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The energy eigen functions of free particle are given. These are found to be eigen functions momentum also. The energy eigen functions have infinite degeneracy. There an eigen function corresponding to each momentum direction.
The free particle Hamiltonian in three dimensions is \begin{equation}\label{EQ16} H =-\frac{\hbar^2}{2m} \nabla^2\end{equation}The free particle solution in three dimensions corresponding to energy eigen value $E$ are given by \begin{equation}u_E(x,y) = N \exp(ik_1x+ik_2y+ik_3z)\end{equation} where \begin{equation} E=\frac{\hbar^2k^2}{2m}, \qquad k^2 = k_i^2+k_2^2+k_3^2 \end{equation} and $k_1,k_2,k_3$ are otherwise arbitrary. If we write $\hat{k} = k \hat{n}$, where $\hat{n}$ is a unit vector, the solution $u_E(x,y,z)$ can be written as \begin{equation}u_E(x,y,z) = N \exp(i\vec{k}\cdot\vec{r})= N \exp(ik \hat{n}\cdot\vec{r}).\end{equation} These energy eigen functions are also eigen functions of momentum operator with eigen value $\hbar \vec{k}$. For a fixed energy there are infinite number of solutions, one corresponding to each direction of momentum. In other words the energy eigen functions have infinite degeneracy. One can impose a momentum delta function normalisation and the corresponding eigen functions are
\begin{equation} \psi_\kbf(\xbf) = \frac{1}{(2\pi)^{3/2}} e^{i\kbf\cdot\rbf}\end{equation} where the vector \(\rbf=(x,y,z)\) takes all possible values. Similar results hold for a free particle in any dimension.
