Let us consider what effect the wave-particle duality has on attempts to measure simultaneously the $x$ coordinate and the $x$ component of linear momentum of a microscopic particle. We start with a beam of particles with momentum $p$, traveling in the $y$ direction, and we let the beam fall on a narrow slit. Behind this slit is a photographic plate. See Fig. 1.1.
Particles that pass through the slit of width $w$ have an uncertainty $w$ in their $x$ coordinate at the time of going through the slit. Calling this spread in $x$ values $\Delta x$, we have $\Delta x=w$.
Since microscopic particles have wave properties, they are diffracted by the slit producing (as would a light beam) a diffraction pattern on the plate. The height of the graph in Fig. 1.1 is a measure of the number of particles reaching a given point. The diffraction pattern shows that when the particles were diffracted by the slit, their direction of motion was changed so that part of their momentum was transferred to the $x$ direction. The $x$ component of momentum $p_{x}$ equals the projection of the momentum vector $\mathbf{p}$ in the $x$ direction. A particle deflected upward by an angle $\alpha$ has $p_{x}=p \sin \alpha$. A particle deflected downward by $\alpha$ has $p_{x}=-p \sin \alpha$. Since most of the particles undergo deflections in the range $-\alpha$ to $\alpha$, where $\alpha$ is the angle to the first minimum in the diffraction pattern, we shall take one-half the spread of momentum values in the central diffraction peak as a measure of the uncertainty $\Delta p_{x}$ in the $x$ component of momentum: $\Delta p_{x}=p \sin \alpha$.
Hence at the slit, where the measurement is made,
\( \begin{equation<em>} \Delta x \Delta p_{x}=p w \sin \alpha \tag{1.6} \end{equation</em>} \)
FIGURE 1.1 Diffraction of electrons by a slit.
The angle $\alpha$ at which the first diffraction minimum occurs is readily calculated. The condition for the first minimum is that the difference in the distances traveled by particles passing through the slit at its upper edge and particles passing through the center of the slit should be equal to $\frac{1}{2} \lambda$, where $\lambda$ is the wavelength of the associated wave. Waves originating from the top of the slit are then exactly out of phase with waves originating from the center of the slit, and they cancel each other. Waves originating from a point in the slit at a distance $d$ below the slit midpoint cancel with waves originating at a distance $d$ below the top of the slit. Drawing $A C$ in Fig. 1.2 so that $A D=C D$, we have the difference in path length as $B C$. The distance from the slit to the screen is large compared with the slit width. Hence $A D$ and $B D$ are nearly parallel. This makes the angle $A C B$ essentially a right angle, and so angle $B A C=\alpha$. The path difference $B C$ is then $\frac{1}{2} w \sin \alpha$. Setting $B C$ equal to $\frac{1}{2} \lambda$, we have $w \sin \alpha=\lambda$, and Eq. (1.6) becomes $\Delta x \Delta p_{x}=p \lambda$. The wavelength $\lambda$ is given by the de Broglie relation $\lambda=h / p$, so $\Delta x \Delta p_{x}=h$. Since the uncertainties have not been precisely defined, the equality sign is not really justified. Instead we write
\( \begin{equation<em>} \Delta x \Delta p_{x} \approx h \tag{1.7} \end{equation</em>} \)
indicating that the product of the uncertainties in $x$ and $p_{x}$ is of the order of magnitude of Planck's constant.
Although we have demonstrated (1.7) for only one experimental setup, its validity is general. No matter what attempts are made, the wave-particle duality of microscopic "particles" imposes a limit on our ability to measure simultaneously the position and momentum of such particles. The more precisely we determine the position, the less accurate is our determination of momentum. (In Fig. 1.1, $\sin \alpha=\lambda / w$, so narrowing the slit increases the spread of the diffraction pattern.) This limitation is the uncertainty principle, discovered in 1927 by Werner Heisenberg.
Because of the wave-particle duality, the act of measurement introduces an uncontrollable disturbance in the system being measured. We started with particles having a precise value of $p_{x}$ (zero). By imposing the slit, we measured the $x$ coordinate of the particles to an accuracy $w$, but this measurement introduced an uncertainty into the $p_{x}$ values of the particles. The measurement changed the state of the system.