WAVE PACKETS

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WAVE PACKETS

Finally, let’s think once more about how an electron can be a particle and a wave at the same time. Armed with the idea of a:wave function, we can be a little more specific about how to reconcile these seemingly incompatible aspects of particle behavior. First we note that a particle with a definite wavelength A. also has a definite momentum p, as

WAVE PACKETS

WAVE PACKETS

shown by the de Broglie relation A = hlp. For such a state, there is no uncertainty in momentum:!J.p =O.The uncertainty principle, Eq. (41-11), says that Ax !J.P. ~ h12Tr. If Sp is zero, !J.x must be infinite. If we know the particle’s momentum precisely, we have no idea at all where the particle is. Such a state is represented by a sinusoidal wave function with no beginning and no end. We can superpose two or more sinusoidal functions to make a wave function that is
more localized in space. To keep things simple, we’ll imagine doing this only in one dimension (x) and at one instant of time. Our wave functions are then functions only of the spatial coordinate x, so we denote them as ‘II. In our discussion of beats in Section 21-4, we superposed two sinusoidal waves with slightly different frequencies (Fig.
21-6). The result was a wave that had a lumpy character that the individual waves do not possess. Imagine doing the same thing with two particle waves. Superposing two waves with slightly different wavelengths (Fig. 41-lOa) gives the wave shown in Fig.
41-10b. A particle represented by this function is more likely to be found in some regions than in others, but the particle’s momentum no longer has a definite value because we began with two different wavelengths. It’s not hard to imagine superposing two additional sinusoidal waves with different wavelengths so as to reinforce alternate lumps in Fig. 41-10b and cancel out the inbetween ones. Finally, if we superpose waves with a very large number of different wavelengths, we can construct a wave with only one lump (Fig. 41-11). Then, finally, we have something that begins to look like both a particle and a wave. It is a particle in
the sense that it is localized in space; if we look from a distance, it may look like a point.

But it also has a periodic structure that is characteristic of a wave. Such a wave pulse is called a wave packet. We can represent such a superposition by an expression such as

WAVE PACKETS

WAVE PACKETS

where cos (2Trx IA) is a sinusoidal wave with wavelength A. The integral represents a superposition in which we add a very large number of such waves with different values
of A, each with an amplitude A(A) that depends on A. It turns out that there is a very important relation between the two functions IjI(x) and A(A). It is shown qualitatively in Fig. 41-12. If the function A(A) is sharply peaked, as in Fig. 41-12a, we are superposing only a narrow range of wavelengths. The resulting wave pulse IjI(x) is then relatively broad (Fig. 41-12b). But if we use a wider range of values of A, so that the function A(A) is broader (Fig. 41-12c), then the wave pulse is more narrowly localized (Fig. 41-12d). What we are seeing is the uncertainty principle in action. A narrow range of A means a narrow range of P. and thus a small !J.P.; the result is a relatively large S», A broad range of A corresponds to a large !J.P., and the resulting tJ.x is smaller.

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WAVE PACKETS