{"id":168,"date":"2019-04-05T21:09:45","date_gmt":"2019-04-06T01:09:45","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&p=168"},"modified":"2019-04-12T18:53:50","modified_gmt":"2019-04-12T22:53:50","slug":"2-6-wave-particle-duality","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/2-6-wave-particle-duality\/","title":{"raw":"2.6 Wave-Particle Duality","rendered":"2.6 Wave-Particle Duality"},"content":{"raw":"
\r\n
\r\n

Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nBy the end of this section, you will be able to:\r\n
    \r\n \t
  • Identify phenomena in which electromagnetic waves behave like a beam of photons and particles behave like waves<\/li>\r\n \t
  • Describe the physics principles behind electron microscopy<\/li>\r\n \t
  • Summarize the evolution of scientific thought that led to the development of quantum mechanics<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nThe energy of radiation detected by a radio-signal receiving antenna comes as the energy of an electromagnetic wave. The same energy of radiation detected by a photocurrent in the photoelectric effect comes as the energy of individual photon particles. Therefore, the question arises about the nature of electromagnetic radiation: Is a photon a wave or is it a particle? Similar questions may be asked about other known forms of energy. For example, an electron that forms part of an electric current in a circuit behaves like a particle moving in unison with other electrons inside the conductor. The same electron behaves as a wave when it passes through a solid crystalline structure and forms a diffraction image. Is an electron a wave or is it a particle? The same question can be extended to all particles of matter\u2014elementary particles, as well as compound molecules\u2014asking about their true physical nature. At our present state of knowledge, such questions about the true nature of things do not have conclusive answers. All we can say is that<\/span>\u00a0<\/span>wave-particle duality<\/span>\u00a0<\/span>exists in nature: Under some experimental conditions, a particle appears to act as a particle, and under different experimental conditions, a particle appears to act a wave. Conversely, under some physical circumstances electromagnetic radiation acts as a wave, and under other physical circumstances, radiation acts as a beam of photons.<\/span>\r\n\r\n<\/header><\/div>\r\n

    This dualistic interpretation is not a new physics concept brought about by specific discoveries in the twentieth century. It was already present in a debate between Isaac Newton and Christiaan Huygens about the nature of light, beginning in the year 1670. According to Newton, a beam of light is a collection of corpuscles of light. According to Huygens, light is a wave. The corpuscular hypothesis failed in 1803, when Thomas Young announced his\u00a0<\/span>double-slit interference experiment<\/span>\u00a0<\/span>with light (see\u00a0<\/span>Figure 2.23), which firmly established light as a wave. In James Clerk Maxwell\u2019s theory of electromagnetism (completed by the year 1873), light is an electromagnetic wave. Maxwell\u2019s classical view of radiation as an electromagnetic wave is still valid today; however, it is unable to explain blackbody radiation and the photoelectric effect, where light acts as a beam of photons.<\/p>\r\n\r\n

    \r\n
    \r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]\"Picture Figure 2.23 Young\u2019s double-slit experiment explains the interference of light by making an analogy with the interference of water waves. Two waves are generated at the positions of two slits in an opaque screen. The waves have the same wavelengths. They travel from their origins at the slits to the viewing screen placed to the right of the slits. The waves meet on the viewing screen. At the positions marked \u201cMax\u201d on the screen, the meeting waves are in-phase and the combined wave amplitude is enhanced. At positions marked \u201cMin,\u201d the combined wave amplitude is zero. For light, this mechanism creates a bright-and-dark fringe pattern on the viewing screen.[\/caption]<\/figure>\r\n<\/div>\r\n

    A similar dichotomy existed in the interpretation of electricity. From Benjamin Franklin\u2019s observations of electricity in 1751 until J.J. Thomson\u2019s discovery of the electron in 1897, electric current was seen as a flow in a continuous electric medium. Within this theory of electric fluid, the present theory of electric circuits was developed, and electromagnetism and electromagnetic induction were discovered. Thomson\u2019s experiment showed that the unit of negative electric charge (an electron) can travel in a vacuum without any medium to carry the charge around, as in electric circuits. This discovery changed the way in which electricity is understood today and gave the electron its particle status. In Bohr\u2019s early quantum theory of the hydrogen atom, both the electron and the proton are particles of matter. Likewise, in the Compton scattering of X-rays on electrons, the electron is a particle. On the other hand, in electron-scattering experiments on crystalline structures, the electron behaves as a wave.<\/p>\r\n

    A skeptic may raise a question that perhaps an electron might always be nothing more than a particle, and that the diffraction images obtained in electron-scattering experiments might be explained within some macroscopic model of a crystal and a macroscopic model of electrons coming at it like a rain of ping-pong balls. As a matter of fact, to investigate this question, we do not need a complex model of a crystal but just a couple of simple slits in a screen that is opaque to electrons. In other words, to gather convincing evidence about the nature of an electron, we need to repeat the Young double-slit experiment with electrons. If the electron is a wave, we should observe the formation of interference patterns typical for waves, such as those described in\u00a0<\/span>Figure 2.23, even when electrons come through the slits one by one. However, if the electron is a not a wave but a particle, the interference fringes will not be formed.<\/p>\r\n

    The very first double-slit experiment with a beam of electrons, performed by Claus J\u00f6nsson in Germany in 1961, demonstrated that a beam of electrons indeed forms an interference pattern, which means that electrons collectively behave as a wave. The first double-slit experiments with\u00a0<\/span>single<\/em>\u00a0<\/span>electrons passing through the slits one-by-one were performed by Giulio\u00a0<\/span>Pozzi<\/span>\u00a0<\/span>in 1974 in Italy and by Akira\u00a0<\/span>Tonomura<\/span>\u00a0<\/span>in 1989 in Japan. They show that interference fringes are formed gradually, even when electrons pass through the slits individually. This demonstrates conclusively that electron-diffraction images are formed because of the wave nature of electrons. The results seen in double-slit experiments with electrons are illustrated by the images of the interference pattern in\u00a0<\/span>Figure 2.24.<\/p>\r\n\r\n

    \r\n
    \r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"975\"]\"Picture Figure 2.24 Computer-simulated interference fringes seen in the Young double-slit experiment with electrons. One pattern is gradually formed on the screen, regardless of whether the electrons come through the slits as a beam or individually one-by-one.[\/caption]<\/figure>\r\n<\/div>\r\n
    <\/header>
    \r\n
    \r\n

    EXAMPLE\u00a02<\/span><\/span>.15<\/span><\/span><\/h3>\r\n<\/header>
    \r\n

    Double-Slit Experiment with Electrons<\/strong><\/span><\/p>\r\nIn one experimental setup for studying interference patterns of electron waves, two slits are created in a gold-coated silicon membrane. Each slit is 62-nm wide and\u00a0<\/span>4-\u03bcm<\/span><\/span><\/span>\u00a0<\/span>long, and the separation between the slits is 272 nm. The electron beam is created in an electron gun by heating a tungsten element and by accelerating the electrons across a 600-V potential. The beam is subsequently collimated using electromagnetic lenses, and the collimated beam of electrons is sent through the slits. Find the angular position of the first-order bright fringe on the viewing screen.\r\n\r\nStrategy<\/strong><\/span>\r\n\r\nRecall that the angular position\u00a0<\/span>\u03b8<\/span><\/span><\/span>\u00a0<\/span>of the\u00a0<\/span>n<\/em>th order bright fringe that is formed in Young\u2019s two-slit interference pattern (discussed in a previous chapter) is related to the separation,\u00a0<\/span>d<\/em>, between the slits and to the wavelength,\u00a0<\/span>\u03bb,<\/span><\/span><\/span>\u00a0<\/span>of the incident light by the equation\u00a0<\/span>dsin\u03b8=n\u03bb,<\/span><\/span><\/span>\u00a0<\/span>where\u00a0<\/span>n=0,\u00b11,\u00b12,....<\/span><\/span><\/span>\u00a0<\/span>The separation is given and is equal to\u00a0<\/span>d=272nm.<\/span><\/span><\/span>\u00a0<\/span>For the first-order fringe, we take\u00a0<\/span>n=1.<\/span><\/span><\/span>\u00a0<\/span>The only thing we now need is the wavelength of the incident electron wave.\r\n\r\nSince the electron has been accelerated from rest across a potential difference of\u00a0<\/span>\u0394V=600V,<\/span><\/span><\/span>\u00a0<\/span>its kinetic energy is\u00a0<\/span>K=e\u0394V=600eV.<\/span><\/span><\/span>\u00a0<\/span>The rest-mass energy of the electron is\u00a0<\/span>E0=511keV.<\/span><\/span><\/span>\r\n\r\nWe compute its de Broglie wavelength as that of a nonrelativistic electron because its kinetic energy\u00a0<\/span>K<\/span><\/span><\/span>\u00a0<\/span>is much smaller than its rest energy\u00a0<\/span>E0,<\/span><\/span><\/span>\u00a0<\/span>K\u226aE0.<\/span><\/span><\/span>\r\n\r\nSolution<\/strong><\/span>\r\n\r\nThe electron\u2019s wavelength is<\/span>\r\n

    \r\n
    \r\n\r\n\u03bb=hp=h2meK=h2E0\/c2K=hc2E0K=1.241\u00d710\u22126eV\u00b7m2(511keV)(600eV)=0.050nm.<\/span><\/span>\r\n\r\nThis<\/span>\u00a0<\/span>\u03bb<\/span><\/span>\u00a0<\/span>is used to obtain the position of the first bright fringe:<\/span>\r\n\r\nsin\u03b8=1\u00b7\u03bbd=0.050nm272nm=0.000184\u21d2\u03b8=0.010\u00b0.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
    \r\n
    \r\n\r\nSignificance<\/strong><\/span>\r\n\r\nNotice that this is also the angular resolution between two consecutive bright fringes up to about<\/span>\u00a0<\/span>n=1000.<\/span><\/span>\u00a0<\/span>For example, between the zero-order fringe and the first-order fringe, between the first-order fringe and the second-order fringe, and so on.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n
    \r\n

    CHECK YOUR UNDERSTANDING\u00a02<\/span><\/span>.14<\/span><\/p>\r\n\r\n<\/header>\r\n

    For the situation described in<\/span>\u00a0<\/span>Example 2.15, find the angular position of the fifth-order bright fringe on the viewing screen.<\/span><\/header><\/div>\r\n<\/div>\r\n

    The wave-particle dual nature of matter particles and of radiation is a declaration of our inability to describe physical reality within one unified classical theory because separately neither a classical particle approach nor a classical wave approach can fully explain the observed phenomena. This limitation of the classical approach was realized by the year 1928, and a foundation for a new statistical theory, called quantum mechanics, was put in place by Bohr, Edwin\u00a0<\/span>Schr\u00f6dinger<\/span>, Werner\u00a0<\/span>Heisenberg<\/span>, and Paul\u00a0<\/span>Dirac<\/span>. Quantum mechanics takes de Broglie\u2019s idea of matter waves to be the fundamental property of all particles and gives it a statistical interpretation. According to this interpretation, a wave that is associated with a particle carries information about the probable positions of the particle and about its other properties. A single particle is seen as a moving\u00a0<\/span>wave packet<\/em><\/span>\u00a0<\/span>such as the one shown in\u00a0<\/span>Figure 2.25. We can intuitively sense from this example that if a particle is a wave packet, we will not be able to measure its exact position in the same sense as we cannot pinpoint a location of a wave packet in a vibrating guitar string. The uncertainty,\u00a0<\/span>\u0394x,<\/span><\/span><\/span>\u00a0<\/span>in measuring the particle\u2019s position is connected to the uncertainty,\u00a0<\/span>\u0394p,<\/span><\/span><\/span>\u00a0<\/span>in the simultaneous measuring of its linear momentum by Heisenberg\u2019s uncertainty principle:<\/p>\r\n\r\n

    \r\n
    \u0394x\u0394p\u226512\u210f.<\/span><\/span><\/div>\r\n
    [2.63]<\/span><\/div>\r\n<\/div>\r\n

    Heisenberg\u2019s principle expresses the law of nature that, at the quantum level, our perception is limited. For example, if we know the exact position of a body (which means that\u00a0<\/span>\u0394x=0<\/span><\/span><\/span>\u00a0<\/span>in\u00a0<\/span>Equation 2.63) at the same time we cannot know its momentum, because then the uncertainty in its momentum becomes infinite (because\u00a0<\/span>\u0394p\u22650.5\u210f\/\u0394x<\/span><\/span><\/span>\u00a0<\/span>in\u00a0<\/span>Equation 2.63). The\u00a0<\/span>Heisenberg uncertainty principle<\/span>\u00a0<\/span>sets the limit on the precision of\u00a0<\/span>simultaneous<\/em>\u00a0<\/span>measurements of position and momentum of a particle; it shows that the best precision we can obtain is when we have an equals sign (=<\/span><\/span><\/span>) in\u00a0<\/span>Equation 2.63, and we cannot do better than that, even with the best instruments of the future. Heisenberg\u2019s principle is a consequence of the wave nature of particles.<\/p>\r\n\r\n

    \r\n
    \r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]\"Graphic Figure 2.25 In this graphic, a particle is shown as a wave packet and its position does not have an exact value.[\/caption]<\/figure>\r\n<\/div>\r\n

    We routinely use many electronic devices that exploit wave-particle duality without even realizing the sophistication of the physics underlying their operation. One example of a technology based on the particle properties of photons and electrons is a charge-coupled device, which is used for light detection in any instrumentation where high-quality digital data are required, such as in digital cameras or in medical sensors. An example in which the wave properties of electrons is exploited is an electron microscope.<\/p>\r\n

    In 1931, physicist Ernst\u00a0<\/span>Ruska<\/span>\u2014building on the idea that magnetic fields can direct an electron beam just as lenses can direct a beam of light in an optical microscope\u2014developed the first prototype of the electron microscope. This development originated the field of\u00a0<\/span>electron microscopy<\/span>. In the\u00a0<\/span>transmission electron microscope (TEM)<\/span>, shown in\u00a0<\/span>Figure 2.26, electrons are produced by a hot tungsten element and accelerated by a potential difference in an electron gun, which gives them up to 400 keV in kinetic energy. After leaving the electron gun, the electron beam is focused by electromagnetic lenses (a system of condensing lenses) and transmitted through a specimen sample to be viewed. The image of the sample is reconstructed from the transmitted electron beam. The magnified image may be viewed either directly on a fluorescent screen or indirectly by sending it, for example, to a digital camera or a computer monitor. The entire setup consisting of the electron gun, the lenses, the specimen, and the fluorescent screen are enclosed in a vacuum chamber to prevent the energy loss from the beam. Resolution of the TEM is limited only by spherical aberration (discussed in a previous chapter). Modern high-resolution models of a TEM can have resolving power greater than 0.5 \u00c5 and magnifications higher than 50 million times. For comparison, the best resolving power obtained with light microscopy is currently about 97 nm. A limitation of the TEM is that the samples must be about 100-nm thick and biological samples require a special preparation involving chemical \u201cfixing\u201d to stabilize them for ultrathin slicing.<\/p>\r\n\r\n

    \r\n
    \"Picture<\/span><\/figure>\r\n
    Figure\u00a02<\/span>.26<\/span>\u00a0<\/span><\/span>TEM: An electron beam produced by an electron gun is collimated by condenser lenses and passes through a specimen. The transmitted electrons are projected on a screen and the image is sent to a camera. (credit: modification of work by Dr. Graham Beards)<\/span><\/em><\/div>\r\n<\/div>\r\n

    Such limitations do not appear in the\u00a0<\/span>scanning electron microscope (SEM)<\/span>, which was invented by Manfred von Ardenne in 1937. In an SEM, a typical energy of the electron beam is up to 40 keV and the beam is not transmitted through a sample but is scattered off its surface. Surface topography of the sample is reconstructed by analyzing back-scattered electrons, transmitted electrons, and the emitted radiation produced by electrons interacting with atoms in the sample. The resolving power of an SEM is better than 1 nm, and the magnification can be more than 250 times better than that obtained with a light microscope. The samples scanned by an SEM can be as large as several centimeters but they must be specially prepared, depending on electrical properties of the sample.<\/p>\r\n

    High magnifications of the TEM and SEM allow us to see individual molecules. High resolving powers of the TEM and SEM allow us to see fine details, such as those shown in the SEM micrograph of pollen at the beginning of this chapter (Figure 2.1).<\/p>\r\n\r\n

    <\/header>
    \r\n
    \r\n

    EXAMPLE\u00a02<\/span><\/span>.16<\/span><\/span><\/h3>\r\n<\/header>
    \r\n

    Resolving Power of an Electron Microscope<\/strong><\/span><\/p>\r\nIf a 1.0-pm electron beam of a TEM passes through a\u00a0<\/span>2.0-\u03bcm<\/span><\/span><\/span>\u00a0<\/span>circular opening, what is the angle between the two just-resolvable point sources for this microscope?\r\n\r\nSolution<\/strong><\/span>\r\n\r\nWe can directly use a formula for the resolving power,\u00a0<\/span>\u0394\u03b8,<\/span><\/span><\/span>\u00a0<\/span>of a microscope (discussed in a previous chapter) when the wavelength of the incident radiation is\u00a0<\/span>\u03bb=1.0pm<\/span><\/span><\/span>\u00a0<\/span>and the diameter of the aperture is\u00a0<\/span>D=2.0\u03bcm:<\/span><\/span><\/span>\r\n

    \r\n
    \r\n\r\n\u0394\u03b8=1.22\u03bbD=1.221.0pm2.0\u03bcm=6.10\u00d710\u22127rad=3.50\u00d710\u22125degree.<\/span><\/span>\r\n\r\nSignificance<\/strong><\/span>\r\n\r\nNote that if we used a conventional microscope with a 400-nm light, the resolving power would be only<\/span>\u00a0<\/span>14\u00b0,<\/span><\/span>\u00a0<\/span>which means that all of the fine details in the image would be blurred.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n
    \r\n

    CHECK YOUR UNDERSTANDING\u00a02<\/span><\/span>.15<\/span><\/p>\r\n\r\n<\/header>\r\n

    \r\n
    Suppose that the diameter of the aperture in Example 2.16 is halved. How does it affect the resolving power?<\/span><\/div>\r\n<\/header><\/div>\r\n<\/div>\r\n \r\n\r\n \r\n
    Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>","rendered":"
    \n
    \n
    <\/div>\n<\/header>\n
    \n

    Learning Objectives<\/p>\n<\/header>\n

    \n

    By the end of this section, you will be able to:<\/p>\n

      \n
    • Identify phenomena in which electromagnetic waves behave like a beam of photons and particles behave like waves<\/li>\n
    • Describe the physics principles behind electron microscopy<\/li>\n
    • Summarize the evolution of scientific thought that led to the development of quantum mechanics<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

      The energy of radiation detected by a radio-signal receiving antenna comes as the energy of an electromagnetic wave. The same energy of radiation detected by a photocurrent in the photoelectric effect comes as the energy of individual photon particles. Therefore, the question arises about the nature of electromagnetic radiation: Is a photon a wave or is it a particle? Similar questions may be asked about other known forms of energy. For example, an electron that forms part of an electric current in a circuit behaves like a particle moving in unison with other electrons inside the conductor. The same electron behaves as a wave when it passes through a solid crystalline structure and forms a diffraction image. Is an electron a wave or is it a particle? The same question can be extended to all particles of matter\u2014elementary particles, as well as compound molecules\u2014asking about their true physical nature. At our present state of knowledge, such questions about the true nature of things do not have conclusive answers. All we can say is that<\/span>\u00a0<\/span>wave-particle duality<\/span>\u00a0<\/span>exists in nature: Under some experimental conditions, a particle appears to act as a particle, and under different experimental conditions, a particle appears to act a wave. Conversely, under some physical circumstances electromagnetic radiation acts as a wave, and under other physical circumstances, radiation acts as a beam of photons.<\/span><\/p>\n

      This dualistic interpretation is not a new physics concept brought about by specific discoveries in the twentieth century. It was already present in a debate between Isaac Newton and Christiaan Huygens about the nature of light, beginning in the year 1670. According to Newton, a beam of light is a collection of corpuscles of light. According to Huygens, light is a wave. The corpuscular hypothesis failed in 1803, when Thomas Young announced his\u00a0<\/span>double-slit interference experiment<\/span>\u00a0<\/span>with light (see\u00a0<\/span>Figure 2.23), which firmly established light as a wave. In James Clerk Maxwell\u2019s theory of electromagnetism (completed by the year 1873), light is an electromagnetic wave. Maxwell\u2019s classical view of radiation as an electromagnetic wave is still valid today; however, it is unable to explain blackbody radiation and the photoelectric effect, where light acts as a beam of photons.<\/p>\n

      \n
      \n
      \"Picture
      Figure 2.23 Young\u2019s double-slit experiment explains the interference of light by making an analogy with the interference of water waves. Two waves are generated at the positions of two slits in an opaque screen. The waves have the same wavelengths. They travel from their origins at the slits to the viewing screen placed to the right of the slits. The waves meet on the viewing screen. At the positions marked \u201cMax\u201d on the screen, the meeting waves are in-phase and the combined wave amplitude is enhanced. At positions marked \u201cMin,\u201d the combined wave amplitude is zero. For light, this mechanism creates a bright-and-dark fringe pattern on the viewing screen.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n

      A similar dichotomy existed in the interpretation of electricity. From Benjamin Franklin\u2019s observations of electricity in 1751 until J.J. Thomson\u2019s discovery of the electron in 1897, electric current was seen as a flow in a continuous electric medium. Within this theory of electric fluid, the present theory of electric circuits was developed, and electromagnetism and electromagnetic induction were discovered. Thomson\u2019s experiment showed that the unit of negative electric charge (an electron) can travel in a vacuum without any medium to carry the charge around, as in electric circuits. This discovery changed the way in which electricity is understood today and gave the electron its particle status. In Bohr\u2019s early quantum theory of the hydrogen atom, both the electron and the proton are particles of matter. Likewise, in the Compton scattering of X-rays on electrons, the electron is a particle. On the other hand, in electron-scattering experiments on crystalline structures, the electron behaves as a wave.<\/p>\n

      A skeptic may raise a question that perhaps an electron might always be nothing more than a particle, and that the diffraction images obtained in electron-scattering experiments might be explained within some macroscopic model of a crystal and a macroscopic model of electrons coming at it like a rain of ping-pong balls. As a matter of fact, to investigate this question, we do not need a complex model of a crystal but just a couple of simple slits in a screen that is opaque to electrons. In other words, to gather convincing evidence about the nature of an electron, we need to repeat the Young double-slit experiment with electrons. If the electron is a wave, we should observe the formation of interference patterns typical for waves, such as those described in\u00a0<\/span>Figure 2.23, even when electrons come through the slits one by one. However, if the electron is a not a wave but a particle, the interference fringes will not be formed.<\/p>\n

      The very first double-slit experiment with a beam of electrons, performed by Claus J\u00f6nsson in Germany in 1961, demonstrated that a beam of electrons indeed forms an interference pattern, which means that electrons collectively behave as a wave. The first double-slit experiments with\u00a0<\/span>single<\/em>\u00a0<\/span>electrons passing through the slits one-by-one were performed by Giulio\u00a0<\/span>Pozzi<\/span>\u00a0<\/span>in 1974 in Italy and by Akira\u00a0<\/span>Tonomura<\/span>\u00a0<\/span>in 1989 in Japan. They show that interference fringes are formed gradually, even when electrons pass through the slits individually. This demonstrates conclusively that electron-diffraction images are formed because of the wave nature of electrons. The results seen in double-slit experiments with electrons are illustrated by the images of the interference pattern in\u00a0<\/span>Figure 2.24.<\/p>\n

      \n
      \n
      \"Picture
      Figure 2.24 Computer-simulated interference fringes seen in the Young double-slit experiment with electrons. One pattern is gradually formed on the screen, regardless of whether the electrons come through the slits as a beam or individually one-by-one.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n
      \n
      <\/header>\n
      \n
      \n
      \n

      EXAMPLE\u00a02<\/span><\/span>.15<\/span><\/span><\/h3>\n<\/header>\n
      \n

      Double-Slit Experiment with Electrons<\/strong><\/span><\/p>\n

      In one experimental setup for studying interference patterns of electron waves, two slits are created in a gold-coated silicon membrane. Each slit is 62-nm wide and\u00a0<\/span>4-\u03bcm<\/span><\/span><\/span>\u00a0<\/span>long, and the separation between the slits is 272 nm. The electron beam is created in an electron gun by heating a tungsten element and by accelerating the electrons across a 600-V potential. The beam is subsequently collimated using electromagnetic lenses, and the collimated beam of electrons is sent through the slits. Find the angular position of the first-order bright fringe on the viewing screen.<\/p>\n

      Strategy<\/strong><\/span><\/p>\n

      Recall that the angular position\u00a0<\/span>\u03b8<\/span><\/span><\/span>\u00a0<\/span>of the\u00a0<\/span>n<\/em>th order bright fringe that is formed in Young\u2019s two-slit interference pattern (discussed in a previous chapter) is related to the separation,\u00a0<\/span>d<\/em>, between the slits and to the wavelength,\u00a0<\/span>\u03bb,<\/span><\/span><\/span>\u00a0<\/span>of the incident light by the equation\u00a0<\/span>dsin\u03b8=n\u03bb,<\/span><\/span><\/span>\u00a0<\/span>where\u00a0<\/span>n=0,\u00b11,\u00b12,….<\/span><\/span><\/span>\u00a0<\/span>The separation is given and is equal to\u00a0<\/span>d=272nm.<\/span><\/span><\/span>\u00a0<\/span>For the first-order fringe, we take\u00a0<\/span>n=1.<\/span><\/span><\/span>\u00a0<\/span>The only thing we now need is the wavelength of the incident electron wave.<\/p>\n

      Since the electron has been accelerated from rest across a potential difference of\u00a0<\/span>\u0394V=600V,<\/span><\/span><\/span>\u00a0<\/span>its kinetic energy is\u00a0<\/span>K=e\u0394V=600eV.<\/span><\/span><\/span>\u00a0<\/span>The rest-mass energy of the electron is\u00a0<\/span>E0=511keV.<\/span><\/span><\/span><\/p>\n

      We compute its de Broglie wavelength as that of a nonrelativistic electron because its kinetic energy\u00a0<\/span>K<\/span><\/span><\/span>\u00a0<\/span>is much smaller than its rest energy\u00a0<\/span>E0,<\/span><\/span><\/span>\u00a0<\/span>K\u226aE0.<\/span><\/span><\/span><\/p>\n

      Solution<\/strong><\/span><\/p>\n

      The electron\u2019s wavelength is<\/span><\/p>\n

      \n
      \n

      \u03bb=hp=h2meK=h2E0\/c2K=hc2E0K=1.241\u00d710\u22126eV\u00b7m2(511keV)(600eV)=0.050nm.<\/span><\/span><\/p>\n

      This<\/span>\u00a0<\/span>\u03bb<\/span><\/span>\u00a0<\/span>is used to obtain the position of the first bright fringe:<\/span><\/p>\n

      sin\u03b8=1\u00b7\u03bbd=0.050nm272nm=0.000184\u21d2\u03b8=0.010\u00b0.<\/span><\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      Significance<\/strong><\/span><\/p>\n

      Notice that this is also the angular resolution between two consecutive bright fringes up to about<\/span>\u00a0<\/span>n=1000.<\/span><\/span>\u00a0<\/span>For example, between the zero-order fringe and the first-order fringe, between the first-order fringe and the second-order fringe, and so on.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n

      \n
      \n

      CHECK YOUR UNDERSTANDING\u00a02<\/span><\/span>.14<\/span><\/p>\n<\/header>\n

      \n
      For the situation described in<\/span>\u00a0<\/span>Example 2.15, find the angular position of the fifth-order bright fringe on the viewing screen.<\/span><\/header>\n<\/div>\n<\/div>\n

      The wave-particle dual nature of matter particles and of radiation is a declaration of our inability to describe physical reality within one unified classical theory because separately neither a classical particle approach nor a classical wave approach can fully explain the observed phenomena. This limitation of the classical approach was realized by the year 1928, and a foundation for a new statistical theory, called quantum mechanics, was put in place by Bohr, Edwin\u00a0<\/span>Schr\u00f6dinger<\/span>, Werner\u00a0<\/span>Heisenberg<\/span>, and Paul\u00a0<\/span>Dirac<\/span>. Quantum mechanics takes de Broglie\u2019s idea of matter waves to be the fundamental property of all particles and gives it a statistical interpretation. According to this interpretation, a wave that is associated with a particle carries information about the probable positions of the particle and about its other properties. A single particle is seen as a moving\u00a0<\/span>wave packet<\/em><\/span>\u00a0<\/span>such as the one shown in\u00a0<\/span>Figure 2.25. We can intuitively sense from this example that if a particle is a wave packet, we will not be able to measure its exact position in the same sense as we cannot pinpoint a location of a wave packet in a vibrating guitar string. The uncertainty,\u00a0<\/span>\u0394x,<\/span><\/span><\/span>\u00a0<\/span>in measuring the particle\u2019s position is connected to the uncertainty,\u00a0<\/span>\u0394p,<\/span><\/span><\/span>\u00a0<\/span>in the simultaneous measuring of its linear momentum by Heisenberg\u2019s uncertainty principle:<\/p>\n

      \n
      \u0394x\u0394p\u226512\u210f.<\/span><\/span><\/div>\n
      [2.63]<\/span><\/div>\n<\/div>\n

      Heisenberg\u2019s principle expresses the law of nature that, at the quantum level, our perception is limited. For example, if we know the exact position of a body (which means that\u00a0<\/span>\u0394x=0<\/span><\/span><\/span>\u00a0<\/span>in\u00a0<\/span>Equation 2.63) at the same time we cannot know its momentum, because then the uncertainty in its momentum becomes infinite (because\u00a0<\/span>\u0394p\u22650.5\u210f\/\u0394x<\/span><\/span><\/span>\u00a0<\/span>in\u00a0<\/span>Equation 2.63). The\u00a0<\/span>Heisenberg uncertainty principle<\/span>\u00a0<\/span>sets the limit on the precision of\u00a0<\/span>simultaneous<\/em>\u00a0<\/span>measurements of position and momentum of a particle; it shows that the best precision we can obtain is when we have an equals sign (=<\/span><\/span><\/span>) in\u00a0<\/span>Equation 2.63, and we cannot do better than that, even with the best instruments of the future. Heisenberg\u2019s principle is a consequence of the wave nature of particles.<\/p>\n

      \n
      \n
      \"Graphic
      Figure 2.25 In this graphic, a particle is shown as a wave packet and its position does not have an exact value.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n

      We routinely use many electronic devices that exploit wave-particle duality without even realizing the sophistication of the physics underlying their operation. One example of a technology based on the particle properties of photons and electrons is a charge-coupled device, which is used for light detection in any instrumentation where high-quality digital data are required, such as in digital cameras or in medical sensors. An example in which the wave properties of electrons is exploited is an electron microscope.<\/p>\n

      In 1931, physicist Ernst\u00a0<\/span>Ruska<\/span>\u2014building on the idea that magnetic fields can direct an electron beam just as lenses can direct a beam of light in an optical microscope\u2014developed the first prototype of the electron microscope. This development originated the field of\u00a0<\/span>electron microscopy<\/span>. In the\u00a0<\/span>transmission electron microscope (TEM)<\/span>, shown in\u00a0<\/span>Figure 2.26, electrons are produced by a hot tungsten element and accelerated by a potential difference in an electron gun, which gives them up to 400 keV in kinetic energy. After leaving the electron gun, the electron beam is focused by electromagnetic lenses (a system of condensing lenses) and transmitted through a specimen sample to be viewed. The image of the sample is reconstructed from the transmitted electron beam. The magnified image may be viewed either directly on a fluorescent screen or indirectly by sending it, for example, to a digital camera or a computer monitor. The entire setup consisting of the electron gun, the lenses, the specimen, and the fluorescent screen are enclosed in a vacuum chamber to prevent the energy loss from the beam. Resolution of the TEM is limited only by spherical aberration (discussed in a previous chapter). Modern high-resolution models of a TEM can have resolving power greater than 0.5 \u00c5 and magnifications higher than 50 million times. For comparison, the best resolving power obtained with light microscopy is currently about 97 nm. A limitation of the TEM is that the samples must be about 100-nm thick and biological samples require a special preparation involving chemical \u201cfixing\u201d to stabilize them for ultrathin slicing.<\/p>\n

      \n
      \"Picture<\/span><\/figure>\n
      Figure\u00a02<\/span>.26<\/span>\u00a0<\/span><\/span>TEM: An electron beam produced by an electron gun is collimated by condenser lenses and passes through a specimen. The transmitted electrons are projected on a screen and the image is sent to a camera. (credit: modification of work by Dr. Graham Beards)<\/span><\/em><\/div>\n<\/div>\n

      Such limitations do not appear in the\u00a0<\/span>scanning electron microscope (SEM)<\/span>, which was invented by Manfred von Ardenne in 1937. In an SEM, a typical energy of the electron beam is up to 40 keV and the beam is not transmitted through a sample but is scattered off its surface. Surface topography of the sample is reconstructed by analyzing back-scattered electrons, transmitted electrons, and the emitted radiation produced by electrons interacting with atoms in the sample. The resolving power of an SEM is better than 1 nm, and the magnification can be more than 250 times better than that obtained with a light microscope. The samples scanned by an SEM can be as large as several centimeters but they must be specially prepared, depending on electrical properties of the sample.<\/p>\n

      High magnifications of the TEM and SEM allow us to see individual molecules. High resolving powers of the TEM and SEM allow us to see fine details, such as those shown in the SEM micrograph of pollen at the beginning of this chapter (Figure 2.1).<\/p>\n

      \n
      <\/header>\n
      \n
      \n
      \n

      EXAMPLE\u00a02<\/span><\/span>.16<\/span><\/span><\/h3>\n<\/header>\n
      \n

      Resolving Power of an Electron Microscope<\/strong><\/span><\/p>\n

      If a 1.0-pm electron beam of a TEM passes through a\u00a0<\/span>2.0-\u03bcm<\/span><\/span><\/span>\u00a0<\/span>circular opening, what is the angle between the two just-resolvable point sources for this microscope?<\/p>\n

      Solution<\/strong><\/span><\/p>\n

      We can directly use a formula for the resolving power,\u00a0<\/span>\u0394\u03b8,<\/span><\/span><\/span>\u00a0<\/span>of a microscope (discussed in a previous chapter) when the wavelength of the incident radiation is\u00a0<\/span>\u03bb=1.0pm<\/span><\/span><\/span>\u00a0<\/span>and the diameter of the aperture is\u00a0<\/span>D=2.0\u03bcm:<\/span><\/span><\/span><\/p>\n

      \n
      \n

      \u0394\u03b8=1.22\u03bbD=1.221.0pm2.0\u03bcm=6.10\u00d710\u22127rad=3.50\u00d710\u22125degree.<\/span><\/span><\/p>\n

      Significance<\/strong><\/span><\/p>\n

      Note that if we used a conventional microscope with a 400-nm light, the resolving power would be only<\/span>\u00a0<\/span>14\u00b0,<\/span><\/span>\u00a0<\/span>which means that all of the fine details in the image would be blurred.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n

      \n
      \n

      CHECK YOUR UNDERSTANDING\u00a02<\/span><\/span>.15<\/span><\/p>\n<\/header>\n

      \n
      \n
      Suppose that the diameter of the aperture in Example 2.16 is halved. How does it affect the resolving power?<\/span><\/div>\n<\/header>\n<\/div>\n<\/div>\n

       <\/p>\n

       <\/p>\n

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