The Particle-Wave-Dualism

2.7 The Particle-Wave-Dualism

The most important example of two physical properties which can not been measured at the same time are location and momentum of a particle. The unitary transformation which couples both operators is the Fourier-Transformation. We will not use Eq. (2.19) to get an exact result but will just calculate an approximation.
We look at a state that is a superposition of a couple of free electrons with similar momenta:

\begin{equation*} \psi(x) = \int\limits_{p_x-\frac{\Delta p}{2}}^{p_x+\frac{\Delta p}{2}} e^{\frac{i x p_x}{\hbar}} dx =\frac{2 \hbar}{x} \sin\frac{x \Delta p}{2 \hbar} e^{\frac{i x p_x}{\hbar}} \end{equation*}

(2.21)

The uncertainty of the momentum is obviously \(\Delta p\). \(||\psi||^2\) has its first zeros at \(\frac{x \Delta p}{2 \hbar} e^{\frac{i x p_x}{\hbar}}\). It’s main weight has the width \(\pi\). This leads to the uncertainty relation

\begin{equation*} \Delta x \Delta p = h \end{equation*}

(2.22)

The more exactly the position of a particle is defined the less information one has concerning its momentum (and vice versa).