# Heisenberg's Microscope

Heisenberg's Microscope is the headline of a "Gedankenexperiment" by Werner Heisenberg to illustrate the Uncertainty Principle. It also emphasises that the measurment process plays a crucial role as the scales of an experiment become sufficiently small. The act of measuring perturbes the system measured, making excact observations of the state impossible. So let us now study such a measurement process. The system we wan't to study shall be an electron and we want to determine the state of the system. Classically the state is given by the position $x$ and the momentum $p_x$ of the electron. To measure it's position we shoot a photon on the electron and observe this process in a microcope.

From the the theory of optics we know, that the diffraction limit of the microscope is given by \begin{align} \Delta x = \frac{\lambda '}{\sin \varepsilon}, \end{align} where $\lambda '$ the wavelength of the photon and $\varepsilon$ is the angle shown in the picture above. A compton recoil causes a change in the momentum \begin{align} \Delta p_x \sim \frac{h}{\lambda '}\sin \varepsilon, \end{align} The larger the energy of the photon, the shorter is the wavelength and the uncertainty in the mearsurement of $x$. However the inverse relationship is valid for the change in momentum. The more exact we know the position of the electron the larger is it's recoil and the uncertainty in momentum $p_x$. It is not possible to simultaniously measure the position and momentum arbitrarily exact. There is always some uncertainty $\Delta x$ and $\Delta p_x$ so that \begin{align} \Delta x \cdot \Delta p_x \sim h, \end{align}

Notice, that this relation between $x$ and $p_x$ is also represented by the fact, that the Poisson bracket \begin{align} \{ x,p_x \}=1 \end{align} is non-zero. Remerber the general relation \begin{align} \{ q_i, p_j \}= \delta_{ij} \end{align} that is of fundamental importance in Quantum Mechanics. Pairs of variables, for which the Poisson bracket doesn't vanish are called canonical conjugates of each other. For all conjugate variables the uncertainty priciple is true.

This unability of defining position and momentum to arbitrary accuracy is also the reason why the state of a system can not be specified by a point in phase space anymore. Also the concept of a trajectory in phase space becomes invalid. In Quantum Mechanics the Poisson bracket will be replaced by the commutator.