Operators representing physical properties

2.4 Operators representing physical properties

The Energy-Operator: Hamilton Operator, Hamiltonian

\begin{equation*} A|\psi\rangle = \frac{\hbar}{i}\frac{\partial}{\partial t} |\psi\rangle = E \psi\rangle \label{time_ev_op} \end{equation*}

(2.7)

\begin{equation*} \mbox{Eigenvector:} \qquad e^{i\omega t} \label{energy_eigenvector} \end{equation*}

(2.8)

\begin{equation*} \mbox{Eigenvalue:} \qquad E = \hbar \omega \label{energy_eigenvalue} \end{equation*}

(2.9)

A synonym for the Hamilton Operator is time evolution operator. The meaning of this expression will be explained later.
The Momentum-Operator:

\begin{equation*} A|\psi\rangle = P_x |\psi\rangle = \frac{\hbar}{i}\frac{\partial}{\partial x} |\psi\rangle = p_x |\psi\rangle \end{equation*}

(2.10)

\begin{equation*} \mbox{Eigenvector:} \qquad e^{i k_x x} \end{equation*}

(2.11)

\begin{equation*} \mbox{Eigenvalue:} \qquad p_x = \hbar k_x \end{equation*}

(2.12)

The Eigenstates are plane waves. In this representation the Eigenvectors just show the space dependence. Since according to Eq. (2.9) each state has a time dependence, we find that each Momentum-Eigenvector reflects the properties of a plane wave.
The wave length is

\begin{equation*} \lambda = \frac{2 \pi}{|k_x|} \end{equation*}

(2.13)

The Space-Operator:

\begin{equation*} A|\psi\rangle = X |\psi\rangle = x |\psi\rangle = a |\psi\rangle \end{equation*}

(2.14)

i.e we find

\begin{equation*} (x - a) |\psi\rangle = 0 \end{equation*}

(2.15)

For \(x \neq a\) follows \( |\psi\rangle = 0\). In order to get a state, which can be normalized, we must set:

\begin{equation*} |\psi\rangle = \delta(x-a) \end{equation*}

(2.16)

Thus we find for the expectation value of the space operator:

\begin{equation*} \langle x\rangle = \langle \psi|X|\psi\rangle = \int_{-\infty}^{+\infty} \psi^*(x) x \psi(x) dx = \int_{-\infty}^{+\infty} x ||\psi||^2 dx = \int_{-\infty}^{+\infty} x P(x) dx \end{equation*}

(2.17)

\begin{equation*} P(x) dx = ||\psi||^2 dx \end{equation*}

(2.18)

is obviously the probability for finding a particle in the interval \((x, x+dx)\).