Chapter 10: Detecting Radiation from Space

Chapter 1
How Science Works

  • The Scientific Method
  • Evidence
  • Measurements
  • Units and the Metric System
  • Measurement Errors
  • Estimation
  • Dimensions
  • Mass, Length, and Time
  • Observations and Uncertainty
  • Precision and Significant Figures
  • Errors and Statistics
  • Scientific Notation
  • Ways of Representing Data
  • Logic
  • Mathematics
  • Geometry
  • Algebra
  • Logarithms
  • Testing a Hypothesis
  • Case Study of Life on Mars
  • Theories
  • Systems of Knowledge
  • The Culture of Science
  • Computer Simulations
  • Modern Scientific Research
  • The Scope of Astronomy
  • Astronomy as a Science
  • A Scale Model of Space
  • A Scale Model of Time
  • Questions

Chapter 2
Early Astronomy

  • The Night Sky
  • Motions in the Sky
  • Navigation
  • Constellations and Seasons
  • Cause of the Seasons
  • The Magnitude System
  • Angular Size and Linear Size
  • Phases of the Moon
  • Eclipses
  • Auroras
  • Dividing Time
  • Solar and Lunar Calendars
  • History of Astronomy
  • Stonehenge
  • Ancient Observatories
  • Counting and Measurement
  • Astrology
  • Greek Astronomy
  • Aristotle and Geocentric Cosmology
  • Aristarchus and Heliocentric Cosmology
  • The Dark Ages
  • Arab Astronomy
  • Indian Astronomy
  • Chinese Astronomy
  • Mayan Astronomy
  • Questions

Chapter 3
The Copernican Revolution

  • Ptolemy and the Geocentric Model
  • The Renaissance
  • Copernicus and the Heliocentric Model
  • Tycho Brahe
  • Johannes Kepler
  • Elliptical Orbits
  • Kepler's Laws
  • Galileo Galilei
  • The Trial of Galileo
  • Isaac Newton
  • Newton's Law of Gravity
  • The Plurality of Worlds
  • The Birth of Modern Science
  • Layout of the Solar System
  • Scale of the Solar System
  • The Idea of Space Exploration
  • Orbits
  • History of Space Exploration
  • Moon Landings
  • International Space Station
  • Manned versus Robotic Missions
  • Commercial Space Flight
  • Future of Space Exploration
  • Living in Space
  • Moon, Mars, and Beyond
  • Societies in Space
  • Questions

Chapter 4
Matter and Energy in the Universe

  • Matter and Energy
  • Rutherford and Atomic Structure
  • Early Greek Physics
  • Dalton and Atoms
  • The Periodic Table
  • Structure of the Atom
  • Energy
  • Heat and Temperature
  • Potential and Kinetic Energy
  • Conservation of Energy
  • Velocity of Gas Particles
  • States of Matter
  • Thermodynamics
  • Entropy
  • Laws of Thermodynamics
  • Heat Transfer
  • Thermal Radiation
  • Wien's Law
  • Radiation from Planets and Stars
  • Internal Heat in Planets and Stars
  • Periodic Processes
  • Random Processes
  • Questions

Chapter 5
The Earth-Moon System

  • Earth and Moon
  • Early Estimates of Earth's Age
  • How the Earth Cooled
  • Ages Using Radioactivity
  • Radioactive Half-Life
  • Ages of the Earth and Moon
  • Geological Activity
  • Internal Structure of the Earth and Moon
  • Basic Rock Types
  • Layers of the Earth and Moon
  • Origin of Water on Earth
  • The Evolving Earth
  • Plate Tectonics
  • Volcanoes
  • Geological Processes
  • Impact Craters
  • The Geological Timescale
  • Mass Extinctions
  • Evolution and the Cosmic Environment
  • Earth's Atmosphere and Oceans
  • Weather Circulation
  • Environmental Change on Earth
  • The Earth-Moon System
  • Geological History of the Moon
  • Tidal Forces
  • Effects of Tidal Forces
  • Historical Studies of the Moon
  • Lunar Surface
  • Ice on the Moon
  • Origin of the Moon
  • Humans on the Moon
  • Questions

Chapter 6
The Terrestrial Planets

  • Studying Other Planets
  • The Planets
  • The Terrestrial Planets
  • Mercury
  • Mercury's Orbit
  • Mercury's Surface
  • Venus
  • Volcanism on Venus
  • Venus and the Greenhouse Effect
  • Tectonics on Venus
  • Exploring Venus
  • Mars in Myth and Legend
  • Early Studies of Mars
  • Mars Close-Up
  • Modern Views of Mars
  • Missions to Mars
  • Geology of Mars
  • Water on Mars
  • Polar Caps of Mars
  • Climate Change on Mars
  • Terraforming Mars
  • Life on Mars
  • The Moons of Mars
  • Martian Meteorites
  • Comparative Planetology
  • Incidence of Craters
  • Counting Craters
  • Counting Statistics
  • Internal Heat and Geological Activity
  • Magnetic Fields of the Terrestrial Planets
  • Mountains and Rifts
  • Radar Studies of Planetary Surfaces
  • Laser Ranging and Altimetry
  • Gravity and Atmospheres
  • Normal Atmospheric Composition
  • The Significance of Oxygen
  • Questions

Chapter 7
The Giant Planets and Their Moons

  • The Gas Giant Planets
  • Atmospheres of the Gas Giant Planets
  • Clouds and Weather on Gas Giant Planets
  • Internal Structure of the Gas Giant Planets
  • Thermal Radiation from Gas Giant Planets
  • Life on Gas Giant Planets?
  • Why Giant Planets are Giant
  • Gas Laws
  • Ring Systems of the Giant Planets
  • Structure Within Ring Systems
  • The Origin of Ring Particles
  • The Roche Limit
  • Resonance and Harmonics
  • Tidal Forces in the Solar System
  • Moons of Gas Giant Planets
  • Geology of Large Moons
  • The Voyager Missions
  • Jupiter
  • Jupiter's Galilean Moons
  • Jupiter's Ganymede
  • Jupiter's Europa
  • Jupiter's Callisto
  • Jupiter's Io
  • Volcanoes on Io
  • Saturn
  • Cassini Mission to Saturn
  • Saturn's Titan
  • Saturn's Enceladus
  • Discovery of Uranus and Neptune
  • Uranus
  • Uranus' Miranda
  • Neptune
  • Neptune's Triton
  • Pluto
  • The Discovery of Pluto
  • Pluto as a Dwarf Planet
  • Dwarf Planets
  • Questions

Chapter 8
Interplanetary Bodies

  • Interplanetary Bodies
  • Comets
  • Early Observations of Comets
  • Structure of the Comet Nucleus
  • Comet Chemistry
  • Oort Cloud and Kuiper Belt
  • Kuiper Belt
  • Comet Orbits
  • Life Story of Comets
  • The Largest Kuiper Belt Objects
  • Meteors and Meteor Showers
  • Gravitational Perturbations
  • Asteroids
  • Surveys for Earth Crossing Asteroids
  • Asteroid Shapes
  • Composition of Asteroids
  • Introduction to Meteorites
  • Origin of Meteorites
  • Types of Meteorites
  • The Tunguska Event
  • The Threat from Space
  • Probability and Impacts
  • Impact on Jupiter
  • Interplanetary Opportunity
  • Questions

Chapter 9
Planet Formation and Exoplanets

  • Formation of the Solar System
  • Early History of the Solar System
  • Conservation of Angular Momentum
  • Angular Momentum in a Collapsing Cloud
  • Helmholtz Contraction
  • Safronov and Planet Formation
  • Collapse of the Solar Nebula
  • Why the Solar System Collapsed
  • From Planetesimals to Planets
  • Accretion and Solar System Bodies
  • Differentiation
  • Planetary Magnetic Fields
  • The Origin of Satellites
  • Solar System Debris and Formation
  • Gradual Evolution and a Few Catastrophies
  • Chaos and Determinism
  • Extrasolar Planets
  • Discoveries of Exoplanets
  • Doppler Detection of Exoplanets
  • Transit Detection of Exoplanets
  • The Kepler Mission
  • Direct Detection of Exoplanets
  • Properties of Exoplanets
  • Implications of Exoplanet Surveys
  • Future Detection of Exoplanets
  • Questions

Chapter 10
Detecting Radiation from Space

  • Observing the Universe
  • Radiation and the Universe
  • The Nature of Light
  • The Electromagnetic Spectrum
  • Properties of Waves
  • Waves and Particles
  • How Radiation Travels
  • Properties of Electromagnetic Radiation
  • The Doppler Effect
  • Invisible Radiation
  • Thermal Spectra
  • The Quantum Theory
  • The Uncertainty Principle
  • Spectral Lines
  • Emission Lines and Bands
  • Absorption and Emission Spectra
  • Kirchoff's Laws
  • Astronomical Detection of Radiation
  • The Telescope
  • Optical Telescopes
  • Optical Detectors
  • Adaptive Optics
  • Image Processing
  • Digital Information
  • Radio Telescopes
  • Telescopes in Space
  • Hubble Space Telescope
  • Interferometry
  • Collecting Area and Resolution
  • Frontier Observatories
  • Questions

Chapter 11
Our Sun: The Nearest Star

  • The Sun
  • The Nearest Star
  • Properties of the Sun
  • Kelvin and the Sun's Age
  • The Sun's Composition
  • Energy From Atomic Nuclei
  • Mass-Energy Conversion
  • Examples of Mass-Energy Conversion
  • Energy From Nuclear Fission
  • Energy From Nuclear Fusion
  • Nuclear Reactions in the Sun
  • The Sun's Interior
  • Energy Flow in the Sun
  • Collisions and Opacity
  • Solar Neutrinos
  • Solar Oscillations
  • The Sun's Atmosphere
  • Solar Chromosphere and Corona
  • Sunspots
  • The Solar Cycle
  • The Solar Wind
  • Effects of the Sun on the Earth
  • Cosmic Energy Sources
  • Questions

Chapter 12
Properties of Stars

  • Stars
  • Star Names
  • Star Properties
  • The Distance to Stars
  • Apparent Brightness
  • Absolute Brightness
  • Measuring Star Distances
  • Stellar Parallax
  • Spectra of Stars
  • Spectral Classification
  • Temperature and Spectral Class
  • Stellar Composition
  • Stellar Motion
  • Stellar Luminosity
  • The Size of Stars
  • Stefan-Boltzmann Law
  • Stellar Mass
  • Hydrostatic Equilibrium
  • Stellar Classification
  • The Hertzsprung-Russell Diagram
  • Volume and Brightness Selected Samples
  • Stars of Different Sizes
  • Understanding the Main Sequence
  • Stellar Structure
  • Stellar Evolution
  • Questions

Chapter 13
Star Birth and Death

  • Star Birth and Death
  • Understanding Star Birth and Death
  • Cosmic Abundance of Elements
  • Star Formation
  • Molecular Clouds
  • Young Stars
  • T Tauri Stars
  • Mass Limits for Stars
  • Brown Dwarfs
  • Young Star Clusters
  • Cauldron of the Elements
  • Main Sequence Stars
  • Nuclear Reactions in Main Sequence Stars
  • Main Sequence Lifetimes
  • Evolved Stars
  • Cycles of Star Life and Death
  • The Creation of Heavy Elements
  • Red Giants
  • Horizontal Branch and Asymptotic Giant Branch Stars
  • Variable Stars
  • Magnetic Stars
  • Stellar Mass Loss
  • White Dwarfs
  • Supernovae
  • Seeing the Death of a Star
  • Supernova 1987A
  • Neutron Stars and Pulsars
  • Special Theory of Relativity
  • General Theory of Relativity
  • Black Holes
  • Properties of Black Holes
  • Questions

Chapter 14
The Milky Way

  • The Distribution of Stars in Space
  • Stellar Companions
  • Binary Star Systems
  • Binary and Multiple Stars
  • Mass Transfer in Binaries
  • Binaries and Stellar Mass
  • Nova and Supernova
  • Exotic Binary Systems
  • Gamma Ray Bursts
  • How Multiple Stars Form
  • Environments of Stars
  • The Interstellar Medium
  • Effects of Interstellar Material on Starlight
  • Structure of the Interstellar Medium
  • Dust Extinction and Reddening
  • Groups of Stars
  • Open Star Clusters
  • Globular Star Clusters
  • Distances to Groups of Stars
  • Ages of Groups of Stars
  • Layout of the Milky Way
  • William Herschel
  • Isotropy and Anisotropy
  • Mapping the Milky Way
  • Questions

Chapter 15
Galaxies

  • The Milky Way Galaxy
  • Mapping the Galaxy Disk
  • Spiral Structure in Galaxies
  • Mass of the Milky Way
  • Dark Matter in the Milky Way
  • Galaxy Mass
  • The Galactic Center
  • Black Hole in the Galactic Center
  • Stellar Populations
  • Formation of the Milky Way
  • Galaxies
  • The Shapley-Curtis Debate
  • Edwin Hubble
  • Distances to Galaxies
  • Classifying Galaxies
  • Spiral Galaxies
  • Elliptical Galaxies
  • Lenticular Galaxies
  • Dwarf and Irregular Galaxies
  • Overview of Galaxy Structures
  • The Local Group
  • Light Travel Time
  • Galaxy Size and Luminosity
  • Mass to Light Ratios
  • Dark Matter in Galaxies
  • Gravity of Many Bodies
  • Galaxy Evolution
  • Galaxy Interactions
  • Galaxy Formation
  • Questions

Chapter 16
The Expanding Universe

  • Galaxy Redshifts
  • The Expanding Universe
  • Cosmological Redshifts
  • The Hubble Relation
  • Relating Redshift and Distance
  • Galaxy Distance Indicators
  • Size and Age of the Universe
  • The Hubble Constant
  • Large Scale Structure
  • Galaxy Clustering
  • Clusters of Galaxies
  • Overview of Large Scale Structure
  • Dark Matter on the Largest Scales
  • The Most Distant Galaxies
  • Black Holes in Nearby Galaxies
  • Active Galaxies
  • Radio Galaxies
  • The Discovery of Quasars
  • Quasars
  • Types of Gravitational Lensing
  • Properties of Quasars
  • The Quasar Power Source
  • Quasars as Probes of the Universe
  • Star Formation History of the Universe
  • Expansion History of the Universe
  • Questions

Chapter 17
Cosmology

  • Cosmology
  • Early Cosmologies
  • Relativity and Cosmology
  • The Big Bang Model
  • The Cosmological Principle
  • Universal Expansion
  • Cosmic Nucleosynthesis
  • Cosmic Microwave Background Radiation
  • Discovery of the Microwave Background Radiation
  • Measuring Space Curvature
  • Cosmic Evolution
  • Evolution of Structure
  • Mean Cosmic Density
  • Critical Density
  • Dark Matter and Dark Energy
  • Age of the Universe
  • Precision Cosmology
  • The Future of the Contents of the Universe
  • Fate of the Universe
  • Alternatives to the Big Bang Model
  • Space-Time
  • Particles and Radiation
  • The Very Early Universe
  • Mass and Energy in the Early Universe
  • Matter and Antimatter
  • The Forces of Nature
  • Fine-Tuning in Cosmology
  • The Anthropic Principle in Cosmology
  • String Theory and Cosmology
  • The Multiverse
  • The Limits of Knowledge
  • Questions

Chapter 18
Life On Earth

  • Nature of Life
  • Chemistry of Life
  • Molecules of Life
  • The Origin of Life on Earth
  • Origin of Complex Molecules
  • Miller-Urey Experiment
  • Pre-RNA World
  • RNA World
  • From Molecules to Cells
  • Metabolism
  • Anaerobes
  • Extremophiles
  • Thermophiles
  • Psychrophiles
  • Xerophiles
  • Halophiles
  • Barophiles
  • Acidophiles
  • Alkaliphiles
  • Radiation Resistant Biology
  • Importance of Water for Life
  • Hydrothermal Systems
  • Silicon Versus Carbon
  • DNA and Heredity
  • Life as Digital Information
  • Synthetic Biology
  • Life in a Computer
  • Natural Selection
  • Tree Of Life
  • Evolution and Intelligence
  • Culture and Technology
  • The Gaia Hypothesis
  • Life and the Cosmic Environment

Chapter 19
Life in the Universe

  • Life in the Universe
  • Astrobiology
  • Life Beyond Earth
  • Sites for Life
  • Complex Molecules in Space
  • Life in the Solar System
  • Lowell and Canals on Mars
  • Implications of Life on Mars
  • Extreme Environments in the Solar System
  • Rare Earth Hypothesis
  • Are We Alone?
  • Unidentified Flying Objects or UFOs
  • The Search for Extraterrestrial Intelligence
  • The Drake Equation
  • The History of SETI
  • Recent SETI Projects
  • Recognizing a Message
  • The Best Way to Communicate
  • The Fermi Question
  • The Anthropic Principle
  • Where Are They?

The Quantum Theory


Astronomers study large objects like planets and stars and galaxies, but their understanding depends on knowledge of the microscopic interactions of matter and radiation. For example, we have seen that temperature is a measure of the microscopic motions of atoms and molecules. The electromagnetic spectrum is our window into the universe, but to understand how this radiation is produced, we must revisit the tiny world of the atom. More than 100 years ago, scientists who studied radiation from atoms were faced with two mysteries; one was observational and the other was theoretical.

 

Solar spectrum showing the dark absorption lines.

Astronomers studying the radiation from stars discovered a curious phenomenon. They noticed narrow features in visible spectra of the Sun and other stars. In addition to the smooth thermal spectrum, there were narrow, bright lines at certain wavelengths and narrow, dark lines at other wavelengths. Similar features were seen in the spectra of hot gases in the laboratory. Recall that a spectrum is a map of wavelengths, but since wavelength is inversely related to energy, a spectrum is also a map of energy. These "spectral lines" were a mystery because classical physics held that atoms could have any energy. Since energy can take any value and varies smoothly and continuously, there is no way to explain sharp features at particular energies.

 

Diagram of an idealized Lithium atom, primarily useful to illustrate the nucleus of an atom. This sort of design is scientifically inaccurate in many important respects, but serves as a powerful mandala of the nuclear age.

Ernest Rutherford

The second issue was equally puzzling. Ernest Rutherford developed a model of atoms consisting of a small, dense nucleus (of protons and neutrons) surrounded by a cloud of much lighter electrons. Protons carry a positive electric charge and electrons carry an equal negative electric charge. An atom of the simplest element, hydrogen, has a single proton in the nucleus and a single orbiting electron. It seems like this orbit might continue forever. However, in the classical theory of radiation, any charged particle emits radiation if it is accelerated. (This is how we make radio waves, by sending electrons racing up and down a wire.) A simple calculation showed that the electron in a hydrogen atom should rapidly lose energy and spiral towards the proton, pulled by the electrical attraction between the two particles. The same argument goes for any atom. Classical theory predicted that atoms should collapse in a fraction of a second, yet normal matter is obviously stable. Atoms seemed to mock physicists by their very existence!

 

The Bohr model for the atom, showing the nucleus and different levels that an electron can occupy

Neils Bohr

Max Planck

About 100 years ago, German physicist Max Planck developed a radical new theory that solved both of these mysteries. Just as matter has a fundamental unit called an atom, Planck proposed that energy has a fundamental unit called a quantum (plural: quanta). Energy is not smooth and continuous but it comes in discrete packets called photons. This revolution in physics was called the quantum theory of radiation. Neils Bohr combined Rutherford's picture of the atom — a tiny nucleus with electrons around it — with Planck's quantum theory to show how matter emits and absorbs photons. In some ways, the atom is like a tiny solar system: a relatively massive central object and tiny orbiting particles. But there is an extraordinary difference. In the Solar System, we can imagine a planet occupying any orbit around the Sun; each orbit would have its own velocity and hence energy. However, in an atom, electrons can occupy only specific orbits called energy levels. Physicists describe this by saying that the atom has quantized energy levels because only certain quantities of orbital energy are possible.

 

A visible spectrum of the sun. The black lines in the spectrum are absorption lines. These lines can tell us about the chemical composition of the sun. Similar spectra can be gathered from planets, which allow us to determine their composition.

Iron spectrum

Hydrogen energy diagram

How does the quantum theory explain the mysteries of spectral lines and the stability of the atom? The quantum theory holds that there is a lowest energy that any electron in an atom can have. The electron can not lose energy continuously so it does not spiral in toward the nucleus and the atom does not collapse. The answer to the second mystery is particularly interesting. If an atom could gain or lose any amount of energy it could emit radiation with any wavelength. All these wavelengths would fill in to make a smooth and continuous spectrum. But electrons are held in specific energy levels. So atoms can only gain or lose the energy corresponding to an atom moving between two energy levels. This creates a set of rules for the emission and absorption of radiation in atoms.

Emission is the process where an electron loses energy in the form of a photon. Absorption is the complementary process where an electron gains energy in the form of a photon. Levels or orbits that are farthest from the nucleus have the highest energy. Each transition between levels corresponds to a photon with energy equal to the energy difference between the two levels. The classical picture corresponds to a situation where the energy levels are so finely subdivided that any energy level is possible. In the quantum picture, the energy levels are discrete so an electron can only lose specific amounts of energy. This creates sharp lines in the spectrum since each specific energy loss corresponds to a particular wavelength of radiation. It is like the difference between sliding down a smooth slope and walking down steps. In the classical case, an electron moves to a lower energy level by losing any amount of energy. Consequently, the spectral transitions have any wavelength, and so the spectrum fills in to be smooth and featureless.

As an analogy for the quantized world of the atom, think of two different types of arithmetic. In one system you can add or subtract any fraction or decimal number. Starting with 8, say, you can add ½ or 1/9 or 2.37, or you can subtract 1/18 or 10.4 or 3.006. Do this many times and you will fill in the number line. This is the smooth and continuous world of classical physics. Now think of integer arithmetic. Starting with 8 again, you can add 1 or 4 or 7, or you can subtract 2 or 5 or 6. Do this many times and you will have a set of integer values. This is the quantum world. (An integer is analogous to a quantum and the number line is analogous to a spectrum of energy or wavelength.)

In the everyday world, the quantized nature of energy and matter is not apparent. Crouch down on a beach and you can see the tiny particles of rock that makeup sand. From a distance, the texture appears smooth and uniform, and the structure is not visible. Let's go back to the analogy of the number line. If we step far back from the number line so that we can only see billions or trillions marked off, the individual integers are no longer visible. Seen on this large scale the integer number line appears smooth and continuous. Similarly, transactions in the everyday world involve many trillions of atoms or photons. The "graininess" of the quantum world is not visible to us.


Werner Heisenberg

The quantum theory of radiation may seem strange to you. It was strange and uncomfortable for many physicists too! The theory became accepted because it describes how nature works very well. In the 1930s, physicists were still reeling from the idea of the quantum when German physicist Werner Heisenberg came up with an equally bizarre idea. He showed that quantities like energy, momentum, and position can not be defined with absolute precision. This idea is enshrined in physics as the Heisenberg uncertainty principle. Uncertainty is not the best word to describe this principle; it's better thought of as imprecision. Nature puts a floor on the precision with which we can make measurements. In particular, the product of the uncertainty in jointly measuring position and momentum is limited to Planck's constant: Δx $times; Δp > h/4π. Similarly, the product of the uncertainty in measuring energy and time is limited to Planck's constant: ΔE × Δt > h/4π. Since h is a tiny number this imprecision is not noticeable in the everyday world.

Scientists believe that the solar system analogy of the atom, with particles like hard balls in specific orbits, is a weak one. In the quantum model of the atom, the electron is not a discrete object like a billiard ball. It has a certain probability of being at various positions, so the electron is not just at one position in its orbit like a planet would be. The imprecision described by Heisenberg is only substantial on the scale of atoms or subatomic particles. It does not limit our knowledge of the everyday world. The "fuzziness" of the subatomic world is not visible to us.

Light can be thought of as either a wave or a particle. In the quantum view of the subatomic world, a particle can also be thought of as a wave. Let's look at the difference between the classical and quantum ways of looking at particles and waves. The classical view of a particle is something hard-edged like a marble or a ball bearing. The quantum view of the particle is somewhat fuzzy because it has a probability of being found over a range of space. We can make the same comparison for a wave. A classical wave extends off forever into space in either direction. The quantum view considers the energy of the wave to be more concentrated in space. In the classical view, particles and waves appear to be totally distinct entities. In the quantum view, they are quite similar!

There are two alternative views of electrons in an atom. The classical model of a miniature solar system does not explain why the electrons are confined to certain orbits. But the quantum model is based on the wave properties of particles. A useful analogy for imagining the quantized energy levels of an atom is to think of the set of vibration modes of a plucked string. The vibrations of a guitar string are made of a whole number of waves. In the same way, each energy level corresponds to a whole number (as opposed to a fraction) of waves and the orbits have quantized wavelengths (or energies). The guitar string analogy is also appropriate because the wave's energy is distributed in space, not concentrated like a hard-edged particle. The quantum idea of particles as waves may seem counter-intuitive, but it helps explain why the atom should have quantized energy levels.


Author: Chris Impey