_l) = (3)(12)$
However, there are no two integers $n_h$ and $n_l$ which satisfy this equation.
Let's suppose that:
$(n_h - n_l)(n_h+n_l) = (4)(9)$
However, there are no two integers $n_h$ and $n_l$ which satisfy this equation.
Let's suppose that:
$(n_h - n_l)(n_h+n_l) = (6)(6)$
However, there are no two integers $n_h$ and $n_l$ which satisfy this equation.
Therefore, no pair of adjacent levels has an energy difference equal to the energy of the $n=6$ level.
