Chapter 28

1. Chapter 28 Quantum Theory April 10th, 2013

2. Electron Spin • Electrons have another quantum property that involves their magnetic behavior • This is the electron spin • Classically, the electron spincan be thought of as resulting from a spinning ball of charge

3. Electron Spin, cont. • The spinning ball of charge acts like a collection of current loops • The loops produce a magnetic field • Which acts like a small bar magnet • Therefore, the electron can be attracted to or repelled from the poles of other magnets

4. Electron Spin, Directions • Stern-Gerlach experiment: When a beam of electrons passes near one end of a bar magnet, it can be deflected in two directions: 2 directions only! Up or down! The electron spin is quantized • Classical theory would predict that the spin may point in any direction

5. Electron Spin, Direction, cont. • Classically, the electrons should deflect over a range of angles • Observing only two directions of deflection indicates that there are only two possible orientations for spin • The electron spin is quantized with only two possible values • Quantization of the electron spinapplies to both direction and magnitude • All electrons under all circumstances act as identical bar magnets

6. Quantization of Electron Spin • Classical explanation of electron spin • Circulating charge acts as a current loop • The current loops produce a magnetic field • This result is called the spin magnetic moment • You can also say that the electron has spin angular momentum • The classical ideas do not explain the splitting in only-two directions when the beam of electrons passes near a magnet • Quantum explanation • Only spin up or spin down are possible • Other quantum particles also have spin angular momentum and a resulting magnetic moment

7. Wave Function • In the quantum world, the motion of a particle-wave is described by its wave function • The wave function can be calculated from Schrödinger’s equation • Developed by Erwin Schrödinger, one of the inventors of quantum theory • Schrödinger’s equation plays a role similar to Newton’s laws of motion since it tells how the wave function varies with time • In many situations, the solutions of the Schrödinger equation are similar to standing waves

8. Wave Function Example • An electron is confined to a particular region of space • A classical particle would travel back and forth inside the box • The wave function for the electron is described by standing waves • Two possible waves are shown

9. Wave Function Example, cont. • The wave function solutions correspond to electrons with different kinetic energies • The wavelengths of the standing waves are different • Given by de Broglie’s equation • After finding the wave function, one can calculate the position and velocity of the electron • But does not give a single value • The wave function allows for the calculation of the probability of finding the electron at different locations in space

10. Heisenberg Uncertainty Principle • For a particle-wave, quantum effects place fundamental limits on the precision of measuring position or velocity • For an electron in a box, there is some probability of finding the electron at virtually any spot in the box, but this probability is greater in some positions than in others • The uncertainty, Δx, is approximately the size of the box • This uncertainty is due to the wave nature of the electron

11. Uncertainty, Example • Electrons are incident on a narrow slit • The electron wave is diffracted as it passes through the slit • The interference pattern gives a measure of how the wave function of the electron is distributed throughout space after it passes through the slit • The width of the slit affects the interference pattern • The narrower the slit, the broader the distribution pattern

12. Uncertainties in Position and Momentum • The position of the electron passing through a slit is known with an uncertainty Δx equal to the width of the slit • Since the outgoing electrons have a spread in their momenta along x, there is some uncertainty Δpx in the x component of the momentum • The uncertainties Δx and Δp are absolute limits set by quantum theory remember: p = mv

13. Heisenberg Uncertainty Principle • The Heisenberg Uncertainty Principle gives the lower limit on the product of Δx and Δp • The relationship holds for any quantum situation and for any wave-particle

14. Explaining the Uncertainty Principle • The Heisenberg uncertainty principle means that, in the quantum regime, the uncertainties in x and p are connected • Under the very best of circumstance, the product of Δx and Δp is a constant, proportional to h • If you measure the position of a particle-wave with great accuracy, you must accept a large uncertainty in its momentum • If you know the momentum very accurately, you must accept a large uncertainty in the position of the particle-wave • You cannotmake both uncertainties small at the same time

15. Heisenberg Time-Energy Uncertainty • You can also derive a relation between the uncertainties in the energy ΔE of a particle and the time interval Δt over which this energy is measured or generated • The Heisenberg energy-time uncertainty principle is • The uncertainty in energy measured over a time period is negligibly small for a macroscopic object, but it can be important in atomic and nuclear reactions

16. Heisenberg Uncertainty Principle, final • Quantum theory and the uncertainty principle mean that there is always a trade-off between the uncertainties • It is not possible, even in principle, to have perfect knowledge of both x and p, or E and t. • This suggests that there is always some inherent uncertainty in our knowledge of the physical universe • Quantum theory says that the world is inherently unpredictable • This is in sharp contrast with the perfect predictability of classical physics • Because h is so small, for any macro-scale object, the uncertainties in the real macroscopic measurement will always be much larger than the inherent uncertainties due to the Heisenberg uncertainty relation. • The uncertainties are relevant at the scale of atoms, electrons, nuclei.

17. Third Law of Thermodynamics • According to the Third Law of Thermodynamics, it is not possible to reach the absolute zero of temperature • In a classical kinetic theory picture, the speed of all particles would be zero at absolute zero • There is nothing in classical physics to prevent that • In quantum theory, the Heisenberg uncertainty principle indicates that the uncertainty in the speed of a particle cannot be zero, therefore at absolute zero there is still some motion. This is called the zero-point energy.

18. Tunneling • According to classical physics, an electron trapped in a box cannot escape • A quantum effect called tunneling allows an electron to escape under certain circumstances • Quantum theory allows the electron’s wave function to penetrate a short distance into the wall

19. Tunneling, cont. • The wave function extends a short distance into the classically forbidden region • According to Newton’s mechanics, the electron must stay completely inside the box and cannot go into the wall • If two boxes are very close together so that the walls between them are very thin, the wave function can extend from one box into the next box • The electron has some probability for passing through the wall

20. Scanning Tunneling Microscope • A scanningtunneling microscope (STM) operates by using tunneling • A very sharp tip is positioned near a conducting surface • If the separation is large, the space between the tip and the surface acts as a barrier for electron flow

21. Scanning Tunneling Microscope, cont. • The barrier is similar to a wall since it prevents electrons from leaving the metal • If the tip is brought very close to the surface, an electron may tunnel between them • This produces a tunneling current • By measuring this current as the tip is scanned over the surface, it is possible to construct an image of how atoms are arranged on the surface • The tunneling current is highest when the tip is closest to an atom

22. STM Image Si atoms on the surface of a silicon wafer

23. STM, final • Tunneling plays a dual role in the operation of the STM • The detector current is produced by tunneling • Without tunneling there would be no image • Tunneling is needed to obtain high resolution • The tip is very sharp, but still has some rounding • The electrons can tunnel across many different paths • See fig. 28.17 C • The majority of electrons that tunnel follow the shortest path • The STM can form images of individual atoms although the tip is larger than the atoms

24. Color Vision • Wave theory cannot explain color vision • Light is detected in the retina at the back of the eye • The retina contains rods and cones • Both are light-sensitive cells • When the cells absorb light, they generate an electrical signal that travels to the brain • Rods are more sensitive to low light intensities and are used predominantly at night. Low resolution. • Cones are responsible for color vision, high resolution

25. Rods • About 10% of the light that enters your eye reaches the retina • The other 90% is reflected or absorbed by the cornea and other parts of the eye • The absorption of even a single photon by a rod cell causes the cell to generate a small electrical signal • The signal from an individual cell is not sent directly to the brain • The eye combines the signals from many rod cells before passing the combination signal along the optic nerve

26. Cones • The retina contains three types of cone cells • They respond to light of different colors • The brain deduces the color of light by combining the signals from all three types of cones • Each type of cone cell is most sensitive to a range of frequencies, independent of the light intensity

27. Cones, cont. • The explanation of color vision depends on two aspects of quantum theory • Light arrives at the eye as photons whose energy depends on the frequency of the light • When an individual photon is absorbed by a cone, the energy of the photon is taken up by a pigment molecule within the cell • The energy of the pigment molecule is quantized • Photon absorption is possible because the difference in energy levels in the pigment matches the energy of the photon Up to here

28. Cones, final • In the simplified energy level diagram (A), a pigment molecule can absorb a photon only if the photon energy precisely matches the pigment energy level • More realistically (C), a range of energies is absorbed • Quantum theory and the existence of quantized energies for both photons and pigment molecules lead to color vision

29. The Nature of Quanta, summary • The principles of conservation of energy, momentum, and charge hold true under all circumstances, including quantum mechanics • The energy and momentum of a photon are quantized • Electric charge is quantized in units of ±e • The true nature of electrons and photons are particle-waves

30. Puzzles About Quanta • The relation between gravity and quantum theory is a major unsolved problem • No one knows how Planck’s constant enters the theory of gravitation, or what a quantum theory of gravity looks like • Why are there two kinds of charge, ±e? • Why do the positive and negative charges come in the same quantized units e? • What new things happen in the regime where the micro- and macro-worlds meet? • How do quantum theory and the uncertainty principle apply to living things?

31. Problem 28.56 A beam of electrons is directed along the x axis and through a slit that is parallel to the y axis and 10 µm wide. The electrons then continue on to a screen 1.5 m away. The electrons in the beam have a kinetic energy of 70 eV. (a) What is the de Broglie wavelength of the electrons? (b) What is the uncertainty in the y component of their momentum after passing through the slit? (c) How long does it take the electrons to reach the screen? (d) What is the uncertainty in the y position when they hit the screen?

32. Problem 28.56

33. Problem 28.56

34. Problem 28.56