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Quantum Physics in three units • I: How light interacts with particles • II: Simple systems Quantum Physics Part II – a) Atoms ,their structure, interaction with light – b) Quantum wave functions and quantum tunneling • III: More complex systems – a)Bosons, Fermions, and Bose Condensates – b)Superconductivity and superfluidity – c)Quantum entanglement and quantum computing Review of What We Learned From Quantum Part I • • • • • • • Black Body Radiation Photoelectric Effect Einstein’s Interpretation of these two results Wave-particle duality Diffraction and interference DeBroglie’s Hypothesis of matter waves The uncertainty principle Black Body Radiation • What it is – The electromagnetic radiation given off by a hot object – Does not agree with classical physics – Ultraviolet Catastrophe • Planck’s Solution Bright Line Spectra • Spectra of Atoms – Bright Line, Absorption, From The Sun – Mathematical Models of Results Photoelectric Effect • Shine light on a metal surface • No electrons emitted unless the frequency is above a critical threshold value • Once past the threshold, the amount of emitted electrons is proportional to the optical power – If the energy of light come in quanta – small packets = hf then he can explain the results 1 Einstein’s Interpretation New Discoveries • Light comes in quantized bundles called photons • The energy is give by: E � hf • The total power of a beam of photons is E # photons given by: P� t � t • Bright Line Spectra • Absorption Spectra • Spectrum from the Sun hf • The threshold effect for emitting electrons is due to the work function for the metal: KEmax � hf � � Bright Line Spectra Prism Observer Atomic Spectra: Key to the Structure of the Atom A very thin gas heated in a discharge tube emits light only at characteristic frequencies. Light Source 1752 – Scottish Physicist Thomas Melville • Compares the spectrum for a pure hot gas (not a flame) to the spectrum of a black body • Surprising result: Line Spectra Joseph von Fraunhofer German Optician • Takes a spectrum of the sun • Sees an almost countless number of dark lines superimposed on a continuous background. Modern- Super High Resolution Spectrum 2 Gustav Kirchoff German Physicist 12 March 1824 – 17 October 1887 First to show that the absorption lines from an emission spectrum lines up with the absorption lines in an absorption spectrum. But…By very careful work, he was able to determine that there were some spectral absorption lines on the sun had no matching lines on earth. A worldwide search began to find the elusive element: Result – Discovery of “Helium”. How do the spectra relate to the atoms? Anders Jonas Ångström 13 August 1814– 21 June 1874 - Swedish • In 1862 Angstrom studies spectrum of Hydrogen in great detail. Question: How can we make sense of all of these lines of different frequencies of light? Helium comes from Greek word “Helios” (sun) Johann Jakob Balmer Swiss Mathematician Additional Hydrogen Wavelengths May 1, 1825 – March 12, 1898 Balmer finds that he can represent the frequencies of all observed lines from Hydrogen by a simple formula: � 1 1 � 1 � R� 2 � 2 � �2 n � � � � n1=1 n2=1 n3=1 A constant An integer > 2 Using this equation, Balmer predicts additional lines before they are discovered. Cathode Rays Late 1800’s: Studies were being conducted in what happens when electricity is discharged into a rarefied gas. All wavelengths agree with a generalized model � 1 1 1� � R�� 2 � 2 �� n n � 2 � � 1 1890’s: A result waiting for an explanation! J. J. Thomson British Physicist and Nobel Laureate 18 December 1856 – 30 August 1940 How Thomson Discovers the Electron (11:08) (This is the same guy who discovered Thomson Scattering) Thomson Discovers Electron (2:53) Hypothesis: The Plum Pudding Model of the Atom: 3 Ernest Rutherford 30 August 1871 – 19 October 1937 New Zealand-born British chemist and physicist who became known as the father of nuclear physics. He is considered the greatest experimentalist since Michael Faraday. Lab: ROLLING WITH RUTHERFORD 27.10 Early Models of the Atom Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom. He scattered alpha particles – helium nuclei – from a metal foil and observed the scattering angle. He found that some of the angles were far larger than the plum-pudding model would allow. 27.10 Early Models of the Atom The only way to account for the large angles was to assume that all the positive charge was contained within a tiny volume – now we know that the radius of the nucleus is 1/10000 that of the atom. The work was carried out by one of Hans Geiger’s graduate students, Earnest Marsden. Gold Foil Experiment 9:07 Rutherford’s Result Was A Total Surprise “This is quite the most incredible event that has ever happened in my life. It was almost as if you fired a 15” shell at a piece of tissue paper and it came back and hit you! 1 1Introducing 27.10 Early Models of the Atom Therefore, Rutherford’s model of the atom is mostly empty space: Quantum Theory, J.P. McEvoy, Oscar Zarate 4 Rutherford’s Model Had A Lot of Critics Bohr Model of Atom But the biggest problem with his model was yet to be explained. How Do Radio Transmitters Work? An accelerating charge produces electromagnetic radiation. If the charge oscillates with a specific frequency, then the radiation will have the same frequency. Classical Physics- All accelerating charges produce electromagnetic energy. Niels Bohr (The Great Dane) Danish Physicist Bohr arrives in England in 1911 and initially works with J.J. Thomson. However, the two do not get along with each other. When he arrived he spoke almost no English. He brought a dictionary and the complete works of Charles Dickens to learn the language. The Grandfather of Quantum Physics 7 October 1885 – 18 November 1962 Alice and Bob: How Can Atoms Exist? However, Bohr Hits it Off With Rutherford. Niels Bohr (The Great Dane) 5 Niels Bohr (The Great Dane) Physicists Can Learn From Unit Analysis 1 2 mv 2 Energy: Frequency f Planck’s Constant Angular Momentum �kg ��m�2 � �kg ��m�2 1 �s� �s� �s�2 1 �s� E � hf Units for h : �kg � L � mvr �kg � �m� �m� �s � �m� �m� �s � Planck’s Constant has units of Angular Momentum! Is this just a coincidence??? J. J. Nicholson 1912 Angular Momentum • He attempts to apply a quantum theory to Thomson’s Plum Pudding model. • He decides that the thing to quantize in the atom is angular momentum of the electron. • However, he is unable to reconcile these two ideas. Bohr’s Great Breakthrough • In 1913 Bohr combines three ideas together. – The line spectra formula from Balmer – The quantizing of angular momentum from Nicholson – The need to define stable orbits for Rutherford’s model Bohr’s First Postulate 6 Principle Quantum Number, n The angular momentum can not take on any value (as would be the case for classical physics). The angular momentum must be an integer multiple of h/2π L1 � 1� Finding the radius of the orbit This part is done using classical physics. It is very similar to calculating the orbits of planets around the sun Planets: Gravity Atom: Electromagnetic M �� m Centripetal Force Fc � m planet v 2 r qelectron � q proton � e qnucleus � Ze Provided By FG � Gmsun m planet r 2 Solution for planets L2 � 2� L � m planet v planet r ... r� Ln � n� 2 L GMm 2 Conservation of Angular Momentum is Kepler’s 2nd Law Centripetal Force Provided By m v2 Fc � electron r FE � KZe 2 r2 Solution for atoms L � melectron velectron r r� �2 L2 rn � n2 2 KZm elece 2 KmZe Quantinization of Angular Momentum is Bohr’s 1st Postulate. Adding Energy to Bohr’s Model Bohr defines radius of each orbit rn � �2 Kmelec Ze 2 n2 • Once the radius and the angular momentum are known, it is fairly straightforward to determine the total energy of the atom depending on which orbit the electron is in. • Procedure: – Determine the Kinetic Energy – Determine the Potential Energy – Add them together Bohr Derives the Balmer Formula Bohr’s 2nd Postulate � 1 1 1 � � R�� 2 � 2 �� � � n1 n2 � The value for R calculated by Bohr agrees with the value calculated by Balmer within a few percent. R depends on Planck’s constant, the speed of light, and the fundamental constant of electromagnetic attraction between charged particles. The energy of the atom is quantized. 7 How We Understand the Bohr Atom - 1913 1. The atom is quantized by a single quantum number “n”, which relates to the angular momentum of the state that the electron is in. 2. The same number defines the energy of the atom. 3. Absorption and emission of a photon can only occur if the energy level between two states is exactly equal to energy of the photon being absorbed or emitted. 4. The quantum number, n, defines the “shell” for the electron. Bohr’s Formula for Energy • Overall energy levels: Z is the number of protons in the nucleus. Z2 En � �13.6 2 eV n n is a quantum number. It can be 1, 2, 3, … Chladni Plate Vibrations En � �13.6 1 eV n2 When n=∞, E=0, electron is ionized from atom. What might Chladni patterns look like in 3D? The patterns of the hydrogen atom are complex, but much simpler than these! More complicated structure • Additional spectral lines were observed • It was proposed by Arnold Sommerfeld that these were due to the fact that the orbitals were not simply circular in shape. • A new quantum number was used. It was called the l quantum number or the azimuthal quantum number. • These were called subshells. • For any given quantum number n, the possible subshells range from l=0 to l=n-1 • Again, the angular momentum was determined by the value according to L2 � � 2l �l � 1� 8 Electron cloud, or probability distribution, for n = 2 states in hydrogen Orbital Shapes- Derived Later Orbitals How the Periodic Table relates to the azimuthal quantum number Chemistry • You learned about the l quantum numbers in chemistry. l number 0 1 2 3 orbital type S P D F Overview of Magnetism How Do They Work 6:25 Magnetic Moment and Orbital Angular Momentum Orbital Magnetic Moment �orb � IA v 2�r T 2�r � � � 1 1 T v 2�r T� v Charge: e � � e Lorb �orb � � m 2 9 Still more complicated structure The Zeeman Effect • • • • • The strength of the Zeeman effect depends on magnetic field Pieter Zeeman discovers that if you place an atom in a strong magnetic field, additional transition lines are observed. This leads to an understanding that there are additional energy states. These are defined by the “magnetic quantum number”, m. In the absence of a magnetic field, these additional states are still present. For any azimuthal quantum number, l, it was found that there were possibilities for the m quantum number according to: �l � m � l Measuring the magnetic fields of stars • Since the optical splitting depends on the strength of the magnetic field, observation of the degree of splitting is a way to measure the magnetic field strength in stars. Example of a selection rule • When a photon is emitted or absorbed, the l quantum number must change by ±1. �l � �1 • The reason for this is that the photon has angular momentum. L photon � �1� Three quantum numbers: n, l, m • Bohr builds on Sommerfeld’s work and works out a bunch of details for “selection rules”. • These rules showed that certain transitions between states were not allowed. • We will learn more about forbidden transitions when we get to particle physics. The fourth quantum number (The anomalous Zeeman effect) Wolfgang Pauli – Austrian Theoretical Physicist 25 April 1900 – 15 December 1958 In 1925, additional spectral splitting was observed that could not be explained. It was an accepted fact that often theorists were terrible with experimental equipment. For some reason, Pauli had the reputation that by his just stepping into a laboratory he could make equipment fall apart. A famous physicist, Otto Stern, would not allow him into his lab, but would only talk to him through a closed door. Other, “forbidden,” transitions also occur but with much lower probability. 10 Hidden Rotation Pauli hypothesized that the anomalous Zeeman effect could be explained by a “hidden rotation”. This would result in a fourth quantum number, “s”, which would explain the result. Intrinsic spin of electrons is either “up” or “down”. Electron Spin Spin of an electron • Although it is described as if the electron is spinning on its axis, that is not how it is understood. • Instead, the spin of an electron is said to be an intrinsic property of the electron (like its mass). • We now understand that all fundamental particles have a property called spin. Pauli Exclusion Principle Two electrons can not occupy the same quantum state. Thus, for each combination of n, l, m there are at most two electrons one in the + ½ state and one in the – ½ state. Why We Can Not Walk Through Walls? Alice and Bob Complex Atoms Complex atoms contain more than one electron, so the interaction between electrons must be accounted for in the energy levels. This means that the energy depends on both n and l. A neutral atom has Z electrons, as well as Z protons in its nucleus. Z is called the atomic number. 11 The Exclusion Principle In order to understand the electron distributions in atoms, another principle is needed. This is the Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state. The quantum state is specified by the four quantum numbers; no two electrons can have the same set. The Periodic Table of the Elements We can now understand the organization of the periodic table. Electrons with the same n are in the same shell. Electrons with the same n and l are in the same subshell. The exclusion principle limits the maximum number of electrons in each subshell to 2(2l + 1). Review of What We Learned From Quantum Part I • • • The entire periodic table, all chemical properties, can be explained by the combined work of the cast of characters that we have studied so far. However, this is not the end of the story for quantum theory. Now the story gets even stranger! De Broglie’s Hypothesis applied to atoms h h �� p � mv n� � 2�r For in-phase h n � 2�r mv Black Body Radiation Photoelectric Effect Einstein’s Interpretation of these two results Wave-particle duality Diffraction and interference DeBroglie’s Hypothesis of matter waves The uncertainty principle • • • • A Revolution in Quantum Thought So Who is correct? Bohr? De Broglie? Neither? � mv � � mv � h � 2�r � � � �n � 2� � � 2� � mv n h � mvr 2� This is Bohr’s Quantum Condition! 12 Particle-Wave Duality Bohr-Schrödinger-Heisenberg (6:21) Werner Heisenberg • Challenges electron “orbits” as just being an imaginary tool to visualize the atom. • He treated atoms as simple oscillators in which he could define the momentum, p, and the degree to which the charge, q, was displaced from equilibrium position. • He comes up with a very abstract, complicated algebra. • It explains the observed quantum results, but offers no pictures to visualize the atom. Schrodinger • Defines a type of wave function that can be used to solve for many properties of an atom. • The Schrodinger equation is a complex partial differential equation that can be solved to find this wave function. • Once the wave function is found, it can be used to explain all of the observed results. Imaginary Numbers i � � �1 Complex Numbers One of the foundations of quantum physics Examples of imaginary numbers: 3i, � 3i, 127.2i, � 15.6i �3i �2 � 32 i 2 � �9 �� 3i �2 � �� 3�2 i 2 � �9 Imaginary numbers play an important role in many areas of physics. 13 Complex Numbers A complex number is one in which part of the number is real and part of the number is imaginary. Visualizing Complex Numbers Imaginary Part 1� 2i 2i Example of a complex number: 1� 2i 2 � 1i 1i 2 1 Imaginary Part Real Part Real Part Complex numbers are sometimes used in place of Cartesian coordinates. Complex Conjugate Change the sign of the imaginary part of the complex number. Visualizing Complex Numbers Imaginary Part 2i Example: A � 1 � 2i 1i A � 1 � 2i * 2 � 1i A 2 1 Real Part A* is the complex conjugate of A. Absolute Value of A Complex Number Equivalent to the length of the vector described by the complex number 2 � 1i 2 1 1 �1 � 2i ��1 � 2i � �1 � 2i ��1 � 2i � � 1 � 2i � 2i � 4i 2 � 1 � 4 � 5 Example: Real Part 2 2 � 1i � 1 � 2 � 5 2 The square of the absolute value of a complex number The product of a complex number and its complex conjugate is equal to the square of the absolute value of a complex number. 2i 1i 2 � 1i A* 2 AA* � A 2 14 The phase of a complex number The phase of a complex number Changing the phase of a complex number does not change the magnitude of the complex number 2i 1i 2 � 1i 2i 2 � 1i 1 � 1i 2 1 � 2 Real Part Real Part The complex number 2+1i has a magnitude of 5 and a phase θ. Origin of the Schrodinger Equation Solving The Schrodinger Equation Emily Noether Principle of Least Time Principle of Least Action • Very few exact solutions • Usually done numerically by computer • The function Ψ that you end up with is the wave function. It varies at different places in space. • The probability of finding the electron at a particular place in space is given by Schrodinger Equation William Rowan Hamilton (1805–1865) Irish Physicist H � K � PE The Schrodinger equation is a complex partial differential equation that can be solved to find the wave function. P � � *� Schrodinger’s Wave Function, Ψ • The Schrodinger wave function is not directly observable • Max Born showed that the absolute value squared of the wave function is equal to the probability of finding an object at a particular location. • No more exact answers, said Born. In quantum mechanics all we get are probabilities. How Physicists Use The Schrodinger Equation (1/2) Goal: Predict the outcome of a measurement. • Max Born (1882–1970) • German-British physicist Solve Schrodinger’s equation for a given formula for potential energy, using calculus, and/or using computers to solve the equation. You now have the functions, Ψ(x) and Ψ*(x) • Pick a special operation that you can apply to the function Ψ(x) that will give you a new function. • Example: For position the operator is xΨ • Example: For momentum the operator is You multiply the function by its location to get a new function.. You use calculus to differentiate the function and multiply it by some constants. i� d dx 15 How Physicists Use The Schrodinger Equation (2/2) Wave function for a moving particle • Now multiply that function new function by Ψ* at every point in space. • Carefully add up all of the values for (Ψ* operator Ψ). You have to consider every valid point is space. Anything that is non-zero must be included in this sum. • In reality, this summing process is done by doing an integral with calculus. A wave function which satisfies the non-relativistic Schrödinger equation with PE=0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here. • The result of this process is a real number that represents the observable that you will try to measure. Particle in a Box One of the few exact solutions to Schrodinger’s Equation. Solution to the Particle in a Box All solutions to the equation have Ψ=0 at x=0 and at x=L. Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (BF). In (B-F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wave function. Lowest energy state is not E=0. This is called the zero point energy. The lowest energy of a system can never be zero. � x �� The states (B,C,D) are energy eigenstates, but (E,F) are not. Particle in A Box Visualized 4:17 Particle in a box solution Relationship between total energy, KE, and PE. But PE = 0 everywhere inside the box. E T � K � PE ET � K K� Classical Relationship Between Kinetic Energy and Momentum. This still holds. Expectation value for the “momentum” of the particle = 0 Solution to the Particle in A Box Uncertainty in Position �x � � �x � � Uncertainty in Momentum �p � � �x�p � � p Expectation value for the “location” of the particle = the middle of the box. � p �� 0 Inside the box p2 2m L 2 � n 2� 2 �2 2 3 16 Quantum Tunneling Minute Physics: What is Quantum Tunneling Desktop Physics – Quantum Tunneling Quantum Tunneling and Radioactive Decay 1:04 2:57 Radioactivity 4:17 Quantum Tunneling through a finite barrier (0:26) Touch Screens and Quantum Tunneling (6:26) Simple Harmonic Oscillator Simple Harmonic Oscillator Simple Harmonic Oscillator Simple Harmonic Oscillator 17