The solution of $$ \frac{p^2}{2m}\psi_E(p) = E\psi_E(p) $$ is $$ \psi_E(p) = \delta(p-\sqrt{2mE}) $$
It is zero for every $p$, except for $p=\sqrt{2mE}$ where it has a peak.
See also the definition and visualization of the Dirac $\delta$ function.
Thomas Fritsch
- 41k
- 13
- 75
- 144