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Quantum particles

Just as light-waves sometimes exhibit particle-like properties, it turns out that massive particles sometimes exhibit wave-like properties. For instance, it is possible to obtain a double-slit interference pattern from a stream of mono-energetic electrons passing through two closely spaced narrow slits. Now, the effective wave-length of the electrons can be determined by measuring the width of the light and dark bands in the interference pattern [see Eq. (51)]. It is found that
\lambda = \frac{h}{p}.
\end{displaymath} (56)

The same relation is found for other types of particles. The above wave-length is called the de Broglie wave-length, after Louis de Broglie who first suggested that particles should have wave-like properties in 1923. Note that the de Broglie wave-length is generally pretty small. For instance, that of an electron is
\lambda_e = 1.2\times 10^{-9}\,[E({\rm eV})]^{-1/2}\,{\rm m},
\end{displaymath} (57)

where the electron energy is conveniently measured in units of electron-volts (eV). (An electron accelerated from rest through a potential difference of 1000V acquires an energy of 1000eV, and so on.) The de Broglie wave-length of a proton is
\lambda_p = 2.9\times 10^{-11}\,[E({\rm eV})]^{-1/2}\,{\rm m}.
\end{displaymath} (58)

Given the smallness of the de Broglie wave-lengths of common particles, it is actually quite difficult to do particle interference experiments. In general, in order to perform an effective interference experiment, the spacing of the slits must not be too much greater than the wave-length of the wave. Hence, particle interference experiments require either very low energy particles (since $\lambda\propto E^{-1/2}$), or very closely spaced slits. Usually the ``slits'' consist of crystals, which act a bit like diffraction gratings with a characteristic spacing of order the inter-atomic spacing (which is generally about $10^{-9}$m).

Equation (56) can be rearranged to give

p = \hbar\,k,
\end{displaymath} (59)

which is exactly the same as the relation between momentum and wave-number that we obtained earlier for photons [see Eq. (45)].

Since the relation (45) between momentum and wave-number applies to both photons and massive particles, it seems plausible that the closely related relation (43) between energy and wave angular frequency will also apply to both photons and particles. If this is the case, and we can write

E = \hbar\,\omega
\end{displaymath} (60)

for particle waves, then Eqs. (55) and (59) yield the following dispersion relation for such waves:
\omega = \frac{\hbar\,k^2}{2\,m}.
\end{displaymath} (61)

Now, we saw, in Sect. 3.2, that a plane-wave propagates at the so-called phase-velocity,
v_p = \frac{\omega}{k}.
\end{displaymath} (62)

However, according to the above dispersion relation, a particle plane-wave propagates at
v_p = \frac{p}{2\,m}.
\end{displaymath} (63)

Note, from Eq. (54), that this is only half of the classical particle velocity. Does this imply that the dispersion relation (61) is incorrect? Let us investigate further.

next up previous contents
Next: Wave-packets Up: Wave-particle duality Previous: Classical particles   Contents
Richard Fitzpatrick 2006-12-12