Unit1_Prerequisites.pdf
- 2. • Let’s begin this and take a closer at a number of things which depend upon Quantum Physics for their operation • 1. Toaster • 2. Fluorescent Light • 3. Computer & Mobile Phone • 4. Biological Compass • 5. Transistor • 6. Laser • 7. Microscopy • 8. Global Positioning System (GPS) • 9. Magnetic Resonance Imaging • 10. Telecommunication
- 4. WHAT IS THE DIFFERENCE BETWEEN CLASSICAL PHYSICS AND QUANTUM PHYSICS? Classical physics is causal; • complete knowledge of the past allows computation of the future. • complete knowledge of the future allows precise computation of the past. • Not so in quantum physics. • Objects in quantum physics are neither particles nor waves; they are a strange combination of both. • Given complete knowledge of the past, we can make only probabilistic predictions of the future.
- 5. • In classical physics, two bombs with identical fuses would explode at the same time. • In quantum physics, two absolutely identical radioactive atoms can and generally will explode at very different times. • Two identical atoms of uranium-238 will, on average, undergo radioactive decay separated by billions of years, despite the fact that they are identical.
- 6. • There is a rule that physicists often use to separate classical physics from quantum- If Planck's constant appears in the equations, it is quantum physics. If it doesn't, it is classical physics. • Most physicists believe that quantum physics is the right theory, even though many details are yet to be worked out. • NOTE: Classical physics can be derived from quantum physics in the limit that the quantum properties are hidden. That fact is called the "correspondence principle."
- 7. QUANTUM PHYSICS IS THE REVOLUTION THAT OVERTHREW CLASSICAL PHYSICS. DESCRIBING THE DIFFERENCE BETWEEN THEM IS LIKE DESCRIBING THE DIFFERENCE BETWEEN THE BOLSHEVIKS AND THE TSARS. WHERE DO WE EVEN BEGIN? • On the one hand, we have the Newtonian picture of a clockwork universe. • In this paradigm, all of physical reality is a giant machine that ticks forward in time, changing its configuration predictably according to deterministic laws. • Newton saw his god as a mathematician who constructed the cosmos out of physical elements, setting them in motion according to a small set of simple mathematical laws. • These laws are ultimately responsible for all the complexity and diversity of natural phenomena. Likewise, all phenomena, no matter how complex, can be understood in terms of these simple laws. "All discord," wrote Alexander Pope, is "harmony not understood."
- 8. • On the other hand, we have the quantum universe, which from our perspective, seems to resemble more of a slot machine than a clock. • In the quantum universe, we see the machinery as fundamentally probabilistic. If there is harmony underlying quantum discord, it is inaccessible to the experimenter. • In fact, the quantum revolution goes much deeper than merely introducing probability as a fundamental feature. • It altogether trashes the Newtonian clock, replacing it with a completely alien device built out of much more advanced mathematics. .
- 9. SOME EXAMPLES WHICH COULD NOT BE EXPLAINED WITH CLASSICAL PHYSICS • Particles and fields possess well-defined dynamic variables at all times. • Dynamic variables are the quantities used to describe the motion of objects, such as position, velocity, momentum, and energy. • Classical physics presupposes that the dynamic variables of a system are well-defined and can be measured to perfect precision. • For example, at any given point in time, a classical particle exists at a single point in space and travels with a single velocity. • Even if the exact values of the variables are uncertain, we assume that they exist and only take one specific value.
- 10. • Particles as point-like objects following predictable trajectories. • In classical mechanics, a particle is treated as a dimensionless point. • This point travels from A to B by tracing out a continuous path through the intermediate space. • A billiard ball traces out a straight line as it rolls across the table, a satellite in orbit traces out an ellipse, and so on. • The idea of a definite trajectory presupposes well defined dynamic variables, and so once the first point above is abandoned, the idea of a definite trajectory must be discarded as well.
- 11. • Dynamic variables as continuous real numbers. • In classical physics, dynamic variables are smoothly varying continuous values. • Quantum physics takes its name from the observation that certain quantities, most notably energy and angular momentum, are restricted to certain discrete or 'quantized' values under special circumstances. • The in-between values are forbidden. • Particles and waves as separate phenomena. • Classical physics has one framework for particles and a different framework for waves and fields. • This matches the intuitive notion that a billiard ball and a water wave move from A to B in completely different fashions. • In quantum physics however, these two phenomena are synthesized and treated under a unified, magnificent framework. • All physical entities are particle/wave hybrids
- 12. • Newton's Second Law. A radically different dynamics must be developed that is governed by a very different equation of motion. • Predictability of measurement outcomes. In classical physics, the outcomes of measurements can be predicted perfectly, assuming full knowledge of the system beforehand. • In quantum mechanics, even if you have full knowledge of a system, the outcomes of certain measurements will be impossible to predict.
- 13. QUANTUM ERA • Not quite everything from classical physics was redundant. • The conservation laws were preserved (or Great Conservation Principles as Feynman called them, always capitalized to highlight their centrality to all areas of physics). • Quantum physics conserves things like momentum, energy, and electric charge as perfectly as classical physics. • Conservation laws- more mathematically - Hamiltonian and Lagrangian formulations of classical mechanics. • Erwin Schrodinger chose to adapt the Hamiltonian formalism which led to his eponymous equation. • Richard Feynman adapted Lagrangian mechanics which led to his path integral formulation. • Heisenberg developed his own esoteric approach called matrix mechanics.
- 14. REPLACING NEWTON'S COMPONENTS WITH SCHRODINGER'S: • Particles possess a wave function Ψ(x,t)Ψ(x,t) at all times. The wave function assigns a complex number to each point in space at each moment in time. This function contains within it all available information about the particle. Everything that can be known about the particle's motion is extracted from Ψ (x,t)Ψ(x,t). • To recover dynamic information, we use Born's rule and calculate ΨΨ∗ΨΨ∗ to get the probability density of the particle's position, and we calculate ΨΨ∗ to get the probability density of the particle's momentum, where ϕ(p,t)ϕ(p,t) is the Fourier transform of the wave function. • This is a radically different approach to kinematics than in classical mechanics, which describes particles by listing off the values of the dynamic variables.
- 15. • Trajectories are replaced with wave function evolution. As the wave function changes in time, so do the probabilities of observing particular positions and momenta for the particle. The evolution equation is the time-dependent Schrodinger equation • Dynamic variables are Hermitian matrices. Instead of real-valued, continuously evolving dynamic variables, Schrodinger uses fixed Hermitian matrices (or self-adjoint operators) to represent observable quantities. Each observable such as position, momentum, energy, etc has a corresponding matrix/operator. The eigenvalues of the matrix/operator determines the allowed values of the corresponding observable. The energy levels of atoms, for example, are eigenvalues of the Hamiltonian operator. This is another completely radical shift from how classical physics treats motion.
- 16. • Unification of particles and waves. A mathematical analysis of the Schrodinger equation reveals that it has wavelike solutions, and so particles propagate as waves. This means that we shouldn't picture particles as tiny spheres bouncing around their environment. The closest you can get to visualizing a particle is by visualizing its wave function. As stated previously in the first point above, the wave function assigns a complex number to each point in space. • The time-dependent Schrodinger equation replaces Newton's second law. • .
- 17. • Measurement is random. Even if you have full knowledge of a quantum system prior to measurement (i.e. you know Ψ(x,t)Ψ(x,t)), you still will not be able to predict the outcomes of measurements in general. The outcome of the measurement is probabilistic. • The possible outcomes are determined by the eigenvalues of the operator you are observing (see point 3), and the probability of each outcome is determined by the projection of the wave function onto the eigenvectors of that operator
- 18. THREE FAILURES OF CLASSICAL PHYSICS
- 19. BLACK BODY RADIATION ▪ Any object with a temperature above absolute zero emits light at all wavelengths. ▪ If the object is perfectly black (so it doesn't reflect any light), then the light), light that comes from it is called that blackbody radiation. ▪ The energy of blackbody radiation is not shared evenly by all wavelengths of light. ▪ The spectrum of blackbody radiation (below) shows that some wavelengths get more energy than others. ▪ Three spectra are shown, for three different temperatures. (One of the curves is for the surface temperature of the Sun, 5770 K.)
- 20. SOME EXPERIMENTAL FACTS ABOUT BLACKBODY RADIATION: • The blackbody spectrum depends only on the temperature of the object, and not on what it is made of. An iron horseshoe, a ceramic vase, and a piece of charcoal --- all emit the same blackbody spectrum if their temperatures are the same. • As the temperature of an object increases, it emits more blackbody energy at all wavelengths. • As the temperature of an object increases, the peak wavelength of the blackbody spectrum becomes shorter (bluer). For example, blue stars are hotter than red stars. • The blackbody spectrum always becomes small on the left-hand side (the short wavelength, high-frequency side).
- 21. •The explanation of classical physics: • Light is an electromagnetic wave that is produced when an electric charge vibrates. ( Strictly speaking, "vibrates" means any change in how the charge moves --- speeding up, slowing down, or changing direction.) • Heat is just the kinetic energy of random motion. • In a hot object, electrons vibrate in random directions and produce light as a result. • A hotter object means more energetic vibrations and so more light is emitted by a hotter object --- it glows brighter. • But classical physics could not explain the shape of the blackbody spectrum.
- 22. • The electrons in a hot object can vibrate with a range of frequencies, ranging from very few vibrations per second to a huge number of vibrations per second. • In fact, there is no limit to how great the frequency can be • Classical physics said that each frequency of vibration should have the same energy. • Since there is no limit to how great the frequency can be, there is no limit to the energy of the vibrating electrons at high frequencies. • This means that, according to classical physics, there should be no limit to the energy of the light produced by the electrons vibrating at high frequencies. • WRONG!! Experimentally, the blackbody spectrum always becomes small at the left-hand side (short wavelength, high frequency).
- 23. MAX PLANCK THEORY • At about 1900, Max Planck came up with the solution. • He proposed that the classical idea that each frequency of vibration should have the same energy must be wrong. • Instead, he said that energy is not shared equally by electrons that vibrate with different frequencies. • Planck said that energy comes in clumps. • He called a clump of energy a quantum. • The size of a clump of energy --- a quantum --- depends on the frequency of vibration. • Planck's rule for the a quantum of energy for a vibrating electron: • energy of a quantum = (a calibration constant) x (frequency of vibration)
- 24. • E = hf • where h, the calibration constant, is today called Planck's constant. Its value is about 6 x 10-34, very tiny! • So how does this explain the spectrum of blackbody radiation? Planck said that an electron vibrating with a frequency f could only have an energy of 1 hf, 2 hf,3 hf, 4 hf, ... ; that is, • energy of vibrating electron = (any integer) x hf • But the electron has to have at least one quantum of energy if it is going to vibrate. If it doesn't have at least an energy of 1hf, it will not vibrate at all and can’t produce any light. • At high frequencies the amount of energy in a quantum, hf, is so large that the high-frequency vibrations can never get going! • This is why the blackbody spectrum always becomes small at the left-hand (high frequency) side.
- 25. THE PHOTOELECTRIC EFFECT • When light shines on the surface of a metallic substance, electrons in the metal absorb the energy of the light and they can escape from the metal's surface. • This called the photoelectric effect, and it is used to produce the electric current that runs many solar-powered devices. Using the idea that light is a wave with the energy distributed evenly throughout the wave, classical physicists expected that when using very dim light, it would take some time for enough light energy to build up to eject an electron from a metallic surface. • WRONG!! Experiments show that if light of a certain frequency can eject electrons from a metal, it makesno difference how dim the light is. There is never a time delay. • enough energy to eject an electron from a metallic surface, then electrons will always be ejected immediately after the light is turned on and thephotons hit the metal
- 26. EINSTEIN’S THEORY • In 1905, Albert Einstein came up with the solution. • If Max Planck's idea that energy comes in clumps (quanta) is correct, then light must consist of a stream of clumps of energy. • Each clump of light energy is called a photon, said Einstein, and each photon has an energy equal to hf (Planck's constant times the frequency of the light). • Therefore the energy of light is not evenly distributed along the wave, but is concentrated in the photons. • In 1905, Albert Einstein came up with the solution. • A dimmer light means fewer photons, but simply turning down the light (without changing its frequency) does not alter the energy of an individual photon. • Single photon has enough energy to eject an electron from a metallic surface, then electrons will always be ejected immediately after the light is turned on and the photons hit the metal • So for a specific frequency light, if a singlephoton has enough energy to eject an electron from a metallic surface, then electrons will always be ejected immediately after the light is turned on and thephotons hit the metal.
- 27. THE HYDROGEN ATOM • When a small tube of hydrogen gas is heated, it begins to glow and emit light. • Unlike the blackbody radiation that comes from a hot dense solid or gas, this light consists of just a few colors (wavelengths): a red wavelength, a turquoise, and several violets. • Classical physicists at the beginning of the century thought they should certainly be able to understand hydrogen, since it is the simplest atom. • Hydrogen consists of a positively charged proton at the center, with a negatively charged electron orbiting around it. • The electrical attraction between the positive proton and the negative electron keeps the electron in orbit, just like the gravitational attraction between the Sun and the Earth holds the Earth in orbit. • There was just one problem. Classical physics said that because the orbiting electron is constantly changing direction, it should emit electromagnetic radiation --- light. • As a result, the electron should be continually losing energy. • In fact,physicists calculated that the electron should lose all of its energy and spiral down into the proton in only about 0.000000000001 second! In other words, atoms should not exist longer than a mere 10-12 seconds. WRONG!!
- 29. NIELS BOHR THEORY • Niels Bohr provided an explanation in 1913. In the Bohr model of the hydrogen atom, the electron can't orbit the proton in any size orbit it pleases. • There are only certain allowed orbits, and each allowed orbit has a certain radius and a certain energy. • Bohr invented a rule that allowed him to calculate the size and energy of each orbit. If you are curious, Bohr's rule said that • 2π x (electron mass) x (electron orbital speed) x (orbit radius) = (any integer) x h • The integer would be 1 for the smallest orbit, 2 for the next orbit out, and so on.) • Bohr also made up a new rule to explain the stability of the hydrogen atom --- why it could last longer than 0.000000000001 second. • He said that when an electron is in an allowed orbit, the electron will not produce electromagnetic radiation. • Bohr did not explain why, he just proposed a new law of nature. • His new model of hydrogen gave wavelengths for hydrogen gas that precisely agreed with what was measured.
- 30. QUESTION: IF THE ELECTRONS DO NOT PRODUCE LIGHT WHEN THEY ARE IN THEIR ALLOWED STABLE ORBITS, WHERE IS THE SOURCE OF THE LIGHT THAT COMES FROM HYDROGEN? • Answer: According to Bohr, electrons have more energy when they are in larger orbits. • If an electron falls from a larger orbit down to a smaller orbit, it loses energy. • According to the law of conservation of energy, the energy lost by the electron must go somewhere. • Bohr explained that a photon carries away the lost energy from the hydrogen atom; that is, • photon energy = (electron energy in larger orbit) - (electron energy in smaller orbit) If a photon strikes an atom, the atom can absorb the photon and its energy if (and only if) the photon's energy is exactly equal to thedifference between two orbital energies. In this case, an electron uses the photon's energy to jump from the smaller orbit up to the larger orbit. This is called a quantum jump.
- 31. A photon is absorbed by the atom, whichgains the photon’s energy.The electron usesthis energy to jump up to a higher orbit The electron falls down to a lower orbit and the atom loses energy. A photon carries away the energy lost by the atom.
- 32. WAVES OR PARTICLES? BOTH! • When light passes through a double-slit, an interference pattern consisting of bright bands and dark bands is seen on a screen. This is produced when the waveform one slit combines with the wave from the other slit. If two wave crests meet at the screen, the waves add and you get a bright band. If a wave crest from one slit meets a wave trough from the other slit, the waves cancel and you get a dark band. This proves that light is a wave. • On the other hand, the photoelectric effect proves that light consists of massless particles called photons. • So which is it? Is light a wave or a stream of particles? The answer is "Yes!" • Light acts like a wave if you want to know how it propagates, how it travels from one place to another. To describe how light travels from the double slits to the screen, you have to use the wave characteristics of light. • Light acts like particles (photons) if you want to know how light interacts with matter. To describer how light interacts with the electrons in a metal and how it ejects them from the metal's surface, you have to use the particle characteristics of light. • We say that light exhibits a wave-particle duality. It can behave like either waves or particles (but not both at the same time), depending on the situation.
- 33. • Thinking about the photoelectric effect again, how can a photon (which has no mass) knock an electron about? Einstein used his theory of relativity to show that even massless photons have momentum. Newton defined momentum = (mass) x (velocity) for a particle with mass, but Einstein was able to show that the momentum of a massless photon depends on its wavelength: • The smaller the wavelength, the greater the momentum of the photon.
- 34. DE BROGLIE PRINCIPLE • In 1923, Prince Louis de Broglie of France had an idea. Maybe the wave-particle duality applies to everything in nature. He proposed that everything propagates like a wave, and that everything interacts like a particle. • What do you mean by the wavelength of an electron, or the wavelength of a baseball? • DeBroglie rewrote Einstein's formula for the momentum of a photon and applied it to a particle with mass: • Planck's constant, h, is so tiny that we don't notice the wavelength of a thrown baseball, which is only about 10-35 meters! But an electron's mass is also tiny, so it has a wavelength about 10,000 times shorter than the wavelength of visible light. This is useful, because microscopes that use electron waves instead of light waves can see several thousand times more detail! • The proof that electrons propagate like a wave came when electrons were passed through a double slit and counted as they hit a screen. If the electrons traveled like a stream of particles, they would have simply piled up at two locations behind the two slits. But they didn't. They showed a double-slit interference pattern, bright bands and dark bands just like the ones produced by light waves. Without a doubt, electrons exhibit the wave-particle duality of nature. • In fact, every massive object exhibits the wave-particle duality of nature. It just isn't noticeable on the large scale of our everyday world.