Wave Functions help

1) Ψ1 = sin(𝑘𝑥)
2) Ψ2 = e^𝑖𝑘𝑥 = 𝑐𝑜𝑠(𝑘𝑥) + 𝑖sin(𝑘𝑥)

For each wave function, show that they are eigenfuntions of the hamiltonian in
i) Free Space (V(x)=0).
ii) A 'flat' potential (V(x) = V)

In each case, what is the kinetic and total energy?

I've done all of it, but how do I show in each case, what is the kinetic and total energy?

Thanks!

What's the Hamiltonian operator in each case?

i) ħ^2k^2/2m sin(kx)

ii) sin(kx) [ħ)^2k^2/2m + V]

i) -ħ^2 i^2 k^2/2m e^ikx

ii) e^ikx [-ħ^2 i^2 k^2/2m + V]

Ignoring constants, the Scrodinger equation will be:

$\hat{H} \Psi_n = \left[ -\dfrac12 \dfrac{\text{d}^2}{\text{d}x^2} + V(x) \right] \Psi_n = \text{E}_n \Psi_n$

$- \dfrac12 \dfrac{\text{d}^2}{\text{d}x^2}$ is the kinetic energy operator part of the Hamiltonian.

So for $V=0$, how much potential energy is there?

1/2?

V(x) is the potential energy operator so when it is zero the potential energy will be...

0