Ch.7 Modern Physics - 1.ppt
- 2. What is Modern Physics? 1- Introduction • Modern physics only came in the 1900’s. • Physicists discovered that Newtonian mechanics did not apply when objects were very small or moved very fast! • If things are confined to very small dimensions (nanometer-scale), then QUANTUM mechanics is necessary. • If things move very fast (close to the speed of light), then RELATIVISTIC mechanics is necessary.
- 3. 2- The wave-Particle duality Young's double slit experiment This was one of the defining characteristics of waves Interference of light waves
- 4. The electron exhibits a dual nature, with both particle-like behavior and wave- like behavior. Very short exposure 14 photon impacts Longer exposure ~150 photon impacts Much longer exposure a few thousand photon impacts
- 5. • Waves can exhibit particle-like characteristics, and particles can exhibit wave-like characteristics.
- 6. 3- Blackbody radiation and Planck's constant The electromagnetic radiation emitted by a perfect blackbody at two different temperatures. We see the glow of hot objects because they emit electromagnetic waves in the visible region of the spectrum. A perfect blackbody at a constant temperature absorbs and reemits all the electromagnetic radiation falls on it.
- 7. • In 1900 Planck calculated the blackbody radiation curves, using a model that represents a blackbody as a large number of atomic oscillators. Planck assumed that the energy E of an atomic oscillator could have only the discrete values of • E = 0, h f, 2 h f, 3 h f, and so on. • E = n h f n = 0, 1, 2, 3, . . . , n is a positive integer f is the frequency of vibration (in hertz) h is a constant called Planck's constant.
- 8. • Experiment has shown that Planck's constant has a value of: h = 6.6260755 x 10-34 j.s • The energy of a system can have only certain definite values, and nothing in between, the energy is said to be quantized. Max Planck (1858-1947)
- 9. 4- The photoelectric effect Light consists of photons Electrons emitted from a metal surface when light shines on it. Light shines on a metal surface, and electrons are ejected from the surface. These photoelectrons are drawn to the positive collector, thus producing a current
- 10. • Einstein proposed that light of frequency f could be regarded as a collection of discrete packets of energy (photons), each packet containing an amount of energy E given by : E = h f where h is the Planck's constant.
- 11. • According to Einstein, when light shines on a metal, a photon can give up its energy to an electron in the metal. • If the photon has enough energy to do work of removing the electron from the metal, the electron can be ejected. • The work required depends on how strongly the electron is held. • For the strongly held electrons, the necessary work has a minimum value W0 and is called the work function of the metal.
- 12. • If a photon has energy in excess of the work needed to remove an electron, the excess energy appears as kinetic energy of the ejected electron. • Thus the least strongly held electrons are ejected with the maximum kinetic energy KEmax .
- 13. • Einstein applied the conservation of energy principle and proposed the following relation to describe the photoelectric effect: h f = KEmax + Wo Photon Maximum Minimum energy kinetic energy work needed of ejected e to eject e
- 14. Photons of light can eject electrons from a metal when the light frequency is above fo Applications: The moving photoelectrons constitute a current that change as the intensity of the light changes. For example, Safety feature of automatic door openers
- 15. 5- The Compton effect • Compton used the photon model to explain his research on the scattering of X rays by the electrons in graphite.
- 16. In an experiment performed by Compton, an X-ray photon collides with a stationary electron. X-ray photon strikes an electron in a piece of graphite. Like two billiard balls colliding on a pool table, the X-ray photon in one direction, and the recoil electron in another direction after the collision.
- 17. • The scattered photon has a frequency f ' that is smaller than the frequency f of the incident photon, indicating that the photon loses energy during the collision. • the difference between the two frequencies depends on the angle θ at which the scattered photon leaves the collision. • The phenomenon in which an X-ray photon is scattered from an electron, the scattered photon having a smaller frequency than the incident photon, is called the Compton effect.
- 18. • The electron is assumed to be initially at rest and essentially free. • In the collision between a photon and an electron, the total kinetic energy and the total linear momentum are the same before and after the collisions, So according to the principle of conservation of energy: • h f = h f ' + KE energy of energy of kinetic energy incident scattered of recoil photon photon electron
- 19. • Since λ ' = c / f ' , the wavelength of the scattered X-rays is larger than that of the incident X-rays. For an initially stationary electron, conservation of linear momentum requires that: Dividing those two equations, we find that
- 20. • Since a photon travels at the speed of light, v = c and . Therefore, the momentum of a photon . But the energy of a photon is • , while the wavelength is , the magnitude of the momentum is
- 21. • Compton showed that the difference between the wavelength λ' of the scattered photon and the wavelength λ of the incident photon is related to the scattering angle θ by • The quantity h/mc is referred to as the Compton wavelength of the electron, and has the value h/mc = 2.43 x 10-12 m .
- 22. • Since cos θ varies between +1 and -1, the shift (λ' – λ) in the wavelength can vary between zero and (2h / mc), depending on the value of θ, a fact observed by Compton.
- 23. 6- The de Broglie Wavelength • In 1923 Louis de Broglie made the suggestion that since light waves could exhibit particle- like behavior, particles of matter should exhibit wave- like behavior. • De Broglie proposed that all moving matter has a wavelength associated with it, just as wave does. Louis V. de Broglie (1892-1987)
- 24. • De Broglie made the explicit proposal that the wavelength λ of a particle is given by the same relation that applies to a photon : • h is the Planck's constant and p is the magnitude of the relativistic momentum of the particle. • The effects of this wavelength are observable only for particles whose masses are very small, on the order of the mass of an electron or a neutron.
- 25. 28e- 104 e- Young's double slit experiment for electrons. The characteristic fringe pattern becomes recognizable only after a sufficient number of electrons have struck the screen.
- 26. • Bright fringes occur where there is a high probability of electrons striking the screen, and dark fringes occur where there is a low probability. • Particle waves are waves of probability, waves whose magnitude at a point in space gives an indication of the probability that the particle will be found at that point.
- 27. • This probability is proportional to the square of the magnitude Ψ (psi) of the wave. Ψ is referred to as the wave function of the particle. • In 1925 Erwin Schrodinger and Werner Heisenberg independently developed theoretical frameworks for determining the wave function; they established a new branch of physics called Quantum Mechanics
- 28. 7- The Heisenberg Uncertainty principle • Since there are number of bright fringes, there is more than one place where each electron has some probability of hitting. Any given electron can strike the screen in only one place after passing through the double slit. • As a result, it is not possible to specify in advance exactly where on the screen an individual electron will hit.
- 29. • Because the wave nature of particles is important, we lose the ability to predict with 100 % certainly the path that a single particle will follow. Instead only the average behavior of large numbers of particles is predictable, and the behavior of any individual particle is uncertain.
- 30. A pattern due to the wave nature of the electrons and is analogous to that produced by light waves. Fig. shows the slit and locates the first dark fringe on either side of the central bright fringe. The central fringe is bright because electrons strike the screen over the entire region between the dark fringes.
- 31. • the extent to which the electrons are diffracted is given by the angle θ in the drawing. To reach locations within the central fringe, some electrons must have acquired momentum in the y direction, despite the fact that they enter the slit traveling along the x direction and have no momentum in the y direction to start with.
- 32. • The y component of the momentum may be as large as Δ py . The notation Δ py indicates the difference between the maximum value of the y component of the momentum after the electron passes through the slit and its value of zero before the electron passes the slit. Δ py indicates the uncertainty in the y component of the momentum, in that a diffracted electron may have the value from zero to Δ py .
- 33. • We relate Δ py to the width W of the slit. • We take the equation of the de Broglie wave length λ, sin θ = λ / W • If θ is small , sin θ ≈ tan θ , • from Fig. tan θ = Δ py / px , where px is the x component of the momentum of the electron. Therefore, Δ py / px ≈ h / λ according to de Broglie's equation, so that
- 34. • a smaller slit width “a” leads to a larger uncertainty in the y component of the electron's momentum. Since the electron can pass through anywhere over the width W, the uncertainty in the y position of the electron is Δy = a . • Substituting for a shows that Δ py ≈ h / Δ y , or (Δpy) (Δy) ≈ h .
- 35. • The Heisenberg uncertainty principle: • For momentum and position: • Δy = uncertainty in a particle's position along the y direction, • Δpy = uncertainty in the y component of the linear momentum of the particle
- 36. For energy and time: • ΔE = uncertainty in the energy of a particle when the particle is in a certain state, • Δt = time interval during which the particle is in the state
- 37. • The Heisenberg uncertainty principle places limits on the accuracy with which the momentum and position of a particle can be specified simultaneously, it states that: • it is impossible to specify precisely both the momentum and position of a particle in the same time. • The same apply for uncertainty that deals with energy and time
- 38. 8- The Shrödinger equation • The wave function for de Broglie waves must satisfy an equation developed by Shrödinger. • One of the methods of quantum mechanics is to determine a solution to this equation, which in turn yields the allowed wave functions and energy levels of the system. Erwin Schrödinger (1887-1961
- 39. • the general form of the wave equation for waves traveling along the x axis: • Where v is the wave speed and where the wave function ψ depends on x and t. • We consider systems in which the total energy E remains constant.
- 40. • Since E = h f, the frequency of de Broglie wave also remains constant. In this case, we can express the wave function ψ (x,t) as the product of a term that depends only on x and a term that depends only on t: ψ (x, t) = ψ (x) cos(ω t) Where v is the wave speed and where the wave function ψ depends on x and t.
- 41. We consider systems in which the total energy E remains constant. Since E = h f, the frequency of de Broglie wave also remains constant. In this case, we can express the wave function ψ (x,t) as the product of a term that depends only on x and a term that depends only on t: ψ (x, t) = ψ (x) cos(ω t)
- 42. Recall that ω = 2 π f = 2 π v / λ and, for de Broglie waves, p = h / λ . Therefore,
- 43. • Furthermore, we can express the total energy E as the sum of the kinetic energy and the potential energy: E = K + U = ( p2 /2 m ) + U So that P2 = 2 m ( E – U ) And
- 44. • This is the famous Schrödinger equation as it applies to a particle confined to moving along the x axis. Because this equation is independent of time, it is commonly referred to as the time-independent Schrödinger equation.