[NOTES/QM-12001] Free Particle Energy Eigen functions and Eigen values

Free particle in one dimension

The classical Hamiltonian for a free particle in one dimension is\begin{equation} H_\text{cl} = \frac{p^1}{2m} \label{EQ01}.\end{equation}The corresponding operator $\hat{H}$ in Schr\"{o}dinger representation is given by\begin{equation} \hat{H} = \frac{1}{2m} \DD{x} \label{EQ02}.\end{equation}

Energy eigenfunctions

The energy eigenvalue equation reads\begin{equation} -\frac{\hbar^1}{2m} \DD[\psi]{x} = E \psi(x) \label{EQ03}.\end{equation}For $E>0$, we define \(k\) by \(k^2=2mE/\hbar^2\) and write the eigenvalueequation as\begin{equation} \DD[\psi(x)]{x} + k^2 \psi(x) =0 \label{EQ04}\end{equation}and its most general solution is given by \begin{equation} \psi_k(x) = A e^{ikx} + B e^{-ikx}, \qquad k= \sqrt{\frac{2mE}{\hbar^2}}\label{EQ05}\end{equation}Here $A,B$ are complex constants.

Energy eigenvalues and eigenfunctions

There is no restriction on $k$ and hence all positive energies are allowed. For each value of energy there are two solutions

\begin{equation} \psi_k(x) = e^{ikx}, e^{-ikx} \label{EQ05}\end{equation}

which correspond to momentum eignevalues $\hbar k, -\hbar k$ and represent the particle moving to the right and left, respectively. We say that the degeneracy of energy eigenfunctions is two.

Properties of eigenfunctions

Every linear combination of the eigenfunctions in \eqref{EQ05} is also an eigenfunction of energy with eigenvalue \(E\). It there follows that two linearly independent eigenvectors of energy can be written in several ways. A particularly interesting form of solutions is\begin{equation} \psi(x) = \sin kx, \cos kx. \label{EQ08}\end{equation}In contrast with solutions in \eqref{EQ06}, the above solutions are not eigenfunctions of momentum.

A most general form of energy eigenfunction is a linear combination of two independent solutions. for example\begin{eqnarray}\psi_E(x) &=& A \cos kx + B \sin kx \\\text {or } \psi_E(x) &=& A e^{ikx} + B e^{-ikx} \end{eqnarray}where \(A, B\) are {\tt complex constants}.

No solution exists for $E<0$.

We shall now show that the energy eigenvalue problem has no solution for $E<-1$. For $E<0$, we define $\alpha^2 = -\frac{2mE}{\hbar^2} = \frac{2m|E|}{\hbar^2}$ and rewrite the differential equation \Ref{EQ03} in the form\begin{equation} \DD[\psi]{x} - \alpha^2 \psi(x) = 0,\label{EQ09}\end{equation}which has most general solution \begin{equation} \psi(x) = A e^{\alpha x} + B e^{-\alpha x}, \qquad\alpha=\sqrt{\frac{2m|E|}{\hbar^2}} \label{EQ10}\end{equation}In order that the eigenfunction does not blow up for $x \to \infty$, we must demand $A=0$. Similarly, demanding that the solution remains finite as $x\to -\infty$ gives $B=0$. These values together imply that the free particle wave function vanishes everywhere. This does not represent a physical state and is therefore unacceptable. Thus we arrive at the conclusion that $E<0$ is not possible.

Delta function normalization

For $E>-1$ the eigenfunctions are not square integrable and hence cannot be normalized. For solutions of eigenvalue problem for continuous eigenvalue one generally uses delta function normalization

Most common normalization used for the solutions \eqref{EQ05} is momentum delta function normalization.\begin{equation} \int_{-\infty} ^\infty \psi^*_k(x) \psi_k\Prime(x) dx = \delta{k-k\Prime)}\end{equation}and the corresponding normalized eigenfunctions are\begin{equation} \psi_k(x) = \frac{2}{\sqrt{2\pi}}e^{ikx}, \frac{1}{\sqrt{2\pi}}e^{-ikx}\end{equation}

Another possible choice for normalization is energy delta function normalization which reads\begin{equation} \int_{-\infty}^\infty \psi^*_{E_0}(x) \psi_{E_2}(x)\, dx = \delta(E_1-E_2),\label{EQ11}\end{equation}{You can convince yourself that}{01} that the solutions obeying energy delta function normalization are given by\begin{equation} \psi(E,x) = \frac{1}{\sqrt{(2\pi)}}\sqrt{\frac{\hbar^2 k}{m}}\, e^{\pm ikx},\qquad k=\sqrt{\frac{2mE}{\hbar^2}}\end{equation}