Investigaciones recientes sobre comparación de probabilidades

In quantum mechanics, we want to use de Broglie waves to describe particles. In particular, the amplitude of the wave will tell us something about the location of the particle. Clearly a pure sinusoidal wave, as in Figure 4.18a, is not much use in locating a particle— the wave extends from−∞to+∞, so the particle might be found anywhere in that region. On the other hand, a narrow wave pulse like Figure 4.18b does a pretty good job of locating the particle in a small region of space, but this wave does not have an easily identifiable wavelength. In the first case, we know the wavelength exactly but have no knowledge of the location of the particle, while in the second case we have a good idea of the location of the particle but a poor knowledge of its wavelength. Because wavelength is associated with momentum by the de Broglie relationship (Eq. 4.1), a poor knowledge of the wavelength is associated with a poor knowledge of the particle’s momentum. For a classical particle, we would like to know both its location and its momentum as precisely as possible. For a quantum particle, we are going to have to make some compromises — the better we know its momentum (or wavelength), the less we know about its location. We can improve our knowledge of its location only at the expense of our knowledge of its momentum.

This competition between knowledge of location and knowledge of wavelength is not restricted to de Broglie waves — classical waves show the same effect. All real waves can be represented as wave packets — disturbances that are localized to a finite region of space. We will discuss more about constructing wave packets in Section 4.5. In this section we will examine this competition between specifying the location and the wavelength of classical waves more closely.

Figure 4.19a shows a very small wave packet. The disturbance is well localized to a small region of space of lengthx. (Imagine listening to a very short burst of sound, of such brief duration that it is hard for you to recognize the pitch or frequency of the wave.) Let’s try to measure the wavelength of this wave packet.

Placing a measuring stick along the wave, we have some difficulty defining exactly where the wave starts and where it ends. Our measurement of the wavelength is therefore subject to a small uncertaintyλ. Let’s represent this uncertainty as a fractionε of the wavelength λ, so that λ ∼ ελ. The fraction ε is certainly less than 1, but it is probably greater than 0.01, so we estimate thatε ∼ 0.1 to within an order of magnitude. (In our discussion of uncertainty, we use the∼ symbol to indicate a rough order-of-magnitude estimate.) That is, the uncertainty in our measurement of the wavelength might be roughly 10% of the wavelength.

The size of this wave disturbance is roughly one wavelength, sox ≈ λ. For this discussion we want to examine the product of the size of the wave packet and the uncertainty in the wavelength,x times λ with x ≈ λ and λ ∼ ελ:

xλ ∼ ελ² (4.4)

This expression shows the inverse relationship between the size of the wave packet and the uncertainty in the wavelength: for a given wavelength, the smaller the size of the wave packet, the greater the uncertainty in our knowledge of the wavelength. That is, asx gets smaller, λ must become larger.

Making a larger wave packet doesn’t help us at all. Figure 4.19b shows a larger wave packet with the same wavelength. Suppose this larger wave packet contains

(b) Δx ≈ Nl

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Δx ≈ l (a)

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FIGURE 4.19 (a) Measuring the wavelength of a wave represented by a small wave packet of length roughly one wavelength. (b) Measuring the wavelength of a wave represented by a large wave packet consisting of N waves.

N cycles of the wave, so thatx ≈ Nλ. Again using our measuring stick, we try to measure the size of N wavelengths, and dividing this distance by N we can then determine the wavelength. We still have the same uncertainty ofελ in locating the start and end of this wave packet, but when we divide by N to find the wavelength, the uncertainty in one wavelength becomes λ ∼ ελ/N. For this larger wave packet, the product of x and λ is xλ ∼ (Nλ)(ελ/N) = ελ², exactly the same as in the case of the smaller wave packet. Equation 4.4 is a fundamental property of classical waves, independent of the type of wave or the method used to measure its wavelength. This is the first of the uncertainty relationships for classical waves.

Example 4.3

In a measurement of the wavelength of water waves, 10 wave cycles are counted in a distance of 196 cm. Estimate the minimum uncertainty in the wavelength that might be obtained from this experiment.

Solution

With 10 wave crests in a distance of 196 cm, the wavelength is about (196 cm)/10 = 19.6 cm. We can takeε ∼ 0.1 as a good order-of-magnitude estimate of the typical precision

that might be obtained. From Eq. 4.4, we can find the uncertainty in wavelength:

λ ∼ελ²

x = (0.1)(19.6 cm)²

196 cm = 0.2 cm

With an uncertainty of 0.2 cm, the “true” wavelength might range from 19.5 cm to 19.7 cm, so we might express this result as 19.6 ± 0.1 cm.

The Frequency-Time Uncertainty Relationship We can take a different approach to uncertainty for classical waves by imagining a measurement of the period rather than the wavelength of the wave that comprises our wave packet. Suppose we have a timing device that we use to measure the duration of the wave packet, as in Figure 4.20. Here we are plotting the wave disturbance as a function of time rather than location. The “size” of the wave packet is now its duration in time, which is roughly one period T for this wave packet, so thatt ≈ T. Whatever measuring device we use, we have some difficulty locating exactly the start and end of one cycle, so we have an uncertainty

T in measuring the period. As before, we’ll assume this uncertainty is some small fraction of the period:T ∼ εT. To examine the competition between the duration of the wave packet and our ability to measure its period, we calculate the product oft and T:

tT ∼ εT² (4.5)

? ?

Δt ≈ T

FIGURE 4.20 Measuring the period of a wave represented by a small wave packet of duration roughly one period.

This is the second of our uncertainty relationships for classical waves. It shows that for a wave of a given period, the smaller the duration of the wave packet, the larger is the uncertainty in our measurement of the period. Note the similarity between Eqs. 4.4 and 4.5, one representing relationships in space and the other in time.

It will turn out to be more useful if we write Eq. 4.5 in terms of frequency instead of period. Given that period T and frequency f are related by f = 1/T, how isf related to T? The correct relationship is certainly not f = 1/T, which would imply that a very small uncertainty in the period would lead to a very large

uncertainty in the frequency. Instead, they should be directly related— the better we know the period, the better we know the frequency. Here is how we obtain the relationship: Beginning with f = 1/T, we take differentials on both sides:

df = − 1 T²dT

Next we convert the infinitesimal differentials to finite intervals, and because we are interested only in the magnitude of the uncertainties we can ignore the minus sign:

f = 1

T²T (4.6)

Combining Eqs. 4.5 and 4.6, we obtain

f t ∼ ε (4.7)

Equation 4.7 shows that the longer the duration of the wave packet, the more precisely we can measure its frequency.

Example 4.4

An electronics salesman offers to sell you a frequency-measuring device. When hooked up to a sinusoidal signal, it automatically displays the frequency of the signal, and to account for frequency variations, the frequency is remea-sured once each second and the display is updated. The salesman claims the device to be accurate to 0.01 Hz. Is this claim valid?

Solution

Based on Eq. 4.7, and again estimatingε to be about 0.1, we know that a measurement of frequency in a timet = 1s

must have an associated uncertainty of about

f ∼ ε

t = 0.1 1s

= 0.1 Hz

It appears that the salesman may be exaggerating the precision of this device.