Contents

The Particle-Wave Duality Reviewed

Particle-wave duality—the fact that all particles have wave properties—is one of the cornerstones of quantum mechanics. We first came across it in the treatment of photons, those particles of EM radiation that exhibit both particle and wave properties, but not at the same time. Later it was noted that particles of matter have wave properties as well. The dual properties of particles and waves are found for all particles, whether massless like photons, or having a mass like electrons. (See Figure 29.27.)

Part a shows a moving electron represented as a small spherical ball enclosing a wave. An arrow shows the direction of the moving electron. The speed of electron is v. Part b shows a moving photon as a small ellipse enclosing a wave. An arrow shows the direction of the moving photon. The speed of photon is c.

Figure 29.27 On a quantum-mechanical scale (i.e., very small), particles with and without mass have wave properties. For example, both electrons and photons have wavelengths but also behave as particles.

There are many submicroscopic particles in nature. Most have mass and are expected to act as particles, or the smallest units of matter. All these masses have wave properties, with wavelengths given by the de Broglie relationship $λ = h / p λ = h / p size 12{λ = h/p} {}$ . So, too, do combinations of these particles, such as nuclei, atoms, and molecules. As a combination of masses becomes large, particularly if it is large enough to be called macroscopic, its wave nature becomes difficult to observe. This is consistent with our common experience with matter.

Some particles in nature are massless. We have only treated the photon so far, but all massless entities travel at the speed of light, have a wavelength, and exhibit particle and wave behaviors. They have momentum given by a rearrangement of the de Broglie relationship, $p = h / λ p = h / λ size 12{p = h/λ} {}$ . In large combinations of these massless particles (such large combinations are common only for photons or EM waves), there is mostly wave behavior upon detection, and the particle nature becomes difficult to observe. This is also consistent with experience. (See Figure 29.28.)

A massive rock is shown on the left. A massless wave is shown on the right. The propagation of the wave is shown in three dimensional planes, with the variation of two components, E and B. E is a sine wave in one plane with small arrows showing the direction of vibrations. B is a sine wave in a plane perpendicular to the E wave. The B wave has arrows to show the vibrations of particles in the B plane. The waves are shown intersecting each other at the junction of the planes because E and B are perpendicular to each other. The direction of propagation of the wave is shown perpendicular to both E and B waves.

Figure 29.28 On a classical scale (macroscopic), particles with mass behave as particles and not as waves. Particles without mass act as waves and not as particles.

The particle-wave duality is a universal attribute. It is another connection between matter and energy. Not only has modern physics been able to describe nature for high speeds and small sizes, it has also discovered new connections and symmetries. There is greater unity and symmetry in nature than was known in the classical era—but they were dreamt of. A beautiful poem written by the English poet William Blake some two centuries ago contains the following four lines:

To see the World in a Grain of Sand

And a Heaven in a Wild Flower

Hold Infinity in the palm of your hand

And Eternity in an hour

Integrated Concepts

The problem set for this section involves concepts from this chapter and several others. Physics is most interesting when applied to general situations involving more than a narrow set of physical principles. For example, photons have momentum, hence the relevance of Linear Momentum and Collisions. The following topics are involved in some or all of the problems in this section:

Problem-Solving Strategy

Identify which physical principles are involved.
Solve the problem using strategies outlined in the text.

Example 29.10 illustrates how these strategies are applied to an integrated-concept problem.

Example 29.10

Recoil of a Dust Particle after Absorbing a Photon

The following topics are involved in this integrated concepts worked example:

Photons (quantum mechanics)

Linear Momentum

Table 29.2 Topics

A 550-nm photon (visible light) is absorbed by a $1 . 00-μg 1 . 00-μg size 12{1 "." "00-μg"} {}$ particle of dust in outer space. (a) Find the momentum of such a photon. (b) What is the recoil velocity of the particle of dust, assuming it is initially at rest?

Strategy Step 1

To solve an integrated-concept problem, such as those following this example, we must first identify the physical principles involved and identify the chapters in which they are found. Part (a) of this example asks for the momentum of a photon, a topic of the present chapter. Part (b) considers recoil following a collision, a topic of Linear Momentum and Collisions.

Strategy Step 2

The following solutions to each part of the example illustrate how specific problem-solving strategies are applied. These involve identifying knowns and unknowns, checking to see if the answer is reasonable, and so on.

Solution for (a)

The momentum of a photon is related to its wavelength by the equation:

p = \frac{h}{λ} . p = \frac{h}{λ} . size 12{p= {  {h}  over  {λ} } } {}

29.52

Entering the known value for Planck’s constant $h h size 12{h} {}$ and given the wavelength $λ λ size 12{λ} {}$ , we obtain

\begin{array}{l} p & = & \frac{6.63 \times 10^{- 34} J \cdot s}{550 \times 10^{–9} m} \\ = & 1.21 \times 10^{- 27} kg \cdot m/s . \end{array} \begin{array}{l} p & = & \frac{6.63 \times 10^{- 34}}{550 \times 10^{–9}} \\ = 1.21 \times 10^{- 27} . \end{array} alignl { stack {
 size 12{p= {  {6 "." "63"´"10" rSup { size 8{-"34"} } " J" cdot s}  over  {5 "." "50"´"10" rSup { size 8{ +- 9} } " m"} } }  {} #
=1 "." "21"´"10" rSup { size 8{-"27"} } " kg" cdot "m/s" "."  {}
} } {}

29.53

Discussion for (a)

This momentum is small, as expected from discussions in the text and the fact that photons of visible light carry small amounts of energy and momentum compared with those carried by macroscopic objects.

Solution for (b)

Conservation of momentum in the absorption of this photon by a grain of dust can be analyzed using the equation:

p_{1} + p_{2} = {p'}_{1} + {p'}_{2} (F_{net} = 0) . p_{1} + p_{2} = {p'}_{1} + {p'}_{2} (F_{net} = 0) . size 12{p rSub { size 8{1} } +p rSub { size 8{2} } =p rSub { size 8{1} } '+p rSub { size 8{2} } '"      " \( F rSub { size 8{"net"} } =0 \) } {}

29.54

The net external force is zero, since the dust is in outer space. Let 1 represent the photon and 2 the dust particle. Before the collision, the dust is at rest (relative to some observer); after the collision, there is no photon (it is absorbed). So conservation of momentum can be written

p_{1} = {p'}_{2} = mv, p_{1} = {p'}_{2} = mv, size 12{p rSub { size 8{1} } =p rSub { size 8{2} } ' = ital "mv"} {}

29.55

where $p_{1} p_{1} size 12{p rSub { size 8{1} } } {}$ is the photon momentum before the collision and ${p'}_{2} {p'}_{2} size 12{p rSub { size 8{2} } ' } {}$ is the dust momentum after the collision. The mass and recoil velocity of the dust are $m m size 12{m} {}$ and $v v size 12{v} {}$ , respectively. Solving this for $v v size 12{v} {}$ , the requested quantity, yields

v = \frac{p}{m}, v = \frac{p}{m}, size 12{v= {  {p}  over  {m} } } {}

29.56

where $p p size 12{p} {}$ is the photon momentum found in part (a). Entering known values (noting that a microgram is $10^{- 9} kg 10^{- 9} size 12{"10" rSup { size 8{ - 9} } " kg"} {}$ ) gives

\begin{array}{l} v & = & \frac{1.21 \times 10^{- 27} kg \cdot m/s}{1.00 \times 10^{- 9} kg} \\ = & 1.21 \times 10^{–18} m/s. \end{array} \begin{array}{l} v & = & \frac{1.21 \times 10^{- 27}}{1.00 \times 10^{- 9}} \\ = 1.21 \times 10^{–18} \end{array} alignl { stack {
 size 12{v= {  {1 "." "21"´"10" rSup { size 8{-"27"} } " kg" cdot "m/s"}  over  {1 "." "00"´"10" rSup { size 8{ +- 9} } " kg"} } }  {} #
=1 "." "21"´"10" rSup { size 8{-"18"} } " m/s" "."  {}
} } {}

29.57

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.