Question

Show that the wavefunction Ψ(x) = N sin kx + iN cos kx is an eigen function of the momentum operator and determine its eigen value.

Answer 1

The momentum operator is $\hat{p} =- i\hbar\dfrac{\partial}{\partial x}$. By definition, the eigen function has the following property:

$$\hat{p}\Psi(x) = \lambda\Psi(x)$$

where $\lambda$ is the corresponding eigenvalue. Substituting the function and peforming the differentiation, obtain:

$$\hat{p}\Psi(x) = - i\hbar\dfrac{\partial}{\partial x} (N\sin(kx) + iN\cos(kx)) = - i\hbar(Nkcos(kx) - iNk\sin(kx)) =\\ =-i\hbar Nkcos(kx) - \hbar Nk\sin(kx)$$

Factoring out $-\hbar k$ , have:

$$\hat{p}\Psi(x) = -\hbar k(N\sin(kx) + iN\cos(kx)) = -\hbar k\Psi(x)$$

Thus, this finction is indeed the eigen function of the momentum operator. Comparing the last equation with the first one, find the eigenvalue:

$$\lambda = -\hbar k$$

Answer. $\Psi(x)$ is the eigenfunction of the momentum operator with the eigenvalue $\lambda = -\hbar k$.