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College Physics for AP® Courses

30.8 Quantum Numbers and Rules

College Physics for AP® Courses30.8 Quantum Numbers and Rules
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  1. Preface
  2. 1 Introduction: The Nature of Science and Physics
    1. Connection for AP® Courses
    2. 1.1 Physics: An Introduction
    3. 1.2 Physical Quantities and Units
    4. 1.3 Accuracy, Precision, and Significant Figures
    5. 1.4 Approximation
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
  3. 2 Kinematics
    1. Connection for AP® Courses
    2. 2.1 Displacement
    3. 2.2 Vectors, Scalars, and Coordinate Systems
    4. 2.3 Time, Velocity, and Speed
    5. 2.4 Acceleration
    6. 2.5 Motion Equations for Constant Acceleration in One Dimension
    7. 2.6 Problem-Solving Basics for One Dimensional Kinematics
    8. 2.7 Falling Objects
    9. 2.8 Graphical Analysis of One Dimensional Motion
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  4. 3 Two-Dimensional Kinematics
    1. Connection for AP® Courses
    2. 3.1 Kinematics in Two Dimensions: An Introduction
    3. 3.2 Vector Addition and Subtraction: Graphical Methods
    4. 3.3 Vector Addition and Subtraction: Analytical Methods
    5. 3.4 Projectile Motion
    6. 3.5 Addition of Velocities
    7. Glossary
    8. Section Summary
    9. Conceptual Questions
    10. Problems & Exercises
    11. Test Prep for AP® Courses
  5. 4 Dynamics: Force and Newton's Laws of Motion
    1. Connection for AP® Courses
    2. 4.1 Development of Force Concept
    3. 4.2 Newton's First Law of Motion: Inertia
    4. 4.3 Newton's Second Law of Motion: Concept of a System
    5. 4.4 Newton's Third Law of Motion: Symmetry in Forces
    6. 4.5 Normal, Tension, and Other Examples of Force
    7. 4.6 Problem-Solving Strategies
    8. 4.7 Further Applications of Newton's Laws of Motion
    9. 4.8 Extended Topic: The Four Basic Forces—An Introduction
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  6. 5 Further Applications of Newton's Laws: Friction, Drag, and Elasticity
    1. Connection for AP® Courses
    2. 5.1 Friction
    3. 5.2 Drag Forces
    4. 5.3 Elasticity: Stress and Strain
    5. Glossary
    6. Section Summary
    7. Conceptual Questions
    8. Problems & Exercises
    9. Test Prep for AP® Courses
  7. 6 Gravitation and Uniform Circular Motion
    1. Connection for AP® Courses
    2. 6.1 Rotation Angle and Angular Velocity
    3. 6.2 Centripetal Acceleration
    4. 6.3 Centripetal Force
    5. 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
    6. 6.5 Newton's Universal Law of Gravitation
    7. 6.6 Satellites and Kepler's Laws: An Argument for Simplicity
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  8. 7 Work, Energy, and Energy Resources
    1. Connection for AP® Courses
    2. 7.1 Work: The Scientific Definition
    3. 7.2 Kinetic Energy and the Work-Energy Theorem
    4. 7.3 Gravitational Potential Energy
    5. 7.4 Conservative Forces and Potential Energy
    6. 7.5 Nonconservative Forces
    7. 7.6 Conservation of Energy
    8. 7.7 Power
    9. 7.8 Work, Energy, and Power in Humans
    10. 7.9 World Energy Use
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  9. 8 Linear Momentum and Collisions
    1. Connection for AP® courses
    2. 8.1 Linear Momentum and Force
    3. 8.2 Impulse
    4. 8.3 Conservation of Momentum
    5. 8.4 Elastic Collisions in One Dimension
    6. 8.5 Inelastic Collisions in One Dimension
    7. 8.6 Collisions of Point Masses in Two Dimensions
    8. 8.7 Introduction to Rocket Propulsion
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  10. 9 Statics and Torque
    1. Connection for AP® Courses
    2. 9.1 The First Condition for Equilibrium
    3. 9.2 The Second Condition for Equilibrium
    4. 9.3 Stability
    5. 9.4 Applications of Statics, Including Problem-Solving Strategies
    6. 9.5 Simple Machines
    7. 9.6 Forces and Torques in Muscles and Joints
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  11. 10 Rotational Motion and Angular Momentum
    1. Connection for AP® Courses
    2. 10.1 Angular Acceleration
    3. 10.2 Kinematics of Rotational Motion
    4. 10.3 Dynamics of Rotational Motion: Rotational Inertia
    5. 10.4 Rotational Kinetic Energy: Work and Energy Revisited
    6. 10.5 Angular Momentum and Its Conservation
    7. 10.6 Collisions of Extended Bodies in Two Dimensions
    8. 10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  12. 11 Fluid Statics
    1. Connection for AP® Courses
    2. 11.1 What Is a Fluid?
    3. 11.2 Density
    4. 11.3 Pressure
    5. 11.4 Variation of Pressure with Depth in a Fluid
    6. 11.5 Pascal’s Principle
    7. 11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
    8. 11.7 Archimedes’ Principle
    9. 11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
    10. 11.9 Pressures in the Body
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  13. 12 Fluid Dynamics and Its Biological and Medical Applications
    1. Connection for AP® Courses
    2. 12.1 Flow Rate and Its Relation to Velocity
    3. 12.2 Bernoulli’s Equation
    4. 12.3 The Most General Applications of Bernoulli’s Equation
    5. 12.4 Viscosity and Laminar Flow; Poiseuille’s Law
    6. 12.5 The Onset of Turbulence
    7. 12.6 Motion of an Object in a Viscous Fluid
    8. 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  14. 13 Temperature, Kinetic Theory, and the Gas Laws
    1. Connection for AP® Courses
    2. 13.1 Temperature
    3. 13.2 Thermal Expansion of Solids and Liquids
    4. 13.3 The Ideal Gas Law
    5. 13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
    6. 13.5 Phase Changes
    7. 13.6 Humidity, Evaporation, and Boiling
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  15. 14 Heat and Heat Transfer Methods
    1. Connection for AP® Courses
    2. 14.1 Heat
    3. 14.2 Temperature Change and Heat Capacity
    4. 14.3 Phase Change and Latent Heat
    5. 14.4 Heat Transfer Methods
    6. 14.5 Conduction
    7. 14.6 Convection
    8. 14.7 Radiation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  16. 15 Thermodynamics
    1. Connection for AP® Courses
    2. 15.1 The First Law of Thermodynamics
    3. 15.2 The First Law of Thermodynamics and Some Simple Processes
    4. 15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
    5. 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
    6. 15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
    7. 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
    8. 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  17. 16 Oscillatory Motion and Waves
    1. Connection for AP® Courses
    2. 16.1 Hooke’s Law: Stress and Strain Revisited
    3. 16.2 Period and Frequency in Oscillations
    4. 16.3 Simple Harmonic Motion: A Special Periodic Motion
    5. 16.4 The Simple Pendulum
    6. 16.5 Energy and the Simple Harmonic Oscillator
    7. 16.6 Uniform Circular Motion and Simple Harmonic Motion
    8. 16.7 Damped Harmonic Motion
    9. 16.8 Forced Oscillations and Resonance
    10. 16.9 Waves
    11. 16.10 Superposition and Interference
    12. 16.11 Energy in Waves: Intensity
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
    17. Test Prep for AP® Courses
  18. 17 Physics of Hearing
    1. Connection for AP® Courses
    2. 17.1 Sound
    3. 17.2 Speed of Sound, Frequency, and Wavelength
    4. 17.3 Sound Intensity and Sound Level
    5. 17.4 Doppler Effect and Sonic Booms
    6. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns
    7. 17.6 Hearing
    8. 17.7 Ultrasound
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  19. 18 Electric Charge and Electric Field
    1. Connection for AP® Courses
    2. 18.1 Static Electricity and Charge: Conservation of Charge
    3. 18.2 Conductors and Insulators
    4. 18.3 Conductors and Electric Fields in Static Equilibrium
    5. 18.4 Coulomb’s Law
    6. 18.5 Electric Field: Concept of a Field Revisited
    7. 18.6 Electric Field Lines: Multiple Charges
    8. 18.7 Electric Forces in Biology
    9. 18.8 Applications of Electrostatics
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  20. 19 Electric Potential and Electric Field
    1. Connection for AP® Courses
    2. 19.1 Electric Potential Energy: Potential Difference
    3. 19.2 Electric Potential in a Uniform Electric Field
    4. 19.3 Electrical Potential Due to a Point Charge
    5. 19.4 Equipotential Lines
    6. 19.5 Capacitors and Dielectrics
    7. 19.6 Capacitors in Series and Parallel
    8. 19.7 Energy Stored in Capacitors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  21. 20 Electric Current, Resistance, and Ohm's Law
    1. Connection for AP® Courses
    2. 20.1 Current
    3. 20.2 Ohm’s Law: Resistance and Simple Circuits
    4. 20.3 Resistance and Resistivity
    5. 20.4 Electric Power and Energy
    6. 20.5 Alternating Current versus Direct Current
    7. 20.6 Electric Hazards and the Human Body
    8. 20.7 Nerve Conduction–Electrocardiograms
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  22. 21 Circuits, Bioelectricity, and DC Instruments
    1. Connection for AP® Courses
    2. 21.1 Resistors in Series and Parallel
    3. 21.2 Electromotive Force: Terminal Voltage
    4. 21.3 Kirchhoff’s Rules
    5. 21.4 DC Voltmeters and Ammeters
    6. 21.5 Null Measurements
    7. 21.6 DC Circuits Containing Resistors and Capacitors
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  23. 22 Magnetism
    1. Connection for AP® Courses
    2. 22.1 Magnets
    3. 22.2 Ferromagnets and Electromagnets
    4. 22.3 Magnetic Fields and Magnetic Field Lines
    5. 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
    6. 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
    7. 22.6 The Hall Effect
    8. 22.7 Magnetic Force on a Current-Carrying Conductor
    9. 22.8 Torque on a Current Loop: Motors and Meters
    10. 22.9 Magnetic Fields Produced by Currents: Ampere’s Law
    11. 22.10 Magnetic Force between Two Parallel Conductors
    12. 22.11 More Applications of Magnetism
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
    17. Test Prep for AP® Courses
  24. 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
    1. Connection for AP® Courses
    2. 23.1 Induced Emf and Magnetic Flux
    3. 23.2 Faraday’s Law of Induction: Lenz’s Law
    4. 23.3 Motional Emf
    5. 23.4 Eddy Currents and Magnetic Damping
    6. 23.5 Electric Generators
    7. 23.6 Back Emf
    8. 23.7 Transformers
    9. 23.8 Electrical Safety: Systems and Devices
    10. 23.9 Inductance
    11. 23.10 RL Circuits
    12. 23.11 Reactance, Inductive and Capacitive
    13. 23.12 RLC Series AC Circuits
    14. Glossary
    15. Section Summary
    16. Conceptual Questions
    17. Problems & Exercises
    18. Test Prep for AP® Courses
  25. 24 Electromagnetic Waves
    1. Connection for AP® Courses
    2. 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
    3. 24.2 Production of Electromagnetic Waves
    4. 24.3 The Electromagnetic Spectrum
    5. 24.4 Energy in Electromagnetic Waves
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
    10. Test Prep for AP® Courses
  26. 25 Geometric Optics
    1. Connection for AP® Courses
    2. 25.1 The Ray Aspect of Light
    3. 25.2 The Law of Reflection
    4. 25.3 The Law of Refraction
    5. 25.4 Total Internal Reflection
    6. 25.5 Dispersion: The Rainbow and Prisms
    7. 25.6 Image Formation by Lenses
    8. 25.7 Image Formation by Mirrors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  27. 26 Vision and Optical Instruments
    1. Connection for AP® Courses
    2. 26.1 Physics of the Eye
    3. 26.2 Vision Correction
    4. 26.3 Color and Color Vision
    5. 26.4 Microscopes
    6. 26.5 Telescopes
    7. 26.6 Aberrations
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  28. 27 Wave Optics
    1. Connection for AP® Courses
    2. 27.1 The Wave Aspect of Light: Interference
    3. 27.2 Huygens's Principle: Diffraction
    4. 27.3 Young’s Double Slit Experiment
    5. 27.4 Multiple Slit Diffraction
    6. 27.5 Single Slit Diffraction
    7. 27.6 Limits of Resolution: The Rayleigh Criterion
    8. 27.7 Thin Film Interference
    9. 27.8 Polarization
    10. 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  29. 28 Special Relativity
    1. Connection for AP® Courses
    2. 28.1 Einstein’s Postulates
    3. 28.2 Simultaneity And Time Dilation
    4. 28.3 Length Contraction
    5. 28.4 Relativistic Addition of Velocities
    6. 28.5 Relativistic Momentum
    7. 28.6 Relativistic Energy
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  30. 29 Introduction to Quantum Physics
    1. Connection for AP® Courses
    2. 29.1 Quantization of Energy
    3. 29.2 The Photoelectric Effect
    4. 29.3 Photon Energies and the Electromagnetic Spectrum
    5. 29.4 Photon Momentum
    6. 29.5 The Particle-Wave Duality
    7. 29.6 The Wave Nature of Matter
    8. 29.7 Probability: The Heisenberg Uncertainty Principle
    9. 29.8 The Particle-Wave Duality Reviewed
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  31. 30 Atomic Physics
    1. Connection for AP® Courses
    2. 30.1 Discovery of the Atom
    3. 30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
    4. 30.3 Bohr’s Theory of the Hydrogen Atom
    5. 30.4 X Rays: Atomic Origins and Applications
    6. 30.5 Applications of Atomic Excitations and De-Excitations
    7. 30.6 The Wave Nature of Matter Causes Quantization
    8. 30.7 Patterns in Spectra Reveal More Quantization
    9. 30.8 Quantum Numbers and Rules
    10. 30.9 The Pauli Exclusion Principle
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  32. 31 Radioactivity and Nuclear Physics
    1. Connection for AP® Courses
    2. 31.1 Nuclear Radioactivity
    3. 31.2 Radiation Detection and Detectors
    4. 31.3 Substructure of the Nucleus
    5. 31.4 Nuclear Decay and Conservation Laws
    6. 31.5 Half-Life and Activity
    7. 31.6 Binding Energy
    8. 31.7 Tunneling
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  33. 32 Medical Applications of Nuclear Physics
    1. Connection for AP® Courses
    2. 32.1 Medical Imaging and Diagnostics
    3. 32.2 Biological Effects of Ionizing Radiation
    4. 32.3 Therapeutic Uses of Ionizing Radiation
    5. 32.4 Food Irradiation
    6. 32.5 Fusion
    7. 32.6 Fission
    8. 32.7 Nuclear Weapons
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  34. 33 Particle Physics
    1. Connection for AP® Courses
    2. 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
    3. 33.2 The Four Basic Forces
    4. 33.3 Accelerators Create Matter from Energy
    5. 33.4 Particles, Patterns, and Conservation Laws
    6. 33.5 Quarks: Is That All There Is?
    7. 33.6 GUTs: The Unification of Forces
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  35. 34 Frontiers of Physics
    1. Connection for AP® Courses
    2. 34.1 Cosmology and Particle Physics
    3. 34.2 General Relativity and Quantum Gravity
    4. 34.3 Superstrings
    5. 34.4 Dark Matter and Closure
    6. 34.5 Complexity and Chaos
    7. 34.6 High-Temperature Superconductors
    8. 34.7 Some Questions We Know to Ask
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  36. A | Atomic Masses
  37. B | Selected Radioactive Isotopes
  38. C | Useful Information
  39. D | Glossary of Key Symbols and Notation
  40. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
    18. Chapter 18
    19. Chapter 19
    20. Chapter 20
    21. Chapter 21
    22. Chapter 22
    23. Chapter 23
    24. Chapter 24
    25. Chapter 25
    26. Chapter 26
    27. Chapter 27
    28. Chapter 28
    29. Chapter 29
    30. Chapter 30
    31. Chapter 31
    32. Chapter 32
    33. Chapter 33
    34. Chapter 34
  41. Index

Learning Objectives

By the end of this section, you will be able to:

  • Define quantum number.
  • Calculate the angle of an angular momentum vector with an axis.
  • Define spin quantum number.

Physical characteristics that are quantized—such as energy, charge, and angular momentum—are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers, and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules—all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.

The energy states of bound systems are quantized, because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as En1/n2En1/n2, where n=1, 2, 3, ...n=1, 2, 3, .... We define nn to be the principal quantum number that labels the basic states of a system. The lowest-energy state has n=1n=1, the first excited state has n=2n=2, and so on. Thus the allowed values for the principal quantum number are

n=1, 2, 3, ....n=1, 2, 3, .... size 12{n=1, 2, 3, "." "." "." } {}
30.41

This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of nn size 12{n} {}, as can other characteristics (such as the orbital radii of the hydrogen atom).

The fact that the magnitude of angular momentum is quantized was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum LL size 12{L} {} can have only the values

L=ll+1hl=0, 1, 2, ...,n1,L=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}l=0, 1, 2, ...,n1, size 12{ left (l=0, 1, 2, "." "." "." ,n - 1 right )} {}
30.42

where ll size 12{l} {} is defined to be the angular momentum quantum number. The rule for ll size 12{l} {} in atoms is given in the parentheses. Given nn size 12{n} {}, the value of ll size 12{l} {} can be any integer from zero up to n1n1 size 12{n - 1} {}. For example, if n=4n=4 size 12{n=4} {}, then ll size 12{l} {} can be 0, 1, 2, or 3.

Note that for n=1n=1 size 12{n=1} {}, ll size 12{l} {} can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not h/2π h/2π as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in Figure 30.48. The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg’s uncertainty principle—the electron’s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has n=2n=2 size 12{n=2} {}, so that ll size 12{l} {} can be either 0 or 1, according to the rule in L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} . Similarly, for n=3n=3 size 12{n=3} {}, ll size 12{l} {} can be 0, 1, or 2. It is often most convenient to state the value of ll size 12{l} {}, a simple integer, rather than calculating the value of LL size 12{L} {} from L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}. For example, for l=2l=2 size 12{l=2} {}, we see that

L=22+1h=6h=0.390h=2.58×1034 Js.L=22+1h=6h=0.390h=2.58×1034 Js. size 12{L= sqrt {2 left (2+1 right )} { {h} over {2π} } = sqrt {6} { {h} over {2π} } =0 "." "390"h=2 "." "58" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}
30.43

It is much simpler to state l=2l=2 size 12{l=2} {}.

As recognized in the Zeeman effect, the direction of angular momentum is quantized. We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the zz size 12{z} {}-axis, can have only certain values of LzLz size 12{L rSub { size 8{z} } } {}. The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of LzLz size 12{L rSub { size 8{z} } } {} are

Lz=mlhml=l,l+1, ...,1, 0, 1, ...l1,l,Lz=mlh size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {}ml=l,l+1, ...,1, 0, 1, ...l1,l, size 12{ left (m rSub { size 8{l} } = - l, - l+1, "." "." "." , - 1, 0, 1, "." "." "." l - 1, l right )} {}
30.44

where LzLz size 12{L rSub { size 8{z} } } {} is the zz size 12{z} {}-component of the angular momentum and mlml size 12{m rSub { size 8{l} } } {} is the angular momentum projection quantum number. The rule in parentheses for the values of mlml size 12{m rSub { size 8{l} } } {} is that it can range from ll size 12{ - l} {} to ll size 12{l} {} in steps of one. For example, if l=2l=2 size 12{l=2} {}, then mlml size 12{m rSub { size 8{l} } } {} can have the five values –2, –1, 0, 1, and 2. Each mlml size 12{m rSub { size 8{l} } } {} corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the zz size 12{z} {}-component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in Figure 30.54.

The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.
Figure 30.54 The component of a given angular momentum along the zz-axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for l=1l=1, for which ml=1, 0, and +1ml=1, 0, and +1. The direction of LL is quantized in the sense that it can have only certain angles relative to the zz-axis.

Example 30.3 What Are the Allowed Directions?

Calculate the angles that the angular momentum vector LL size 12{L} {} can make with the zz size 12{z} {}-axis for l=1l=1 size 12{l=1} {}, as illustrated in Figure 30.54.

Strategy

Figure 30.54 represents the vectors LL size 12{L} {} and LzLz size 12{L rSub { size 8{z} } } {} as usual, with arrows proportional to their magnitudes and pointing in the correct directions. LL size 12{L} {} and LzLz size 12{L rSub { size 8{z} } } {} form a right triangle, with LL size 12{L} {} being the hypotenuse and LzLz size 12{L rSub { size 8{z} } } {} the adjacent side. This means that the ratio of LzLz size 12{L rSub { size 8{z} } } {} to LL size 12{L} {} is the cosine of the angle of interest. We can find LL size 12{L} {} and LzLz size 12{L rSub { size 8{z} } } {} using L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} and Lz=mhLz=mh size 12{L rSub { size 8{z} } =m { {h} over {2π} } } {}.

Solution

We are given l=1l=1 size 12{l=1} {}, so that mlml size 12{m rSub { size 8{l} } } {} can be +1, 0, or −1. Thus LL size 12{L} {} has the value given by L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}.

L=ll+1h=2hL=ll+1h=2h size 12{L= { { sqrt {l left (l+1 right )} h} over {2π} } = { { sqrt {2} h} over {2π} } } {}
30.45

LzLz size 12{L rSub { size 8{z} } } {} can have three values, given by Lz=mlhLz=mlh size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {}.

Lz=mlh={ h, ml = +1 0, ml = 0 h, ml = 1 Lz=mlh={ h, ml = +1 0, ml = 0 h, ml = 1
30.46

As can be seen in Figure 30.54, cos θ= L z /L, cos θ= L z /L, and so for ml=+1ml=+1 size 12{m rSub { size 8{l} } "=+"1} {}, we have

cosθ1=LZL=h2h=12=0.707.cosθ1=LZL=h2h=12=0.707. size 12{"cos"θ rSub { size 8{1} } = { {L rSub { size 8{Z} } } over {L} } = { { { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = { {1} over { sqrt {2} } } =0 "." "707"} {}
30.47

Thus,

θ1=cos10.707=45.0º.θ1=cos10.707=45.0º.
30.48

Similarly, for ml=0ml=0 size 12{m rSub { size 8{l} } =0} {}, we find cosθ2=0cosθ2=0 size 12{"cos"θ rSub { size 8{2} } =0} {}; thus,

θ2=cos10=90.0º.θ2=cos10=90.0º. size 12{θ rSub { size 8{2} } ="cos" rSup { size 8{ - 1} } 0="90" "." 0°} {}
30.49

And for ml=1ml=1 size 12{m rSub { size 8{l} } = - 1} {},

cosθ3=LZL=h2h=12=0.707,cosθ3=LZL=h2h=12=0.707, size 12{"cos"θ rSub { size 8{3} } = { {L rSub { size 8{Z} } } over {L} } = { { - { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = - { {1} over { sqrt {2} } } = - 0 "." "707"} {}
30.50

so that

θ3=cos10.707=135.0º.θ3=cos10.707=135.0º. size 12{θ rSub { size 8{3} } ="cos" rSup { size 8{ - 1} } left ( - 0 "." "707" right )="135" "." 0°} {}
30.51

Discussion

The angles are consistent with the figure. Only the angle relative to the zz size 12{z} {}-axis is quantized. LL size 12{L} {} can point in any direction as long as it makes the proper angle with the zz size 12{z} {}-axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle ( θ 1 θ 1 in the example) is for the maximum value of ml=0ml=0, namely ml=lml=l. For that smallest angle,

cosθ=LzL=lll+1,cosθ=LzL=lll+1, size 12{"cos"θ= { {L rSub { size 8{z} } } over {L} } = { {l} over { sqrt {l left (l+1 right )} } } } {}
30.52

which approaches 1 as ll size 12{l} {} becomes very large. If cosθ=1cosθ=1 size 12{"cos"θ=1} {}, then θ=θ=. Furthermore, for large ll, there are many values of mlml, so that all angles become possible as ll gets very large.

Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction

There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have intrinsic spin, roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the magnitude of the intrinsic (internal) spin angular momentum, SS size 12{S} {}, of an electron is given by

S=ss+1h(s=1/2 for electrons),S=ss+1h(s=1/2 for electrons), size 12{s=1/2} {}
30.53

where ss size 12{s} {} is defined to be the spin quantum number. This is very similar to the quantization of LL size 12{L} {} given in L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}, except that the only value allowed for ss size 12{s} {} for electrons is 1/2.

The direction of intrinsic spin is quantized, just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the zz size 12{z} {}-axis, can have only the values

S z = m s h m s = 1 2 , + 1 2 S z = m s h size 12{S rSub { size 8{z} } =m rSub { size 8{s} } { {h} over {2π} } } {} m s = 1 2 , + 1 2 size 12{ left (m rSub { size 8{s} } = - { {1} over {2} } , + { {1} over {2} } right )} {}
30.54

for electrons. SzSz size 12{S rSub { size 8{z} } } {} is the zz size 12{z} {}-component of spin angular momentum and msms size 12{S rSub { size 8{z} } } {} is the spin projection quantum number. For electrons, ss size 12{s} {} can only be 1/2, and msms size 12{m rSub { size 8{s} } } {} can be either +1/2 or –1/2. Spin projection ms=+1/2ms=+1/2 size 12{m rSub { size 8{s} } "=+"1/2} {} is referred to as spin up, whereas ms=1/2ms=1/2 size 12{m rSub { size 8{s} } = - 1/2} {} is called spin down. These are illustrated in Figure 30.53.

Intrinsic Spin

In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles ss size 12{s} {} is half-integral, whereas for others ss size 12{s} {} is integral—there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have s=1/2s=1/2 size 12{s=1/2} {}, whereas photons have s=1s=1 size 12{s=1} {}, and other particles called pions have s=0s=0 size 12{s=0} {}, and so on.

To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form n, l,ml,msn, l,ml,ms —see Table 30.1 For electrons in atoms, the principal quantum number can have the values n=1, 2, 3, ...n=1, 2, 3, .... Once nn is known, the values of the angular momentum quantum number are limited to l=1, 2, 3, ...,n1l=1, 2, 3, ...,n1. For a given value of ll, the angular momentum projection quantum number can have only the values ml=l,l+1, ...,1, 0, 1, ...,l1,lml=l,l+1, ...,1, 0, 1, ...,l1,l. Electron spin is independent of n, l,n, l, and mlml, always having s=1/2s=1/2. The spin projection quantum number can have two values, ms=1/2 or 1/2ms=1/2 or 1/2.

Name Symbol Allowed values
Principal quantum number n n 1, 2, 3, ... 1, 2, 3, ...
Angular momentum l l 0, 1, 2, ... n 1 0, 1, 2, ... n 1
Angular momentum projection m l m l l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l ) l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l )
Spin1 s s 1/2 ( electrons ) 1/2 ( electrons )
Spin projection m s m s 1/2, + 1/2 1/2, + 1/2
Table 30.1 Atomic Quantum Numbers

Figure 30.55 shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements—each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.

The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.
Figure 30.55 Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as n,l,mln,l,ml size 12{ left (n, l, m rSub { size 8{l} } right )} {}. The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.

We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.

PhET Explorations: Stern-Gerlach Experiment

The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.

Figure 30.56 Stern-Gerlach Experiment

Footnotes

  • 1 The spin quantum number s is usually not stated, since it is always 1/2 for electrons
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