Chapter 28 Quantum Theory Lecture 24

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1 Chapter 28 Quantum Theory Lecture Particles, Waves, and Particles-Waves 28.2 Photons 28.3 Wavelike Properties Classical Particles 28.4 Electron Spin 28.5 Meaning of the Wave Function 28.6 Tunneling 28.7 Detection of Photons by the Eye 28.8 The Nature of Quanta: A Few Puzzles

2 Wave Function In the quantum world, the motion of a particle-wave is described by its wave function ( x) The wave function ( x) 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 In many situations, the solutions of the Schrödinger equation are mathematically similar to standing waves Schroedinger ' s h / 2 d ( x) U ( x) ( x) E( x) 2 2m dx KE( x) U ( x) ( x) E( x) Equation ( x) Section 28.5

3 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 (B) ( x) ( x) 2 Schroedinger ' s Equation h / 2 d ( x) U ( x) ( x) E( x) 2 2m dx KE( x) U ( x) ( x) E( x) Section 28.5

4 Wave Function Example, cont. The wave function solutions correspond to electrons with different kinetic energies 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 2 ( x) space ( x) Section 28.5

its position There is some probability for finding the electron at virtually any spot in the box The

5 Heisenberg Uncertainty Principle For a particle-wave, quantum effects place fundamental limits on the precision of measuring position or velocity The standing waves are the electron, so there is an inherent uncertainty in its position There is some probability for finding the electron at virtually any spot in the box The uncertainty, Δx, is approximately the size of the box This uncertainty is due to the wave nature of the electron Section 28.5

6 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 Section 28.5

7 Uncertainty, Example, cont. The initial momentum is in the y direction The diffracted electron acquires a nonzero momentum along x The width of the slit affects the interference pattern The narrower the slit, the broader the distribution pattern The first dark fringe from a single slit occurs at Eq.(25.23) x w

8 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 momentum along x, there is some uncertainty Δp x in the x component of the momentum Section 28.5

9 Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle gives the lower limit on the product of Δx and Δp xp h 4 The uncertainties Δx and Δp are absolute limits set by quantum theory The relationship holds for any quantum situation and for any wave-particle Section 28.5

10 Explaining the Uncertainty Principle The Heisenberg uncertainty principle dictated 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 a particle-wave s position with great accuracy, you must accept a large uncertainty its momentum If you know the momentum very accurately, you must accept a large position uncertainty You cannot make both uncertainties small at the same time h xp 4 Section 28.5

11 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 h Et 4 The uncertainty in energy measured over a time period is negligibly small for a macroscopic object It may be important in atomic and nuclear reactions Section 28.5

12 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 The uncertainty principle provides a justification of the third law of thermodynamics Section 28.5

13 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 Section 28.6

14 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 Section 28.6

15 Scanning Tunneling Microscope (STM) A scanning tunneling 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

16 Scanning Tunneling Microscope, cont. In a scanning tunneling microscope (STM), a very sharp tip is brought very close to a surface, detecting the probability for an electron to tunnel between the tip and surface by measuring the tunneling current

tunneling paths, so there is a large tunneling current when

17 Scanning Tunneling Microscope, cont. The tunneling probability is largest for the shorter tunneling paths, so there is a large tunneling current when tip is directly over an atom The STM is most sensitive to atoms that are directly below the tip

18 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 Section 28.6

19 STM Image STM images of iron atoms on the surface of copper Section 28.6

20 STM Image, cont.

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

21 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 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 Section 28.6

22 Color Vision A complete understanding of human vision depends on the wave theory and the particle theory of light 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 predominately at night Cones are responsible for color vision Section 28.7

23 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 combine the signals from many rod cells before passing the combination signal along the optic nerve About 50 photons within about 0.1 s must be received for the brain to know light as actually arrived Section 28.7

24 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 particular frequency, independent of the light intensity Section 28.7

25 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 Ephoton hf 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 various pigments match the energy of the photon hc Ephoton hf h = x J s = 4.14 x ev s hc

26 Cones, cont. 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 Section 28.7

27 Cones, final Quantum theory and the existence of quantized energies for both photons and pigment molecules lead to color vision

28 The Nature of Quanta The principles of conservation of (1) energy, (2) momentum, and (3) charge are believed to hold true under all circumstances Must allow for the existence of quanta The energy and momentum of a photon come in discrete quantized units hc E hf h Ephoton hf pphoton c c h = x J s = 4.14 x ev s Electric charge also comes in quantized units The true nature of electrons and photons is particlewaves Section 28.8

29 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? Why do the positive and negative charges come in the same quantized units? 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? Section 28.8

Lecture 16 Quantum Physics Chapter 28

Lecture 16 Quantum Physics Chapter 28 Particles vs. Waves Physics of particles p = mv K = ½ mv2 Particles collide and do not pass through each other Conservation of: Momentum Energy Electric Charge Physics