Wave-Particle Duality of Light
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The Interaction of Light and Matter Screen Metal plate Time Wire Learning Objectives Problems with Bohr’s Semiclassical Model of the Atom Wave-Particle Duality: Everything exhibits its wave properties in its propagation, and manifests its particle nature in its interactions de Broglie’s wavelength for electrons in an atom Wave-like Description of the Physical World Probability waves in quantum mechanics Heisenberg’s uncertainty principle Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics: Measurement precision Quantum mechanical tunneling and nuclear fusion in stars Natural widths of spectral lines Schrödinger’s Wave Equation Solutions describe possible quantum states of a physical system Quantum mechanical atom Learning Objectives Pauli’s Exclusion Principle No two electrons (particles) can share the same quantum state Relativistic Wave Equation Solutions to relativistic wave equation for an atom Complex Spectra of Atoms Multielectron atoms with different ionization states Permitted and Non-Permitted Transitions Learning Objectives Problems with Bohr’s Semiclassical Model of the Atom Wave-Particle Duality: Everything exhibits its wave properties in its propagation, and manifests its particle nature in its interactions de Broglie’s wavelength for electrons in an atom Wave-like Description of the Physical World Probability waves in quantum mechanics Heisenberg’s uncertainty principle Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics: Measurement precision Quantum mechanical tunneling and nuclear fusion in stars Natural widths of spectral lines Schrödinger’s Wave Equation Solutions describe possible quantum states of a physical system Quantum mechanical atom Bohr’s Semiclassical Atom Recall that Bohr’s model of the atom combined both classical physics and quantum mechanics, and has as its two central ingredients: - electrons in circular orbits around the the nucleus (classical physics) - the orbital angular momenta of electrons are quantized (quantum mechanics). Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one permitted orbit to another. What are the two problems in Bohr’s model of the atom that cannot be explained using classical physics? why is the angular momentum Bohr’s Semiclassical Atom Recall that Bohr’s model of the atom combined both classical physics and quantum mechanics, and has as its two central ingredients: - electrons in circular orbits around the the nucleus (classical physics) - the orbital angular momenta of electrons are quantized (quantum mechanics). Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one permitted orbit to another. What are the two problems in Bohr’s model of the atom that cannot be explained using classical physics? - why is the angular momentum of the electron quantized? - why does an electron in a permitted orbit not emit electromagnetic waves as demanded by Maxwell’s equations? Bohr’s Semiclassical Atom Recall that Bohr’s model of the atom combined both classical physics and quantum mechanics, and has as its two central ingredients: - electrons in circular orbits around the the nucleus (classical physics) - the orbital angular momenta of electrons are quantized (quantum mechanics). Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one permitted orbit to another. What are the two problems in Bohr’s model of the atom that cannot be explained using classical physics? - why is the angular momentum of the electron quantized? - why does an electron in a permitted orbit not emit electromagnetic waves as demanded by Maxwell’s equations? Learning Objectives Problems with Bohr’s Semiclassical Model of the Atom Wave-Particle Duality: Everything exhibits its wave properties in its propagation, and manifests its particle nature in its interactions de Broglie’s wavelength for electrons in an atom Wave-like Description of the Physical World Probability waves in quantum mechanics Heisenberg’s uncertainty principle Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics: Measurement precision Quantum mechanical tunneling and nuclear fusion in stars Natural widths of spectral lines Schrödinger’s Wave Equation Solutions describe possible quantum states of a physical system Quantum mechanical cat (Schrödinger’s cat) Quantum mechanical atom Wave-Particle Duality Recall the wave-particle duality of light: Light – in the form of electromagnetic waves – shows its wave-like properties as it propagates through space. Light – in the form of photons – shows its particle-like properties as it interacts with matter. If electromagnetic waves – light – can exhibit particlelike properties, can particles (e.g., protons, electrons, neutrons, atoms, molecules, bacteria, plants, animals, humans, planets, stars, galaxies) exhibit wave-like properties? Wave-Particle Duality This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis. Recall that, in 1905, Einstein used Planck’s idea that the energy of an electromagnetic wave is quantized in the manner whereby a quantum of energy is carried by a photon, to explain the photoelectric effect. Louis de Broglie, 1892-1987 Recall that from the Theory of Special Relativity, the energy and momentum of a photon are related by as had been verified in 1922 by Compton through the Compton effect. Wave-Particle Duality Purely from symmetry arguments, de Broglie proposed that all matter (from elementary particles to people, planets, stars, and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by The wavelength given by Eq. (5.17) is known as the de Broglie wavelength. In this view, the wave-particle duality applies to everything in the physical world: - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions Wave-Particle Duality If everything – including us – propagate as waves, should we worry about being diffracted when walking through the door? Wave-Particle Duality If everything – including us – propagate as waves, should we worry about being diffracted when walking through the door? Wave-Particle Duality So, our wave-like properties have very short wavelengths (depending on our mass and speed). Why then do we not have to worry about diffracting? Hint: Conditions for destructive interference for m = 1, 2, 3, … Notice also that the bright fringes in the interference pattern become dimmer for larger θ. L»D Wave-Particle Duality So, our wave-like properties have very short wavelengths (depending on our mass and speed). Why then do we not have to worry about diffracting? The maxima in the interference pattern is concentrated at θ = 0o; i.e., we walk through and appear behind the door in the direction of motion at the door. L»D Wave-Particle Duality In 1927, Clinton J. Davisson and Lester H. Germer directed a beam of electrons on a highly polished single crystal of nickel. The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0.215 nm, about the size of an atom. Clinton J Davisson (1881-1958; left) and Lester H. Germer(1896-1971; right) The electron beam they used had an energy of 54 eV, corresponding to a de Broglie wavelength of 0.167 nm. The first-order (m = 1) maximum should therefore occur at = sin-1 (/d) = 51o, in agreement with their measurements. Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. Metal plate Wire Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. How do electrons interact with the screen? Metal plate Wire Screen Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. How do electrons interact with the screen? particles, which is why points appear on (where electrons interact with screen). Metal plate Wire ScreenAs individual the screen with strike – – the Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. What image would you see if electrons behave purely like Metal plate Wire Screen particles? Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. What image would you see if electrons behave purely like Metalstripes. Two plate Wire Screen particles? Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. What image would you see if electrons propagate with wave-like properties? Metal plate Wire Screen Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. About 10 electrons are emitted a second, each accelerated to ~0.4 c. What image would you see if electrons propagate with wave-like properties? Interference pattern. Each electron passes through both slits and interferes with itself, thus revealing it’s wave-like properties. (Obviously, a particle can only go through one slit.) Metal plate Wire Screen Wave-Particle Duality Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as atoms. Wave-Particle Duality Instead of double slits, experiments using just one slit also have been performed. Wave-Particle Duality Instead of double slits, experiments using just one slit also have been performed. Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment where they directed a beam of neutrons through a single slit. Wave-Particle Duality Recall that Bohr’s model of the atom combined both classical physics and quantum mechanics, and has as its two central ingredients: - electrons in circular orbits around the the nucleus (classical physics) - the orbital angular momenta of electrons are quantized (quantum mechanics). Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one permitted orbit to another. If electrons behave as waves, can you explain why the angular momentum of electrons are quantized (can only have certain permitted orbits)? Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohr’s model of the atom is simply a manifestation of the wave-like nature of the electron. The circumference of an electron’s orbit must be equal to an integral number of wavelengths (i.e., integral number of de Broglie wavelengths) for the electron to undergo constructive interference. Otherwise, the electron will find itself out of phase and suffer destructive interference. Based on this consideration, one can show that electron can only have angular momenta given . Assignment question In this description, we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom, but as standing waves surrounding the nucleus in an atom. by Wave-Particle Duality Wave-Particle Duality Now, using the principles for the wave-particle duality of the physical world - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation? Wave-Particle Duality Now, using the principles for the wave-particle duality of the physical world - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation? Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei, but propagate as waves. Wave-Particle Duality Now, using the principles for the wave-particle duality of the physical world - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths? Wave-Particle Duality Now, using the principles for the wave-particle duality of the physical world - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions When absorbing electromagnetic radiation, an electron, a photon, and the atomic nucleus interact as particles. Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts. Wave-Particle Duality Now, using the principles for the wave-particle duality of the physical world - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths? Wave-Particle Duality Now, using the principles for the wave-particle duality of the physical world - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions When emitting electromagnetic radiation, an electron, a photon, and the atomic nucleus interact as particles. Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts. Learning Objectives Problems with Bohr’s Semiclassical Model of the Atom Wave-Particle Duality: Everything exhibits its wave properties in its propagation, and manifests its particle nature in its interactions de Broglie’s wavelength for electrons in an atom Wave-like Description of the Physical World Probability waves in quantum mechanics Heisenberg’s uncertainty principle Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics: Measurement precision Quantum mechanical tunneling and nuclear fusion in stars Natural widths of spectral lines Schrödinger’s Wave Equation Solutions describe possible quantum states of a physical system Quantum mechanical atom Probability Waves Quantum mechanics describe particles in terms of probability waves. Consider a particle that comprises the following probability wave, Ψ, a sine wave with a precise wavelength propagating along the x-direction Probability wave Ψ : x (x,t) = 0 ei(kx-t) where k = 2 / =2ν The momentum, p = h / , of a particle described by such a wave is known precisely (as the wavelength is known precisely). The probability of finding the particle at a given location x is given by P(x) = * [0 ei(kx-t)] [0 e-i(kx-t)] = |02| which is a constant independent of x or t. Thus, the particle can be found with equal probability at any point along the x-direction: its position is perfectly uncertain; i.e., a sinusoidal wave has no beginning or end. = Probability Waves Consider now a particle that has a probability wave, Ψ, that is equal to the addition of several sine waves with different wavelengths Probability wave Ψ : x Probability Waves Consider now a particle that has a probability wave, Ψ, that is equal to the addition of several sine waves with different wavelengths Probability wave Ψ : x The position of such a particle can be determined with a greater certainty because P(x) = * is large only for a narrow range of locations. On the other hand, because Ψ is now a combination of waves of various wavelengths, the particle’s momentum, p = h / , is less certain. This is nature’s intrinsic tradeoff: the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δp, is inversely related. As one decreases, the other must increase. This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature. Probability Waves and the Two-Slit Experiment As an illustration of nature’s intrinsic tradeoff between the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δp, consider the following. How do you prevent the electron beam from producing an interference pattern? Probability Waves and the Two-Slit Experiment As an illustration of nature’s intrinsic tradeoff between the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δp, consider the following. How do you prevent the electron beam from producing an interference pattern? By precisely controlling the position of the electrons so that they only enter one slit. If we can be sure that an electron only passes through one slit, we no longer see an (double-slit) interference pattern. How do you explain this if electrons behave like waves? Hint: how can you precisely control the the position of an electron, and what are the consequences of doing this? Probability Waves and the Two-Slit Experiment As an illustration of nature’s intrinsic tradeoff between the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δp, consider the following. How do you prevent the electron beam from producing an interference pattern? By precisely controlling the position of the electrons so that they only enter one slit. If we can be sure that an electron only passes through one slit, we no longer see an (double-slit) interference pattern. How can you precisely control the the position of an electron, and what are the consequences of doing this? By using electrical or magnetic deflecting plates. But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction), and so you lost accurate control of the momentum and hence wavelength of these electrons. Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima, hence an (double-slit) interference pattern is lost. Probability Waves and the Two-Slit Experiment As an illustration of nature’s intrinsic tradeoff between the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δp, consider the following. Conversely, if you try to precisely control the momentum of an electron, you will lose good control of its position. As a consequence, an electron can go through both slits (obviously, an electron can go through both slits only if it behaves like a wave), and interfere with itself to produce an (double-slit) interference pattern. (I leave it up to you to imagine how you could try to precisely control the momentum of an electron. It is not trivial!) Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons, why is the minima in the interference pattern not perfectly dark? Metal plate Wire Screen Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons, why is the minima in the interference pattern not perfectly dark? Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons, why is the minima in the interference pattern not perfectly dark? Because the momentum, and hence wavelength, of the electrons cannot be perfectly controlled. Metal plate Wire Screen Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero. Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero. Heisenberg’s Uncertainty Principle In 1927, the German physicist Werner Heisenberg presented the theoretical framework for the inherent “fuzziness” of nature, showing that the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δt, is related by He also showed that the uncertainty of energy measurement, ΔE, and the time interval over which this measurement is taken, Δt, is related by Recall that = = 1.054571596(82) x 10-34 J s. Werner Heisenberg, 1901-1976 Heisenberg’s Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenberg’s microscope. Heisenberg begins by supposing that an electron is like a classical particle, moving in the x direction along a line below the microscope, as in the illustration to the right. Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron. Let λ be the wavelength of the light rays. Then, according to the laws of classical optics, the microscope can only resolve the position of the electron up to an accuracy of This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D Heisenberg’s Uncertainty Principle When an observer perceives an image of the particle, it's because the light rays strike the particle and bounce back through the microscope to their eye. When a photon strikes an electron, the latter recoils (recall the Compton effect) with a momentum proportional to h/λ, where h is Planck's constant. It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment. He writes that "the recoil cannot be exactly known, since the direction of the scattered photon is undetermined within the bundle of rays entering the microscope.” The electron's momentum in the x direction is only determined up to Combining the relations for Δx and Δp, we have Learning Objectives Problems with Bohr’s Semiclassical Model of the Atom Wave-Particle Duality: Everything exhibits its wave properties in its propagation, and manifests its particle nature in its interactions de Broglie’s wavelength for electrons in an atom Wave-like Description of the Physical World Probability waves in quantum mechanics Heisenberg’s uncertainty principle Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics: Measurement precision Quantum mechanical tunneling and nuclear fusion in stars Natural widths of spectral lines Schrödinger’s Wave Equation Solutions describe possible quantum states of a physical system Quantum mechanical atom Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to make an improved image of Betelgeuse: We target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the bandwidth of each of eight images that we plan to make over the frequency range 40–48 GHz) with robust weighting, about two orders of magnitude higher than that previously attained by Lim et al. (1998) with the VLA as shown in Fig. 1. The required integration time is about 3 hrs. With an estimated observing efficiency of 60% (including overheads for absolute flux calibration and pointing checks), we will require a total observing time of 5 hrs. Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to make an improved image of Betelgeuse: We target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the bandwidth of each of eight images that we plan to make over the frequency range 40–48 GHz) with robust weighting, about two orders of magnitude higher than that previously attained by Lim et al. (1998) with the VLA as shown in Fig. 1. The required integration time is about 3 hrs. With an estimated observing efficiency of 60% (including overheads for absolute flux calibration and pointing checks), we will require a total observing time of 5 hrs. Using Eq. (5.20), explain why observers sometimes integrate for hours when observing astronomical objects? Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to make an improved image of Betelgeuse: We target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the bandwidth of each of eight images that we plan to make over the frequency range 40–48 GHz) with robust weighting, about two orders of magnitude higher than that previously attained by Lim et al. (1998) with the VLA as shown in Fig. 1. The required integration time is about 3 hrs. With an estimated observing efficiency of 60% (including overheads for absolute flux calibration and pointing checks), we will require a total observing time of 5 hrs. Using Eq. (5.20), explain why observers sometimes integrate for hours when observing astronomical objects? As the time available for an energy measurement increases, the inherent uncertainty in the result decreases. Thus, making more precise measurements require longer observing times. Measurement Precision Apart from the identified absorption lines as indicated by the yellow line, much of of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength. Quantum Mechanical Tunneling Because of the inherent uncertainty between a particle’s momentum and position, a particle can penetrate a barrier even though the particle’s energy is lower than the barrier potential. If you know the energy of the particle precisely (and it is smaller than the barrier potential), you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small). So, a (small) fraction of particles on one side of a barrier can tunnel through to the other side, an effect called quantum mechanical tunneling. Probability wave Ψ : x Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves. When such a wave enters a medium through which it cannot propagate, its amplitude decays with distance in this medium: the wave becomes evanescent (fades away). If the barrier width is sufficiently small, the amplitude of the wave may not decay away completely before reaching the other side of the barrier, where the wave can once again propagate. Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion. The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon. For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together, their required kinetic energy corresponds to a gas temperature of ~1010 K. Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 1.57 × 107 K. Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (i.e., particle speeds are distributed according to the Maxwell-Boltzmann distribution), not enough protons can overcome their Coulomb repulsion to produce the Sun’s observed luminosity. Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed. Electronic Orbitals in Atoms Heisenberg’s uncertainty principle implies that electrons cannot have well defined orbits as prescribed in Bohr’s model of the atom. Rather, the electron orbits must be imagined as fuzzy clouds of probability, with the clouds being mode “dense” in regions where the electron is more likely to be found. Bohr’s model for the hydrogen atom Modified Bohr’s model for the hydrogen atom Electronic Orbitals in Atoms Heisenberg’s uncertainty principle implies that electrons cannot have well defined orbital angular momenta as prescribed in Bohr’s model of the atom. How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom? Quantum mechanical model for the hydrogen atom Electronic Orbitals in Atoms Heisenberg’s uncertainty principle implies that electrons cannot have well defined orbital angular momenta as prescribed in Bohr’s model of the atom. How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom? Wavelengths closer to the de Broglie wavelengths interfere more constructively, and so there is a higher probability of finding electrons closer to Quantum mechanical model the de Broglie wavelengths. for the hydrogen atom Electronic Orbitals in Atoms Heisenberg’s uncertainty principle implies that electrons cannot have well defined orbital angular momenta as prescribed in Bohr’s model of the atom. How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom? Conversely, wavelengths further from the de Broglie wavelengths interfere more destructively, and so there is a lower probability of finding electrons further Quantum mechanical model from the de Broglie wavelengths. for the hydrogen atom Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely sharp but must have a certain width (natural linewidth), as is indeed observed. Natural line profile is a Lorentzian profile. Modified Bohr’s model for the hydrogen atom Learning Objectives Problems with Bohr’s Semiclassical Model of the Atom Wave-Particle Duality: Everything exhibits its wave properties in its propagation, and manifests its particle nature in its interactions de Broglie’s wavelength for electrons in an atom Wave-like Description of the Physical World Probability waves in quantum mechanics Heisenberg’s uncertainty principle Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics: Measurement precision Quantum mechanical tunneling and nuclear fusion in stars Natural widths of spectral lines Schrödinger’s Wave Equation Solutions describe possible quantum states of a physical system Quantum mechanical atom Wave-Particle Duality Although, to be able to relate to the familiar, we imagine an atom as electrons orbiting a nucleus, a proper description of an atom requires describing electrons as waves. Schrödinger’s Equation Motivated to find a proper wave equation for the electron, in 1926 the Austrian physicist Erwin Schrödinger formulated the wave equation now known as Schrödinger’s equation that describes how the quantum state of a physical system (elementary particles, atoms, molecules, etc.) changes in time. Erwin Schrödinger, 1887-1961 Wavefunctions Ψ(x, t) that satisfy Schrödinger’s equation describe possible quantum states of a physical system. E.g., wavefunctions that satisfy Schrödinger’s equation for a single particle describe the allowed values of the particle’s energy, momentum, etc., as well as its propagation through space. Schrödinger’s equation is to quantum mechanics what Newton’s equations are to classical physics (mechanics). Schrödinger’s Equation Wavefunctions that satisfy Schrödinger’s equation for a single particle describe the allowed values of a particle’s energy, momentum, etc., as well as its propagation through space. Schrödinger’s Equation Wavefunctions that satisfy Schrödinger’s equation for a single particle describe the allowed values of a particle’s energy, momentum, etc., as well as its propagation through space. Plot of the real part of a possible wavefunction for a particle moving at a constant velocity. Quantum Mechanical Atom Schrödinger’s wave equation can be solved analytically for the hydrogen atom, giving exactly the same set of allowed energies as those obtained by Bohr. In addition to the principal quantum number n, Schrödinger found that two other quantum numbers, and , are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron where an is the angular momentum quantum number and n is the principal quantum number. For historical reasons related to how spectral lines were first designated, = 0, 1, 2, 3, 4, 5, etc. are referred to as s, p, d, g, f, h, etc. Erwin Schrödinger, 1887-1961 Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom. For a single-electron atom such as hydrogen, the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate. (Note that this is not the case for a multi-electron atom.) Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different states (n, ) in a hydrogen atom. Each orbital has a characteristic shape reflecting the motion of the electron in that particular orbital, this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital. f g ml = 0 Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number . Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number . For a multi-electron atom like helium, interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies. Quantum Mechanical Atom The projection of the orbital angular momentum in a specified direction (z-axis), the angular momentum vector component, is also referred to as the magnetic quantum number, . The z-component of the angular momentum vector, Lz, can only have values , with equal to any of the integers between and inclusive. Thus, the angular momentum vector can point in different directions. E.g., for n = 1, = 0 and for n = 2, = 0, 1 and for n = 3, = 0, 1, 2 and = 0. = −1, 0, +1 = −2, −1, 0, +1, +2 Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom. In the absence of any preferred direction in space (e.g., as defined by an electric or magnetic field), different orbitals with the same principal quantum number n have the same energy and are said to be degenerate. Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different excited states (n, , ) in a hydrogen atom. The Zeeman Effect An electron in an atom will feel the effect of a magnetic field: the magnitude of this effect depends on the electron’s orbital motion (i.e., magnitude and orientation of the electron’s orbital angular momentum through the magnetic quantum number ) and magnetic field strength B. Electron orbitals with the same n and but different values therefore have (slightly) different energies. The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman. In the example shown, the three frequencies of the split line are given by The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot. The Zeeman effect provides the only direct measure of magnetic field strengths in astrophysics. (There are several indirect methods to estimate magnetic field strengths). spatial dimension along slit slit λ Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen, usually involving even number of unequally spaced spectral lines. This effect is called the anomalous Zeeman effect. Learning Objectives Pauli’s Exclusion Principle No two electrons (particles) can share the same quantum state Relativistic Wave Equation Solutions to relativistic wave equation for an atom Complex Spectra of Atoms Multielectron atoms with different ionization states Permitted and Non-Permitted Transitions Pauli Exclusion Principle Based on the empirical knowledge of the properties of atoms (e.g., from their spectra), in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers. This rule, which at the time did not have a theoretical basis, is now known as the Pauli exclusion principle. Recall that three quantum numbers were known at the time: the principal quantum number n, the angular momentum quantum number , and the magnetic quantum number . The new quantum number that Pauli introduced could take on two possible values. Wolfgang Pauli, 18691955 Quantum Mechanical Atom In 1925, George Uhlenbeck, Samuel Goudsmit, and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron. The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect. The spin angular momentum S is a vector of constant magnitude component are +½ or −½. with a z. The only values for the electron spin quantum number ms Electron Proton Quantum States of the Helium Atom Parahelium/orthohelium corresponds to helium atoms with their two electrons having antiparallel/parallel spins. In orthohelium, one electron is in the 1s state. That state is not shown because the second electron cannot decay to the 1s state. Quantum Mechanical Atom According to Maxwell’s equation, a moving charge generates a magnetic field. Quantum Mechanical Atom According to Maxwell’s equation, a moving charge generates a magnetic field. An orbiting electron therefore generates a magnetic field. A spinning charged sphere is an electrical current, which according to Maxwell’s equations generates a magnetic field. By analogy, a “spinning” electron generates a magnetic field. As a consequence, the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion. This effect is called spin-orbit coupling. Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution, they are found to be closely-spaced doublets. The lines are split due to spin-orbit coupling of the electron, and are known as fine structure lines. (The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course.) 0.016 nm Quantum States of the Sodium Atom Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sun’s spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame. The sodium (D) line is actually a doublet (two closely-spaced lines). Quantum States of the Sodium Atom Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sun’s spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame. The sodium (D) line is actually a doublet (two closely-spaced lines), and is caused by splitting of a single spectral line into two by spin-orbit coupling. ms = +½ ms = −½ Quantum Mechanical Atom The anomalous Zeeman effect, usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field, is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling. Quantum Mechanical Atom The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling. The number of energy levels that results from the application of a magnetic field is beyond the scope of this course. Learning Objectives Pauli’s Exclusion Principle No two electrons (particles) can share the same quantum state Relativistic Wave Equation Solutions to relativistic wave equation for an atom Complex Spectra of Atoms Multielectron atoms with different ionization states Permitted and Non-Permitted Transitions Relativistic Schrödinger Equation In 1928, the English physicist Paul Dirac combined Schrödinger’s equation with Einstein’s theory of special relativity to produce a relativistic wave equation for the electron. Dirac’s solution - naturally included the spin of the electron - naturally explained Pauli’s exclusion principle as being applicable to all particles with spin of an odd integer times (such as electrons, protons, and neutrons) known collectively as fermions - particles (such as photons) that have an integral spin do not obey Pauli’s exclusion principle and are known as bosons - predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic moments) Quantum Mechanical Atom In 1933, Otto Stern and Walther Gerlach measured the effect of nuclear spin. For hydrogen atoms, the nuclear spin quantum number can only have values of +½ or −½. Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling). The transition between these two states emits a photon with a wavelength of 21 cm. Quantum Mechanical Atom In 1933, Otto Stern and Walther Gerlach measured the effect of nuclear spin. For hydrogen atoms, the nuclear spin quantum number can only have values of +½ or −½. Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins. The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm. spin-orbit coupling spin-spin coupling Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy, and permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies. Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy, and permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies. Intensity of 21-cm line Velocity of 21-cm line Learning Objectives Pauli’s Exclusion Principle No two electrons (particles) can share the same quantum state Relativistic Wave Equation Solutions to relativistic wave equation for an atom Complex Spectra of Atoms Multielectron atoms with different ionization states Permitted and Non-Permitted Transitions Complex Spectra of Atoms In summary, an electron in an atom is described by four quantum numbers - principal quantum number, n - orbital quantum number, - magnetic quantum number, = … - spin quantum number, ms = ±½ The electron can interact with itself through spin-orbit coupling. The nucleus also has a spin quantum number. The electron can interact with the nucleus through spin-spin coupling. In an atom/ion with a single electron, obviously the electron does not interact with other electrons. The spectrum of such an atom/ion is hydrogen-like. In a multielectron atom, electrons interact not only with themselves and the nucleus, but also with each other (described as orbit-orbit and spin-spin coupling). The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons. Furthermore, multielectron atoms with different ionization states (e.g., O I, O II, etc.) have different spectra. Complex Spectra of Atoms Finally, different transitions have different likelihoods of occurring. Transitions that have a high likelihood of occurring are known as permitted transitions, and the resulting spectral lines known as permitted lines. The lifetime of an electron at a permitted transition is ≪1 s. Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions, and the resulting spectral lines known as forbidden lines. The lifetime of an electron at a forbidden transition is >1 s. Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line. Complex Spectra of Atoms At optical wavelengths, forbidden transitions are indicated by enclosing square brackets; e.g., [O III]. These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines), but are only seen from gas in space. Why? Complex Spectra of Atoms At optical wavelengths, forbidden transitions are indicated by enclosing square brackets; e.g., [O III]. These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines), but are only seen from gas in space. Why? Forbidden transitions have a relatively low probability of occurring, or equivalently relatively long lifetimes. At relatively high densities, collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions: the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion). At relatively low densities (as in interstellar space), collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions, thus permitting forbidden transitions that produce photons to occur.
