# 29.8 The particle-wave duality reviewed  (Page 3/4)

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$\begin{array}{lll}v& =& \frac{1\text{.}\text{21}×{\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}}{1\text{.}\text{00}×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}\text{kg}}\\ & =& 1\text{.}\text{21}×{\text{10}}^{\text{–18}}\phantom{\rule{0.25em}{0ex}}\text{m/s.}\end{array}$

Discussion

The recoil velocity of the particle of dust is extremely small. As we have noted, however, there are immense numbers of photons in sunlight and other macroscopic sources. In time, collisions and absorption of many photons could cause a significant recoil of the dust, as observed in comet tails.

## Test prep for ap courses

Which of the following describes one of the main features of wave-particle duality?

1. As speed increases, the wave nature of matter becomes more evident.
2. As momentum decreases, the particle nature of matter becomes more evident.
3. As energy increases, the wave nature of matter becomes easier to observe.
4. As mass increases, the wave nature of matter is less easy to observe.

(d)

Explain why Heisenberg’s uncertainty principle limits the precision with which either momentum or position of a subatomic particle can be known, but becomes less applicable for matter at the macroscopic level.

## Section summary

• The particle-wave duality refers to the fact that all particles—those with mass and those without mass—have wave characteristics.
• This is a further connection between mass and energy.

## Conceptual questions

In what ways are matter and energy related that were not known before the development of relativity and quantum mechanics?

## Problems&Exercises

Integrated Concepts

The 54.0-eV electron in [link] has a 0.167-nm wavelength. If such electrons are passed through a double slit and have their first maximum at an angle of $\text{25}\text{.}0º$ , what is the slit separation $d$ ?

0.395 nm

Integrated Concepts

An electron microscope produces electrons with a 2.00-pm wavelength. If these are passed through a 1.00-nm single slit, at what angle will the first diffraction minimum be found?

Integrated Concepts

A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength. (a) What is the average photon energy in joules? (b) How many of these photons are required to increase the temperature of a person’s shoulder by $2\text{.}0º\text{C}$ , assuming the affected mass is 4.0 kg with a specific heat of $0\text{.83 kcal}\text{/kg}\cdot \text{ºC}$ . Also assume no other significant heat transfer. (c) How long does this take?

(a) $1.3×{\text{10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{J}$

(b) $2\text{.}1×{\text{10}}^{\text{23}}$

(c) $1\text{.}4×{\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\text{s}$

Integrated Concepts

On its high power setting, a microwave oven produces 900 W of 2560 MHz microwaves. (a) How many photons per second is this? (b) How many photons are required to increase the temperature of a 0.500-kg mass of pasta by $\text{45}\text{.}0º\text{C}$ , assuming a specific heat of $0\text{.}\text{900 kcal/kg}\cdot \text{ºC}$ ? Neglect all other heat transfer. (c) How long must the microwave operator wait for their pasta to be ready?

Integrated Concepts

(a) Calculate the amount of microwave energy in joules needed to raise the temperature of 1.00 kg of soup from $\text{20}\text{.}0º\text{C}$ to $\text{100}\text{ºC}$ . (b) What is the total momentum of all the microwave photons it takes to do this? (c) Calculate the velocity of a 1.00-kg mass with the same momentum. (d) What is the kinetic energy of this mass?

(a) $3\text{.}\text{35}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}$

(b) $1\text{.}\text{12}×{\text{10}}^{\text{–3}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$

(c) $1\text{.}\text{12}×{\text{10}}^{\text{–3}}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(d) $6.23×{\text{10}}^{\text{–7}}\phantom{\rule{0.25em}{0ex}}\text{J}$

Integrated Concepts

(a) What is $\gamma$ for an electron emerging from the Stanford Linear Accelerator with a total energy of 50.0 GeV? (b) Find its momentum. (c) What is the electron’s wavelength?

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