Chap. 7 Quantum Optics

7.1 Monochromatic radiant exitance and absorptance、 Kirchhoff’s law 7.2 Wien’s formula and Rayleigh-Jeans’s formula 7.3 Planck radiation formula energy quanta 7.4 Photo-electric effect、 photonics 7.5 Compton effect 7.6 wave-corpuscle duality

7.1 Monochromatic radiant exitance and absorptance、Kirchhoff’s law 一、thermal radiation and luminescenc（irrediance） 热辐射和发光 Kinds of radiation: Chemiluminescence, photoluminescence, electroluminescence, Cathode luminescence, thermal radiation. 化学发光、光致发光、场致发光、阴极发光、热辐射 Several physical quantities Monochromatic radiant exitance （单色幅出度） ——物体表面单位面积在单位频率间隔内辐射的功率。 Radiant exitance ——物体表面单位面积辐射的功率。

dW 表示照射到温度为T的物体的单位面积上、频率在+d范围内的辐射能 。 absorptance 吸收比 dW 表示照射到温度为T的物体的单位面积上、频率在+d范围内的辐射能 。 dW‘表示温度为T的物体单位面积所吸收的频率在+d范围内的辐射能。 2. Kirchhoff’s law universal function 普适函数与材料无关 is related to the material

7.2 Wien’s formula and Rayleigh-Jeans’s formula 一、black body/ideal radiator黑体 Black body——absorb all electromagnetic waves in any temperature。 在任何温度状态下、 全部吸收任何波长的电磁波 applying Black body, Universal constant is the monochromatic radiation exitance of black body. T=6000k T=5000k T=3000k Visible light 普适常数就是黑体的单色幅出度。 ∴ Kirchhoff’s law Discussion: 1.In the same temperature，black body radiates most. 2.There is no ideal radiator, so black body model is applied to study 3.Is black body black, and black object is black body?

二、classical radiation laws of black body and their limitation Monochromatic radiation exitance of black body 1. Two experiment laws （1）Stefan-Boltzmann’s law 斯特藩——玻尔兹曼定律 Radiation of black body 黑体的幅出度  = 5.6703210-8 w/(m2K4) Stefan-Boltzmann constant （2）Wien’s displacement law 维恩位移定律 There is a maximum in and its wavelength is, b= 2.897810-3 m·K Wien constant When temperature goes up, maximum moves to short wave.

2. classical radiation laws of black body and their limitation Wien’s formula Rayleigh-Jeans’s law 瑞利——金斯定律（能量均分定理） k= 1.3810-38 J/K 瑞利——金斯线 Boltzmann constant Wien line Ultraviolet disaster 紫外灾难

R=1.5×1011m r=1.39×109m 例7-1（1）如果将恒星表面的辐射近似地看作是黑体辐射， 就可以用测量λmax的方法来估算恒星表面的温度。现测量到太阳 的λmax为510nm，试求它的表面温度。 （2）太阳常数（太阳在单位时间内垂直照射在地球表面 单位面积上的能量）为1352w/m2，日地间的距离为1.5×108km， 太阳直径为1.39×106km，试用这些数据估算一下太阳的温度。 R=1.5×1011m r=1.39×109m

7.3 Planck radiation formula energy quantum In 1900，Planck proposed a hypothesis: 普朗克提出一个假设：（实用主义解释实验， 但由此步入量子化，有质的飞跃。） 辐射体由各种振动频率的谐振子组成，辐射能量连续。 2.每个谐振子能量不连续变化，只能处于某些分立的能量 状态。最小的能量单位E0 即为能量子。E0，2E0，3E0，… h= 6.62617610-34 J·s——Planck constant ——谐振子振动频率 3.谐振子从一个能量状态到另一个能量状态 E02E0 Absorb radiation 吸收外来辐射 2E0E0 Radiate energy 辐射能量

二、Planck formula普朗克公式 According to Planck hypothesis and Boltzmann distribution The probability of oscillator with temperature T and energy E= nE0 is  Average energy of every oscillator: 每个振子平均能量为： The equation of black body radiation changes into: 普朗克黑体辐射公式为：

results：1. Fit well with the experimental curve 2. Short wave， small  Wein’s formula Long wave， large  Raleigh-Jeans’s formula 3. calculate is consistent with experimental law  b accords with experimental law From experiment, we can measure  and b，then apply Planck’s formula to calculate h and k . We found that their values are consistent with experimental results. 由普朗克公式推得h和k，其值与其它 实验结果一样，说明普朗克公式有其正确方面。 From classical to quantum 实现从经典量子的过渡。

7.4 Photo-electric effect 7.4.1Experimental results of photo-electric effect Einstein, in 1905, explained the photoelectric effect based on Planck’s idea of quantum theory of light. 光电效应——电子在光的作用下从金属表面发射出来的现象 逸出来的电子称为光电子 普朗克：振子辐射能量量子化，但辐射场是连续的电磁波 1905年爱因斯坦对光电效应研究电磁场以量子的形式存在 Experimental facility Experimental curve of I-V G V G: galvanometer V I Vg Im V: voltmeter

The following results were observed: Vg Im 1.The number of photo-electrons ejected depends upon the intensity of the incident light. Thus, the photo-electric current depends upon the intensity of the incident light. 饱和电流Im  入射光强 I。 2. The stopping potential Vg depends upon the frequency of the incident light and is independent of the intensity of the incident light, and its value is equal to the maximum kinetic energy of the electrons emitted by the surface. 遏止电压Vg与入射光频率有关， 与I。无关。光电子的最大初动能= eVg

只要  >0 ，不管I。多弱，一照上去，就有光电流产生。 3. Light of frequency greater than the critical frequency ejects electrons of different velocities. 只要  >0 ，不管I。多弱，一照上去，就有光电流产生。 4. For a given metallic surface, there is a smallest value of frequency for which the incident light can eject the photo-electrons out of the metal. Light of frequency smaller than this particular value cannot eject electrons, no matter how long it falls on the surface or how high is its intensity. 入射光频率 <0（某一频率），无论照射多长时间，无光电流产生。 截止频率0 (红限) 5.驰豫时间τ<10-9s

7.4.2 contradiction of photoelectric effect and wave theory of light 光电效应与波动理论的矛盾 Electron energy obtained from the incident light 电子从光波获得的能量 w:自由电子运动到金属表面的能量 w：逸出功(自由电子脱出金属表面所需能量） The maximum of kinetic energy of electron ： Applying wave theory to explain photoelectric effect: 用波动理论解释光电效应

3.4.与驰豫时间τ<10-9s 矛盾 1. 照射光愈强，逸出表面的电子数多，当电压足够大时， 全部电子到达阳极，所以饱和电流Im  入射光强 I0 2. 照射的光强，接受的能量愈多， Vg应与光强有关，实际却与光的频率有关。矛盾 3.照射时间长，积累能量多，只要照射足够长时间，总会有 电子逸出，有电流。实际却是若入射光频率 <0 ， 无论照射多长时间，无光电流产生。矛盾 4.光很弱，必须要照射长时间，才能积累足够的能量， 使电子从金属表面逸出。但实际却只要  >0， 不管I0多弱，一照上去，就有光电流产生。矛盾 3.4.与驰豫时间τ<10-9s 矛盾

7.4.3 Einstein’s quanta explanation 爱因斯坦的量子解释 一、Einstein’s photon hypothesis and photo-electric equation 爱因斯坦的光子假设和光电效应方程 1. Photon hypothesis Planck：吸收、辐射是分立的，电磁波是连续的； 即振子能量量子化，而辐射场仍作连续的。 Einstein：光在传播过程中具有波动性，而在与物质相互 作用过程中，能量集中在光（量）子上。 Each photon has an energy： ——frequency of light h——Planck’s constant

2. photo-electric equation When a photon collides with an electron , it may radiate or absorb energy in the form of a photo energy, this transfer is an “all or none” process. 发射和吸收能量时，以一个光子为最小单位 2. photo-electric equation Ejecting energy Photon energy The maximum kinetic energy of photo-electron光电子最大动能 An electron absorbs energy of a photon 一个电子吸收一个光子能量，一对一吸收

二、quanta explanation of photoelectric effect 对光电效应的量子解释 The intensity of incident light I0  N h ，photoelectrons ejected n  N，when the voltage is large enough, all photoelectrons can reach the anode, so the saturation current Im=ne I0. 2. The higher of frequency， the larger of Vg 3. The energy of photon is large with high frequency, And only when h  W, there are photoelectrons to eject. 0 = c/0 ：the critical wavelength 红限波长 4. When photon energy is larger than ejecting energy h  W ， There is a photon of incident light, there is an electron to escape. 不管入射光多弱，有一个光子，就会有电子逸出，无需时间积累。

三、experiment verification 实验验证 In 1916, Millikan a negative potential to the cylinder And a positive potential to the plate. 密立根用“接触电势差”替代“阳极、阴极”， It is found that the stopping potential Vg is independent of the intensity of the incident light, while depending on the frequency of the incident light. Thus, Einstein’s hypothesis was verified experimentally. 实验上证实了爱因斯坦假设。 -W/e 0  Vg Millikan obtained the Nobel prize in 1923.

四、mass and momentum of photon 光子的质量和动量 Photon is a kind of particle, but it is different from particle. It has wave-corpuscle duality. 光子是一种粒子，但它不同于微粒，具有波粒二象性 Wave character：， Particle property：m，p 1. Mass and energy equation of relativity 相对论的质量和能量公式 （1） m0 ：static mass 静止质量 （2）mass and energy equation 质能公式 Total energy of particle 粒子总能量 Static energy 静止能量 The kinetic energy ( 适用于<<c)

2. Mass and momentum of photon （3）relation of momentum and energy 动量——能量关系 p—momentum 2. Mass and momentum of photon 光子的质量和动量 Static mass of photon： m0=0 Energy of photon： Mass of photon： Momentum of photon：

已知：波长为=200nm的光入射铝表面，铝逸出功W=4.2ev 求：1，出射最快电子动能；2，Vg；3，铝红限波长 解： 1， 2， 3，

N=I/hν 例7-2 波长为200nm的光照射在铝表面上，对铝而言， 移去一个电子所需的能量为4.2eV，试求： （1）射出的最快光电子的动能是多少？ （2）遏止电压是多少？ （3）铝的截止波长是多少？ （4）如果入射光强为2W/m2，则单位时间内打到金属板 单位面积上的光子数为多少？ N=I/hν

7.5 Compton effect 一、experimental phenomenon 无法用波动理论解释 X光管 康普顿效应 一、experimental phenomenon X光管 Scattering material 散射物质 X ray photometer 检测器 0  X光分光计 1923年发现此现象 1927年康普顿获 诺贝尔奖 Experimental results： 1. Scattering light: 0，>0； 无法用波动理论解释 2. =-0, ，； ，Is(0)  ,Is()  3.  is independent of scattering material and wave length of the incident light, and only depended on the scattering angle . Is(0) increased with increasing the atomic number of scattering material 随散射物质的原子序数的增加而增大； While Is()decreased. 随散射物质的原子序数的增加而减小。

二、quanta explanation of Compton effect 康普顿现象的量子解释 Suppose the photon of energy hν strikes a free electron at rest. 光子与散射原子中电子发生弹性碰撞。 Conservation of energy Conservation of momentum  （m: mass of electron） Applying cosine law, we have, =0.00241nm Compton wavelength 康普顿波长

Explanation shows that： (From the upper equation) ，，  is independent of scattering material and wave length of the incident light. 2. c=0.00241nm is consistent with experimental results 0.00236nm. 3. In the region of visible light， /10-5 can be neglected, then phenomenon represents the classical scattering. For X-ray，  0.1nm ,/10-2， phenomenon represents quantum property. 4. In scattering material , there are free electrons and bound electrons.When the incident photon strikes again a free electron, the wavelength of scattered photon is more than the incident. When the photon strikes a bound electron,the scattered photon has the same wavelength and frequency of the incident photon. 实际散射物质中，存在束缚电子，看成光子与原子 碰撞， 0，散射光中有原波长0 。原子序数增大，束缚电子增多，散射光中的Is(0)增大。

例7-3 现有：（a）波长λ=0.1nm的X射线；（b）从137Cs样品 入射方向成90°角的方向去观察散射辐射，问每种情况下： （1）康普顿波长偏移是多少？ （2）相对康普顿波长偏移是多少？ （3）给予反冲电子的动能是多少？

7.6 De Broglie wave and wave-corpuscle duality 德布罗意波和波粒二象性 The dual behavior of photons inspired De Broglie to suggest a model of the atom. 德布罗意波提出假设： 由光具有波粒二象性， Partiles like electrons have wave-corpuscle duality. 一切实物粒子都具有波粒二象性。 Davisson and Germer experiment 戴维孙、革末用实验证实了德布罗意的假设。 v G 0 单晶M K D De Broglie wavelength： In 1929, De Broglie obtained the Nobel prize. 德布罗意1929年获诺贝尔奖

例7-4 在电子显微镜中，电子受到90kV的电压加速，如果要 观察到数量级为10-9cm的分子结构，显微镜的数值孔径应该 多大？ chapter 7