# 量子力學發展史

## 量子力學發展史

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1. 量子力學發展史 近代科學發展之三

2. 物理模型 • 粒子模型 • Allowed us to ignore unnecessary details of an object when studying its behavior • 系統與剛體 • Extension of particle model • 波動模型 • 兩種新模型 • 量子粒子 • 邊界條件下的量子粒子

3. 黑體幅射 • 物體在任何溫度下皆會發出熱幅射(電磁幅射) • 電磁幅射波長會隨物體表面溫度的變化而改變 • 黑體為一理想系統會吸收所有射入的幅射 • 由黑體發射出的電磁幅射稱為幅射

4. 黑體近似 • 黑體模型可近似為開了一小孔洞的金屬空腔 • 離開空腔的電磁幅射其性質將只與空腔的表面溫度有關

5. 黑體實驗結論 • 幅射的總功率與溫度的關係滿足Stefan定律 • Stefan’s Law • P = s A e T4 • For a blackbody, e = 1 • s is the Stefan-Boltzmann constant • s = 5.670 x 10-8 W / m2. K4 • 波長分佈曲線的峰值位置隨溫度升高兒而往短波長方向偏移，Wien位移定律 • Wien’s displacement law • lmax T = 2.898 x 10-3 m.K

6. 黑體幅射強度隨波長的分佈 • 幅射強度隨溫度升高而增強 • 總幅射量隨溫度升高而變大 • The area under the curve • 峰值對應的波長隨溫度升高而變短

7. 紫外危機 • 古典物理的預測與實驗在短波處的結果發生極大的差異 • 此現象稱為紫外危機，尤其古典物理預測當幅射波的波長趨近於零時更會得到無限大的能量，此與實驗觀察完全相反

8. Max Planck(拯救紫外危機的英雄) • 1858 – 1947 • He introduced the concept of “quantum of action” • In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

9. Planck’s Theory of Blackbody Radiation • In 1900, Planck developed a structural model for blackbody radiation that leads to an equation in agreement with the experimental results • He assumed the cavity radiation came from atomic oscillations in the cavity walls • Planck made two assumptions about the nature of the oscillators in the cavity walls

10. Planck’s Theory of Blackbody Radiation • In 1900, Planck developed a structural model for blackbody radiation that leads to an equation in agreement with the experimental results • He assumed the cavity radiation came from atomic oscillations in the cavity walls • Planck made two assumptions about the nature of the oscillators in the cavity walls

11. Planck’s Assumption, 1 • The energy of an oscillator can have only certain discrete values En • En = n h ƒ • n is a positive integer called the quantum number • h is Planck’s constant • ƒ is the frequency of oscillation • This says the energy is quantized • Each discrete energy value corresponds to a different quantum state

12. Planck’s Assumption, 2 • The oscillators emit or absorb energy only in discrete units • They do this when making a transition from one quantum state to another • The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation • An oscillator emits or absorbs energy only when it changes quantum states

13. Energy-Level Diagram • An energy-level diagram shows the quantized energy levels and allowed transitions • Energy is on the vertical axis • Horizontal lines represent the allowed energy levels • The double-headed arrows indicate allowed transitions

14. Correspondence Principle(對應原理) • 當量子系統的量子態總數變大時，量子現象應當會連續地轉變成古典現象 • Quantum effects are not seen on an everyday basis since the energy change is too small a fraction of the total energy • Quantum effects are important and become measurable only on the submicroscopic level of atoms and molecules

15. Photoelectric Effect • The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces • The emitted electrons are called photoelectrons • The effect was first discovered by Hertz

16. Photoelectric Effect Apparatus • When the tube is kept in the dark, the ammeter reads zero • When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter • The current arises from photoelectrons emitted from the negative plate (E) and collected at the positive plate (C)

17. Photoelectric Effect, Results • At large values of DV, the current reaches a maximum value • All the electrons emitted at E are collected at C • The maximum current increases as the intensity of the incident light increases • When DV is negative, the current drops • When DV is equal to or more negative than DVs, the current is zero

18. Photoelectric Effect Feature 1 • Dependence of photoelectron kinetic energy on light intensity • Classical Prediction • Electrons should absorb energy continually from the electromagnetic waves • As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy • Experimental Result • The maximum kinetic energy is independent of light intensity • The current goes to zero at the same negative voltage for all intensity curves

19. Photoelectric Effect Feature 2 • Time interval between incidence of light and ejection of photoelectrons • Classical Prediction • For very weak light, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal • This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal • Experimental Result • Electrons are emitted almost instantaneously, even at very low light intensities • Less than 10-9 s

20. Photoelectric Effect Feature 3 • Dependence of ejection of electrons on light frequency • Classical Prediction • Electrons should be ejected at any frequency as long as the light intensity is high enough • Experimental Result • No electrons are emitted if the incident light falls below some cutoff frequency, ƒc • The cutoff frequency is characteristic of the material being illuminated • No electrons are ejected below the cutoff frequency regardless of intensity

21. Photoelectric Effect Feature 4 • Dependence of photoelectron kinetic energy on light frequency • Classical Prediction • There should be no relationship between the frequency of the light and the electric kinetic energy • The kinetic energy should be related to the intensity of the light • Experimental Result • The maximum kinetic energy of the photoelectrons increases with increasing light frequency

22. Photoelectric Effect Features, Summary • The experimental results contradict all four classical predictions • Einstein extended Planck’s concept of quantization to electromagnetic waves • All electromagnetic radiation can be considered a stream of quanta, now called photons • A photon of incident light gives all its energy hƒ to a single electron in the metal

23. Photoelectric Effect, Work Function • Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax • Kmax = hƒ – f • f is called the work function • The work function represents the minimum energy with which an electron is bound in the metal

24. Some Work Function Values

25. Photon Model Explanation of the Photoelectric Effect • Dependence of photoelectron kinetic energy on light intensity • Kmax is independent of light intensity • K depends on the light frequency and the work function • The intensity will change the number of photoelectrons being emitted, but not the energy of an individual electron • Time interval between incidence of light and ejection of the photoelectron • Each photon can have enough energy to eject an electron immediately

26. Photon Model Explanation of the Photoelectric Effect, cont • Dependence of ejection of electrons on light frequency • There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron • Without enough energy, an electron cannot be ejected, regardless of the light intensity

27. Photon Model Explanation of the Photoelectric Effect, final • Dependence of photoelectron kinetic energy on light frequency • Since Kmax = hƒ – f • As the frequency increases, the kinetic energy will increase • Once the energy of the work function is exceeded • There is a linear relationship between the kinetic energy and the frequency

28. Cutoff Frequency • The lines show the linear relationship between K and ƒ • The slope of each line is h • The absolute value of the y-intercept is the work function • The x-intercept is the cutoff frequency • This is the frequency below which no photoelectrons are emitted

29. Cutoff Frequency and Wavelength • The cutoff frequency is related to the work function through ƒc = f / h • The cutoff frequency corresponds to a cutoff wavelength • Wavelengths greater than lc incident on a material having a work function f do not result in the emission of photoelectrons

30. Applications of the Photoelectric Effect • Detector in the light meter of a camera • Phototube • Used in burglar alarms and soundtrack of motion picture films • Largely replaced by semiconductor devices • Photomultiplier tubes • Used in nuclear detectors and astronomy

31. Arthur Holly Compton • 1892 - 1962 • Director at the lab of the University of Chicago • Discovered the Compton Effect • Shared the Nobel Prize in 1927

32. The Compton Effect, Introduction • Compton and coworkers dealt with Einstein’s idea of photon momentum • Einstein proposed a photon with energy E carries a momentum of E/c = hƒ / c • Compton and others accumulated evidence of the inadequacy of the classical wave theory • The classical wave theory of light failed to explain the scattering of x-rays from electrons

33. Compton Effect, Classical Predictions • According to the classical theory, electromagnetic waves of frequency ƒo incident on electrons should • Accelerate in the direction of propagation of the x-rays by radiation pressure • Oscillate at the apparent frequency of the radiation since the oscillating electric field should set the electrons in motion • Overall, the scattered wave frequency at a given angle should be a distribution of Doppler-shifted values

34. Compton Effect, Observations • Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed

35. Compton Effect, Explanation • The results could be explained by treating the photons as point-like particles having energy hƒ and momentum hƒ / c • Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved • Adopted a particle model for a well-known wave • This scattering phenomena is known as the Compton Effect

36. Compton Shift Equation • The graphs show the scattered x-ray for various angles • The shifted peak, l', is caused by the scattering of free electrons • This is called the Compton shift equation

37. Compton Wavelength • The unshifted wavelength, lo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms • The shifted peak, l', is caused by x-rays scattered from free electrons in the target • The Compton wavelength is

38. Photons and Waves Revisited • Some experiments are best explained by the photon model • Some are best explained by the wave model • We must accept both models and admit that the true nature of light is not describable in terms of any single classical model • Light has a dual nature in that it exhibits both wave and particle characteristics • The particle model and the wave model of light complement each other

39. Louis de Broglie • 1892 – 1987 • Originally studied history • Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons

40. Wave Properties of Particles • Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties • The de Broglie wavelength of a particle is

41. Frequency of a Particle • In an analogy with photons, de Broglie postulated that particles would also have a frequency associated with them • These equations present the dual nature of matter • particle nature, m and v • wave nature, l and ƒ

42. Davisson-Germer Experiment • If particles have a wave nature, then under the correct conditions, they should exhibit diffraction effects • Davission and Germer measured the wavelength of electrons • This provided experimental confirmation of the matter waves proposed by de Broglie

43. Electron Microscope • The electron microscope depends on the wave characteristics of electrons • The electron microscope has a high resolving power because it has a very short wavelength • Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light

44. Quantum Particle • The quantum particle is a new simplification model that is a result of the recognition of the dual nature of light and of material particles • In this model, entities have both particle and wave characteristics • We much choose one appropriate behavior in order to understand a particular phenomenon

45. Ideal Particle vs. Ideal Wave • An ideal particle has zero size • Therefore, it is localized in space • An ideal wave has a single frequency and is infinitely long • Therefore, it is unlocalized in space • A localized entity can be built from infinitely long waves

46. Particle as a Wave Packet • Multiple waves are superimposed so that one of its crests is at x = 0 • The result is that all the waves add constructively at x = 0 • There is destructive interference at every point except x = 0 • The small region of constructive interference is called a wave packet • The wave packet can be identified as a particle

47. Wave Envelope • The blue line represents the envelope function • This envelope can travel through space with a different speed than the individual waves

48. Speeds Associated with Wave Packet • The phase speed of a wave in a wave packet is given by • This is the rate of advance of a crest on a single wave • The group speed is given by • This is the speed of the wave packet itself

49. Speeds, cont • The group speed can also be expressed in terms of energy and momentum • This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent

50. Electron Diffraction, Set-Up