1. In 1909 G.I. Taylor experimented with a very dim light source. His work, and many modern experiments show that even though only one photon passes through a double slit, over time, an interference pattern is still produced - one “particle” at a time. “It would seem that the basic idea of the quantum theory is the impossibility of imagining an isolated quantity of energy without associating with it a certain frequency.” =h/p Louis de Broglie, in 1923, reasoned that if light waves could behave like particles then particles should have a wavelength. Soon after, an experiment by C. J. Davisson and L. H. Germer showed that electrons could produce interference patterns just like those produced by light. n =2dsin 

2. De Broglie’s G.I. Taylor, one of the most distinguished physical scientists of this century, used his deep insight and originality to increase our understanding of phenomena such as the turbulent flow of fluids. His interest in the science of fluid flow was not confined to theory; he was one of the early pioneers of aeronautics, and designed a new type of anchor that was inspired by his passion for sailing. His experiment with photons resulted in his first paper. It leads us to understand that even though one photon encounters the double slits, wave properties still appear, but are displayed in a particle fashion. Prince Louis De Broglie’s proposed the idea that particles can also be thought of as waves for his PhD thesis. It was a difficult sale and required the intervention of Einstein.

3. p= /h If we know the wavelength we are certain about the momentum. But, because a wave is spread out in space, we are uncertain about its position. { If we add together many wavelengths... We are uncertain about the momentum. But, we are now more certain about position. { Heisenberg’s Uncertainty Principle helps us examine the dual nature of light, electrons, and other particles.

4. Particle or Wave? So what is light, particle or wave? Heisenberg’s idea lets us think about a particle as a sum of waves. The more you know about the particle properties, because it is a sum of waves, the less you know about its waviness and visa versa. It is sometimes instructive to have students add waves using a graphing calculator or computer graphing program. I often do this first when talking about beats in my unit on sound.

5. Heisenberg Uncertainty This can be discussed in terms of the Heisenberg Uncertainty principle. If you don’t know the actual location of the photons they produce wave behavior but as soon as you know the location (one slit or the other) the particle properties appear and the interference pattern disappears. In the second set of diagrams each red lines represent one photon that arrives at a photo-detector. In the top diagram it is set up so that the waves destructively interfere for the top detector and constructively interfere for the bottom. In the lower diagram on the right, because you know which path the photon has followed it behaves like a particle and follows one of the two paths. Thus, you can tell if someone is eves dropping on the system by the output of the system.

6. Potential Energy Position Potential energy for a classical harmonic oscillator is continuous and can have a value of zero. Maxwell showed that oscillating electric charge produces an electromagnetic wave. Schrödinger's equation was published in 1926. The solution of this equation for a particular particle and the forces acting on that particle is called the wave function (  ). Quantized energy levels Unlike the classic oscillator, this system has a minimum energy or “zero-point” energy. Zero-point energy

7. Schrödinger Equation More quantum strangeness comes when we consider an harmonic oscillator. If you think of a spring in a classical sense, it can have zero spring potential energy when it stops vibrating. Erwin Schrödinger, in a attempt to explain the waves associated with the electron, developed his wave equation. Given the mass of the particle, and the forces to which it is subjected - the spring for the harmonic oscillator - Schrödinger's equation gives us all the associated waves and in turn, the possible energies. Applying the equation to the harmonic oscillator we find that there is a lowest possible energy or zero-point energy and that the system can’t go below that energy.

8. Schroedinger’s Equation (continued) Schroedinger: If electrons are waves, their postion and motion in space must obey a wave equation. Solutions of wave equations yield wavefunctions, Y, which contain the information required to describe ALL of the properties of the wave. Provides a picture of the electronic distributions of the electrons about an the nucleus of an atom and about the connected nuclei of a molecule.

9. Schr ö dinger ’ s Improvement to Bohr ’ s Model Showed how to obtain Bohr ’ s formula using the Schr ö dinger equation Electron is described by a wave function  Solved for  in the electric potential due to the nucleus of the hydrogen atom

10. Quantum Square Well Approximate electric (roller coaster) potential by a ‘ square well ’ System is then identical to the wave equation for a string that is fixed at both ends

11. Summary Electron does not fly round the nucleus like the Earth around the Sun (Rutherford, Bohr) Depending on which energy level it is in, the electron can take one of a number of stationary probability cloud configurations (Schr ö dinger)

12. Energy Levels in a Box

13. Atoms and molecules can also absorb photons, making a transition from a lower level to a more excited one This photon has been absorbed Energy Ground level Excited level

14. In 1916, Einstein showed that another process, stimulated emission, can occur Before After Absorption Stimulated emission Spontaneous emission

15. Stimulated emission

16. A vibrating rope between two fixed points can only produce standing waves that are a multiple of 1/2. F = πhc/480 d 4 Because of the zero point energy even a complete vacuum is filled with waves of every wavelength… but only certain wavelengths can exist between two closely spaced mirrors in the vacuum. In 1948 Hendrick Casimir predicted that these two mirrors would be pushed together because of the difference in energy.

17. diffraction still limits the size of objects we can resolve. The smaller the object the shorter the wave- length required. Once you have designed optics to correct for aberrations caused by the lenses…

18. Diffraction We can apply what we know about light to help us build a better microscope. Chromic and spherical aberrations have been corrected in modern lens with coatings and combinations of lenses made out of materials with different indexes of refraction. After making those corrections you still need to deal with diffraction. Because of light’s wave nature there is a limit to what an optical microscope can resolve. Ernst Abbe found that the limit for resolving an object because of diffraction was described by the formula shown above. A modern microscope, like the one pictured that uses fluorescence (see next slide), can not overcome this limit.

19. Using Schrödinger's wave equation,  can be calculated for electrons in the Coulomb potential of an atom.   gives us the probability of finding an electron in a particular location. These calculations show discrete energy levels similar to the calculations for a harmonic oscillator. An electron is excited to a higher energy when a photon is absorbed and gives off a photon when it relaxes to a lower energy. An atom fluoresces when a short  photon excites the atom and a longer  photon is given off.

20. Probability Spaces To understand how fluorescence can be used to enhance an image we need to discuss the structure of an atom. Schrödinger's wave equation is used to describe the possible electron waves that can exist inside the Coulomb field of the atom, not unlike the allowed vibrations for an harmonic oscillator. Each allowed wave represents a different energy level the electron can occupy. In a hydrogen atom, as in any atom, the electron jumps between those levels give rise to the spectrum for that atom. Because of Heisenberg's Uncertainty relationship we can’t know exactly where the electron will be but the square of the wave function tells us the probability of finding an electron in a particular location in the atom. These are probability spaces are the orbitals we have come to know from chemistry. In a fluorescent molecule a photon my excite an electron to a higher energy level, that electron relaxes to a slightly lower level, then drops back to the ground state giving off a lower energy photon. The color of the emitted light is characteristic of the substance fluorescing.

21. Energy is “pumped” into the medium, exciting electrons to the metastable state. An electron drops to the ground state and produces a photon. That photon interacts with another excited electron causing it to drop. Meta-stable state Ground state A second photon is produced by stimulated emission. Those photons reflect and continue to stimulate more photons. Coherent Collimated Monochromatic

22. Laser and Einstein Another development important to modern microscopes is the laser. Some energy levels inside the atom are metastable. Normally the average life time of an excited atom is about 10 -8 s. In a metastable state the electrons may remain excited for 10 5 times longer before spontaneously emitting a photon. This allows time to get many atoms into a metastable state and create a population inversion where there are more excited atoms than ground state atoms. In 1917 Einstein introduced the concept of stimulated emission, meaning that a photon can stimulate another atom to emitt an identical photon. “Laser” is an acronym for” light amplification by the stimulated emission of radiation”. There is a mirror at each end of the laser. One mirror is close to 100% reflective, and the other allows a small amount of light to escape. As the photons bounce back and forth between the mirrors they are “amplified” by causing stimulated emissions from other atoms. Because of this process, all the photons in the light emitted have the same energy, phase, polarization, and direction of travel.

23. If you have a very intense light, two photons can induce a single fluorescence. In single photon excitation, fluorescence is produced all along the path of the light. In two-photon excitation the light is only intense enough to produce fluorescence at the focal point.

24. Confocal Microscope To improve the confocal microscope two-photon excitation is used. A laser is pulsed to provide a very intense beam of light with many photons. The photon energy is half the energy needed to excite the fluorescent molecule so two photons are needed to do the excitation. This can only happen because of the intensity of the beam. A disadvantage to the confocal microscope is the fact that the light tends to be dispersed as it travels through the object. This makes it hard to see deeper layers. Fluorescent markers also tend to “bleach” or fade under illumination. The two-photon method eliminates that problem because the beam is only intense enough to get excitation at the focal point so no other molecules fluoresce. This also eliminates the need for the pinhole because light only comes from the one point. Scanning gives you images such as these neurons.

25. Can we break the diffraction limit? Because of diffraction a molecule might look like this. But we would have a pretty good idea of its location. unless we can turn them on one molecule at a time. If we have many molecules in the same location we will not be able to resolve them…

26. Diffraction Limit To break the diffraction limit described by Abbe we must go one step further. Diffraction will make a molecule look blurred out. Even so, we have a good idea where the molecule is. If you have many molecules in one small area it will be difficult to locate individual molecules. Chemical compounds have been developed that can be turned on and off. A laser activates them, giving them the ability to fluoresce. These molecules are attached to the structure of interest, are activated and then when excited tell us the location of that structure. The trick is to illuminate the sample with a very weak excitation beam so that only a few photons strike the sample at a time. In this way only one molecule fluoresces at a time and its location can be inferred. These marker bleach out quickly, turning the fluorescence off again. Using this technique we can use visible light to see things smaller then the typical 2 µm diffraction limit for visible light. In these pictures we see fibers in a sample, first with a conventional microscope and then using this process. Different structures can be seen by using different labeling compounds.

27. Eadweard Muybridge froze the motion of a horse in 1878 - a millisecond event. Harold Edgerton stopped a bullet at the microsecond time scale. If we wish to examine electrons we must take “pictures” at the attosecond level. One attosecond is to 1/2 second as 1/2 second is to the age of the universe.

28. Ultra-fast Photos Light can also be put to work to help us look at events that take place in very short time intervals. Muybridge was motivated to take photos of a running horse to settle the question of whether or not all four hoofs are off the ground at the same time. Edgerton became famous for his stop action photos, but if we wish to freeze the motion of atomic or subatomic particles we need to deal with attosecond flashes of light. It is fun to have students work out their own comparisons for the size of an attosecond.

29. The alternating electric field of laser beam can accelerate an electron out of an atom and then send it crashing back into the atom. The frequencies of light produced as the electron accelerates can interfere to produce a very short pulse of light.

30. Short Pulses One way to make these incredibly short pulses is to “rock” an electron back and forth inside an atom. If we are thinking of light as an electromagnetic wave, the alternating electric field of an intense laser beam can push an electron away from the nucleus and out of an atom and then as the field reverses, send the electron crashing back into the nucleus, much like a ball on a see-saw. The accelerating electron will produce a new electromagnetic wave. This wave will change frequency depending on where the electron is in its oscillation. If done properly all of these wavelengths will add up to make a very short pulse of light.

31. These pulses of light, interfering with the electron itself, can tell you everything you can know about the electron’s wave function. Now, movies of atoms moving and bonds breaking are possible.

32. FS Chemistry These very short pulses of light can then interfere with the wave that is the electron and tell you all about the electron’s place in an atom or molecule. If this is done frame by frame movies of chemistry in action will soon be possible.

33. 飞秒频闪观测谱（ 1987 ） （ Eadweard Muybridge 摄）。

34. 激光化学与飞秒频闪（ 1999 ） 阿梅特 蔡维尔（ Ahmed Zewil ， 1946- ， 埃及）因他的飞秒化学工作， 成了 1999 年诺贝尔化学奖得主。他还得到了提名并成了 Barack Obama 总统的总统科技顾问委员会理事会（ PCAST ）的成员。 BengtNorden 教授在 1999 年瑞典皇家科学院的诺贝尔化学奖颁奖仪式 上说： “ 飞秒化学从根本上改变了我们观测化学反应的方法。围绕在 过渡态周边一百多年的迷雾清晰了，我们现在能直接研究分子中原子 的实际运动。蔡维尔教授，我赏试着解释您的前驱性的工作，如何基 本上改变了我们观测化学反应的方法。从受制于仅用一个隐喻的术语 “ 过渡态 ” 来描述它们，现在我们能同时以时间和空间的图像来描述它 们，它们再也不是不可见的。 ”Zewail 成功地利用了快速飞秒激光光谱 技术，即观察一个分子的原子在化学反应过程中如何移动。飞秒光谱， 泵探针实验 “ 照片 ” （图 23 ），当它们发生化学反应时，用超快激光作 为 “ 闪光 ” 。