The Interaction of Light and Matter

The Interaction of Light and Matter Screen Metal plate Wire Time

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

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?

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 particle-like 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 • 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 Notice also that the bright fringes in the interference pattern become dimmer for larger θ. form = 1, 2, 3, … 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. • 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. Clinton J Davisson (1881-1958; left) and Lester H. Germer(1896-1971; right)

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 ClaussJönsson’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 ClaussJönsson’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? Screen 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 ClaussJönsson’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?As particles, which is why individual points appear on the screen (where electrons with strike – interact with – the screen). Screen 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 ClaussJönsson’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 particles? Screen 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 ClaussJönsson’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 particles? Two stripes. Screen 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 ClaussJönsson’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? Screen 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 ClaussJönsson’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.) Screen Metal plate Wire

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 • 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 by . 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.

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.

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 • 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) =  * = [0ei(kx-t)] [0e-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. (x,t) = 0ei(kx-t) wherek = 2 /   = 2 ν 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

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 • The position of such a particle can be determined with a greater certainty becauseP(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 wave Ψ : x

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? Screen Metal plate Wire

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. Screen Metal plate Wire

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.

The Interaction of Light and Matter