Eigenfunctions, Eigenvalues and Vector Spaces

For any given physical problem, the Schrödinger equation solutions which separate (between time and space), \bgroup\color{black}$\psi(x,t)=u(x)T(t),$\egroup are an extremely important set. If we assume the equation separates, we get the two equations (in one dimension for simplicity)

\begin{displaymath}\bgroup\color{black}i\hbar{\partial T(t)\over\partial t}=E\; T(t)\egroup\end{displaymath}

\begin{displaymath}\bgroup\color{black}Hu(x)=E\; u(x) \egroup\end{displaymath}

The second equation is called the time independent Schrödinger equation. For bound states, there are only solutions to that equation for some quantized set of energies


For states which are not bound, a continuous range of energies is allowed.

The time independent Schrödinger equation is an example of an eigenvalue equation.


If we operate on \bgroup\color{black}$\psi_i$\egroup with \bgroup\color{black}$H$\egroup, we get back the same function \bgroup\color{black}$\psi_i$\egroup times some constant. In this case \bgroup\color{black}$\psi_i$\egroup would be called and Eigenfunction, and \bgroup\color{black}$E_i$\egroup would be called an Eigenvalue. There are usually an infinite number of solutions, indicated by the index \bgroup\color{black}$i$\egroup here.

Operators for physical variables must have real eigenvalues. They are called Hermitian operators. We can show that the eigenfunctions of Hermitian operators are orthogonal (and can be normalized).


(In the case of eigenfunctions with the same eigenvalue, called degenerate eigenfunctions, we can must choose linear combinations which are orthogonal to each other.) We will assume that the eigenfunctions also form a complete set so that any wavefunction can be expanded in them,

\begin{displaymath}\bgroup\color{black}\phi(\vec{x})=\sum\limits_i \alpha_i\psi_i(\vec{x})\egroup\end{displaymath}

where the \bgroup\color{black}$\alpha_i$\egroup are coefficients which can be easily computed (due to orthonormality) by


So now we have another way to represent a state (in addition to position space and momentum space). We can represent a state by giving the coefficients in sum above. (Note that \bgroup\color{black}$\psi_p(x)=e^{i(px-Et)/\hbar}$\egroup is just an eigenfunction of the momentum operator and \bgroup\color{black}$\phi_x(p)=e^{-i(px-Et)/\hbar}$\egroup is just an eigenfunction of the position operator (in p-space) so they also represent and expansion of the state in terms of eigenfunctions.)

Since the \bgroup\color{black}$\psi_i$\egroup form an orthonormal, complete set, they can be thought of as the unit vectors of a vector space. The arbitrary wavefunction \bgroup\color{black}$\phi$\egroup would then be a vector in that space and could be represented by its coefficients.

\begin{displaymath}\bgroup\color{black}\phi=\pmatrix{\alpha_1\cr \alpha_2\cr \alpha_3\cr ...\cr} \egroup\end{displaymath}

The bra-ket \bgroup\color{black}$\langle\phi\vert\psi_i\rangle$\egroup can be thought of as a dot product between the arbitrary vector \bgroup\color{black}$\phi$\egroup and one of the unit vectors. We can use the expansion in terms of energy eigenstates to compute many things. In particular, since the time development of the energy eigenstates is very simple,


we can use these eigenstates to follow the time development of an arbitrary state \bgroup\color{black}$\phi$\egroup

\alpha_3e^{-iE_3t/\hbar}\cr ...\cr} \egroup\end{displaymath}

simply by computing the coefficients \bgroup\color{black}$\alpha_i$\egroup at \bgroup\color{black}$t=0$\egroup.

We can define the Hermitian conjugate \bgroup\color{black}$O^\dagger$\egroup of the operator \bgroup\color{black}$O$\egroup by

\begin{displaymath}\bgroup\color{black}\langle \psi\vert O\vert\psi\rangle=\lang...
...vert O\psi\rangle=\langle O^\dagger\psi\vert\psi\rangle.\egroup\end{displaymath}

Hermitian operators \bgroup\color{black}$H$\egroup have the property that \bgroup\color{black}$H^\dagger=H$\egroup.

Jim Branson 2013-04-22