From a visible light perspective, a body is black if it absorbs all light that strikes it in the visible part of the spectrum.

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4/28 Black Body Radiation From a visible light perspective, a body is black if it absorbs all light that strikes it in the visible part of the spectrum.

1 4/28 Black Body Radiation From a visible light perspective, a body is black if it absorbs all light that strikes it in the visible part of the spectrum. A white body is white because it reflects all of the light that strikes it in the visible part of the spectrum In the context here, a black body is one that absorbs all of the electromagnetic radiation that strikes it the whole electromagnetic spectrum A black body has energy and, consequently, will radiate. Recall heat transfer via radiation. To do experiments, we need to model a black body: Take a metal box, paint the inside black, and put a small hole in the side, and the hole models a black body. Any radiation incident on the hole will go through and be trapped inside (as long as the wavelength is small compared with the size of the hole). If there is energy stored inside the box in the form of electromagnetic standing waves, some will leak out that is the black body radiation. Want to know how much radiation we get from the black body at each individual wavelength make the measurement and get the following plot. Calculate the Black Body Spectrum: Apply the laws of thermodynamics to the radiation inside the box. Note that thermodynamics applies to radiation in the same way it applies to gases. Did not get the same measured spectrum! The plot shows the intensity of the radiation at given wavelength and temperature at given wavelengths. The Rayleigh-Jeans dashed curve shows the thermodynamic calculation, while the solid curve is from experiment. Agreed for long wavelengths, but went to infinity for short wavelengths ultraviolet catastrophe. As we know, this disagreement between theory and experiment means that something is wrong with the theory! Photoelectric Effect Light shined on a metal surface gives electrons enough energy to cross the surface. In the figure, the semicircular

2 cathode has light shined on it, which ejects electrons from it. These electrons travel to the anode and produce an electric current measure by the ammeter. We can measure maximum kinetic energy of the emitted electrons. The battery voltage is adjusted until the current measured by the ammeter goes to zero. When this happens, the potential difference across the tube is just large enough to stop the most energetic electrons ejected from the cathode. Multiplying this potential difference by the fundamental charge will give the energy of these electrons. Expected classically: Should take some time for electrons to absorb enough energy from the incoming light wave to cross the surface. Expect a delay for the current to start. Observed: No delay. Expected Classically: Increasing the intensity of the light should increase the kinetic energies of the emitted electrons. Observed: No increase in electron kinetic energy but more electrons are emitted. Expected Classically: The energy of the emitted electrons should be independent of the frequency of the light. Observed: The kinetic energy of the emitted electrons increases with the frequency of the light. Expected Classically: No matter what the frequency of the light, shining the light long enough on the surface should emit electrons. Observed: Below a certain cutoff frequency, no electrons are emitted no matter how intense the light. Resolutions of Loose Ends in the order resolved. 1. Black Body Radiation Max Planck The classical analysis involved applying thermodynamics to the radiation inside the cavity radiator modeling the black body. Planck did the same thing with one additional assumption: He assumed that the atoms in the cavity walls could only deliver energy to or take energy from the radiation field in discrete chunks. Assumed the energy of a chunk proportional to the frequency: He found agreement with the observed spectrum for This constant is called Planck s Constant.

3 Photoelectric Effect: Einstein When Einstein looked at the observed characteristics of the photoelectric effect, he thought that it looked like the collision between two particles. He assumed that light was composed of particles called photons The energy of each photon was proportional to its frequency. Expect no delay as observed because a photon will strike an electron and immediately drive it across the surface. Increasing the intensity of the light doesn t increase the energy per photon (as long as the frequency doesn t change), it increases the number of photons. Since there are more photons, there will be more electrons ejected. Since each photon has the same energy so will the ejected electrons. Increasing the frequency of the light will increase the energy per photon, which will increase the energy per electron. If the frequency of light is too low, a given photon will not have the energy needed to drive an electron across the surface thus, a cutoff frequency is predicted. Using Planck s constant provided quantitative agreement. Discrete Nature of Atomic Spectra Bohr Two early models of the atom: Thomson s Plum Pudding model and Rutherford s Nuclear Model. Plum Pudding Model: positive jelly with electrons embedded in it. Normally, the electrons sit at rest in the jelly. Atoms emit light when electrons oscillate back and forth around their equilibrium positions in the jelly. The Plum Pudding Model of the hydrogen atom is shown at right. Nuclear Model: Positively charged nucleus with electrons in orbit about it. At the time, it wasn t clear how an atom such as this could emit light. The picture at right shows the nuclear model of the hydrogen atom with the positive nucleus, in this case, being a single proton. The Plum Pudding Model was the preferred model because the nuclear model is unstable classically. The electron in orbit is constantly accelerating, which means constantly produce electromagnetic waves. It would rapidly radiated its energy away and the atom would collapse. Roughly s. As usual, to decide between two scientific ideas, we must do an experiment to see which correctly predicts the results of the experiment.

4 Rutherford Scattering Experiment: Send alpha particles (helium nuclei) at a gold sheet. The plum pudding model predicted that the alpha particles should pass through he sheet undeflected since there would be no concentration of charge for the particles to interact with. The nuclear model predicted that most of the helium nuclei would be undeflected because the gold atoms would be mostly empty space, but some of them would be deflected a lot and some even bounce back if they should come close to the nucleus of the gold atoms. The results of the scattering experiment supported the nuclear model of the atom; some of the helium nuclei suffered large deflections. More importantly, the angular distribution of the deflections agreed with theoretical predictions based on Newton s second law with the force take as force between charged particles as given by Coulomb s law. It still wasn t clear how such an atom could be stable, but the model had to be correct because it correctly predicted the results of the scattering experiment. The first step in the resolution of the problem came from Niels Bohr. Bohr knew that something in the atom was quantized can only take on certain values. He noted that Planck s constant has units of angular momentum. He found that if he assumed the angular momentum in the atom was quantized in units of Planck s constant divided by 2, he correctly predicted the discrete spectrum of hydrogen. That is, he derived the Rydberg formula shown above, and found the Rydberg constant in terms of fundamental constants such as the fundamental charge, the mass of the electron, and Planck s constant. He used Newton s second law applied to the circular motion of the electron in hydrogen with the force being given by Coulomb s law. Bohr found that the electron in the hydrogen atom could only have certain discrete energies; it could have no energy values between these values. The electron in the atom will not emit light while in an orbit associated with one of these discrete values. The atom emits light when an electron jumps from one orbit in the atom to a lower atom in the orbit. The electron loses energy when it does so. To conserve energy, it creates a photon, which is emitted by the atom. The energy of the photon is the same as the difference in energy between to the two orbits. The frequency of the photon is then found from Planck s energy relation. An absorption spectrum is produced when a continuous spectrum passes through a dilute gas. A photon at the right frequency that is, it has energy equal to an energy difference in the atom will be absorbed by the electron in the atom and will cause the electron to jump from a lower energy state to a higher energy state. Since the photon has been absorbed, the wavelength associated with it will be missing in the spectrum.

Bohr s analysis correctly predicted the spectrum of hydrogen, but it didn t answer the question as to why the angular momentum of the electron in the atom should be quantized.

5 Bohr s analysis correctly predicted the spectrum of hydrogen, but it didn t answer the question as to why the angular momentum of the electron in the atom should be quantized. To answer this question, we must answer another question first. Burning Question: What is light? Is it a wave or a particle? Bohr answered this question with the Principle of Complementarity: Light is composed of quantum mechanical objects called photons that some times behaves like a wave and some times like a particle but never both at the same time. DeBroglie Matter Waves if light, long considered to be a wave, can behave like particles, can electrons and protons, long considered to be particles, behave like waves? DeBroglie predicted the wavelength of a particle to be the ratio of Planck s constant to the momentum of the particle: Yes, electrons can behave like waves! This was confirmed by experiments showing electron diffraction from crystal lattices. Revisit Bohr he assumed that angular momentum in the atom was quantized. If we think of the electron orbiting the nucleus as a wave, to do so, it must constructively interfere with itself. This constructive interference condition leads to the quantization of the angular momentum. Schrodinger put together a wave equation that these matter waves had to obey. It was based on conservation of energy and the fact that he knew the solution for a particle confined to a box, because it should be the same as that for a wave on a string fixed at both ends. Solving this equation for the hydrogen atom produced the same spectrum that Bohr had found. In addition, it is possible to solve it approximately for more complex atoms and correctly predicts their spectra as well. One solves for a quantity called the wave function. How to interpret the wave function? We don t know! There are a variety of different interpretations, most of which are too mathematically technical for us. I will give two. 1. The Copenhagen Interpretation: We interpret the wave function as giving the probability that a particle will be at places in space. Actually, it is the square of the wave function. Where it is large, the particle is likely to be there; where it is small, unlikely to be there. When a

6 measurement is made, we find the particle at a specific location we say that the wave function collapses. The figure at right shows the probability distribution for the electron in the ground state of hydrogen. 2. Many Worlds Interpretation: Similar except when a measurement is made. Suppose there are two possibilities for a measurement for a quantity, say, white and black. When the measurement is made, the universe splits in two. In one universe the measurement gives white, and in the other universe, the measurement gives black. We can t decide which of interpretations is correct because they are all designed to be consistent as far as predictions are concerned. Thus, they will all either agree or disagree with any experiment that might be performed. It should be clear that quantum mechanics is strange. Here are some examples. 1. The Heisenberg Uncertainty Principle. Because we are describing a particle as a wave, it is impossible to know both its momentum and position simultaneously to ultimate accuracy. Recall that De Broglie related momentum and wavelength describe the particle as a wave packet. To better define the position, we need to squeeze the wave packet smaller, which provides less room for waves and, therefore, a more poorly defined wavelength and momentum. To improve the knowledge of momentum, we must spread out the wave packet, which means that knowledge of position is reduced. It is impossible in principle to know both the position and momentum to perfect accuracy of a particle at the same time. Two interpretations: 1. Measuring a system disturbs the system: Think of trying to locate a basketball on the floor of gym when it is pitch black. Throw ball bearings until you hear on hit the ball. But this will cause the ball to move. 2. Particles don t have well-defined positions and momenta: A particle might exist in more than one place at the same time and might have more than one momentum at the same time. This appears ridiculous to us with our experience in the macroscopic world, but must be the case in order for quantum physics to correctly predict the outcome of experiments. Note that this interpretation gives us a universe of free will. Feynman s Path Integral Formulation. A particle traveling from point A to point B follows every possible path; interference between these paths produces the most probable straight line path. Suppose we look at law of reflection. Each line in the figure at right shows a path for photons from point A to the

7 mirror and then to point B. If we add up all the paths, the ones far from the equal angles of incidence and reflection cancel, leaving only the path that obeys the law of reflection. Note that this means it is not proper to describe the paths of electrons in an atom by circles around the nucleus. Instead, we must describe them as electron clouds, which display the probability of find the electron at various points. Classically Forbidden Regions Tunneling In classical physics, it is impossible for the kinetic energy of a particle to be less than zero since KE = ½mv 2 and the square of any real number is positive. In quantum mechanics, it is possible for a particle to exist in a classically forbidden region, that is, to tunnel through such a region. In the figure at right, the vertical rectangle represents a wall, and the wiggles represent the wave function. Note that the wave function is not zero within the wall, which means that there is some probability that the particle will be found there. Note, too, the wiggles outside the wall, which means that the particle can be found there as well. We will use this idea to understand certain types of radioactivity. Virtual Particles Quantum mechanics tells us that if something can happen, sooner or later, it will. Quantum mechanics predicts that virtual particles are constantly popping into and out of existence. There is a version of the uncertainty principle that involves time and energy. It is OK for energy to be not conserved as long as it is not conserved over a short enough time. Thus, a particle and its antiparticle can pop into existence and then annihilate each other a very short time later. We know this to be the case because of the Casimir effect. Put two metal plates very close together. Because the wave functions for the virtual particle between the plates must go to zero at the plates (like the fixed ends of a string) the number of virtual particles between the plates will be smaller than the number outside the plates. There should be a small inward pressure on the plates, which produces a force tending to push the plates together. This force has been measured, providing evidence for the existence of virtual particles. The Periodic Table Pauli and Spin We can now understand the periodic table. To move from one element of the table to the next, we add an electron. Add an electron to hydrogen, and helium results. Up to a point... we also have to add a proton and a couple neutrons to the nucleus. But, adding electrons and making appropriate changes in the nucleus will move among the elements on the table. One would expect, however, that all the electrons would go into the same quantum state or energy state because all the electrons would want to be in the lowest energy state. But if they did, all elements would have pretty much the same chemistry. Something must prevent added electrons from going into the same state as other electrons. There is: The Pauli Exclusion Principle: No two electrons can be in the same quantum state. Each added electron goes into the lowest energy available quantum state, that is, the lowest

8 energy state that is unoccupied. Because each added electron is in a different state, the chemistries of the elements are different. What is it about electrons that keep them from occupying the same state? It is difficult to explain, but it has to do with a quantity called spin. While it is not proper to think of an electron as a small spinning ball, it does possess an intrinsic angular momentum that we call spin. The electron has spin ½ in units of Planck s constant divided by 2 ; we call it a spin ½ particle. It turns out that all spin ½ particles (and, in fact, any particles with spin that is one-half of an odd integer) must obey the Pauli Exclusion Principle. Particles with integer spin do not. Photons, for instance, have spin one, that is, one h/2 unit of spin. Terminology: particles with integer spins are called bosons; particles with half odd integer spin are called fermions. Quantum Computing To describe a two-state system that can evolve from one to other, one must assume that it is partly in one state and partly in the other to correctly describe what happens. This means that a radioactive nucleus must be described by a combination of two states, one in which it has not decayed and one in which it has decayed. On the quantum level, the nucleus is partly decayed and partly not! Schrodinger thought this to be absurd. He imagined the following... Put a cat in a closed box along with a container of poison gas and a radioactive nucleus. The system is set up so that when the nucleus decays, the gas will be released and kill the cat. Since the nucleus is partly decayed and partly not, the cat must be partly dead and partly not! (Note: resolved by decoherence.) In a normal computer, a given transistor can ether be on (letting current though a circuit) or off (blocking current). When on, a 1 is represented; when off, a zero. Large arrays of transistors representing ether 1's or 0's can represent binary numbers and circuits can be designed to do binary arithmetic, which can be translated in the decimal arithmetic with which we are familiar. In a quantum computer, each transistor can be partly on and partly off, meaning the binary digit is partly a 1 and partly a 0. This means that two arithmetic processes can be working in parallel. Why stop there? Imagine a quantum computer with fundamental parts that can be in many states: 10, 100, 1000 or more. Each one will be partly in all states, and many parallel processes can be can be carried on at once. Note that, if a quantum computer is ever built, current computer security won t be secure! Einstein and Bohr Bohr embraced the statistical interpretation of quantum mechanics, but Einstein abhorred it. They argued (in print, in papers published in physics journals). Every objection that Einstein found, Bohr was able to reconcile. Eventually, though, after both men had died, Bohr prevailed. His view was supported by experiment. Chapters 33/34 Nuclear Physics Terminology: Represent a nucleus

9 by X = chemical symbol, Z = number of protons in the nucleus,(atomic Number) and A is the total number of neutrons and protons in the nucleus (Mass Number). Z is called the atomic number of the nucleus of the atom of which it is a part. This is the number that is in the upper left corner of the cells in the periodic table.

Planck s Quantum Hypothesis Blackbody Radiation

Planck s Quantum Hypothesis Blackbody Radiation The spectrum of blackbody radiation has been measured(next slide); it is found that the frequency of peak intensity increases linearly with temperature.