The First 100 Years of Quantum Physics Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics The Wave Function Quantum States Wave Function Probability Interpretation Lecture 5 The Wave Function and Quantum Probability Erwin Schrödinger Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics The Wave Function The “wave function” is a mathematical object that incorporates the quantum nature of matter/energy and contains all the information that is, in principle, available about the physical entity to which it relates. A physical quantity, called an observable (i.e., energy, position, momentum, etc.) is represented mathematically by an “operator”. This is a mathematical entity which “operates” in some manner on the wave function. Lecture 5 The Wave Function and Quantum Probability Quantum physics involves a “wave” description of nature. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics The Wave Function If a physical object is known to have a certain value of some observable quantity, its wave function will contain that knowledge. In this case, the wave function is called an “eigenfunction” of that quantity and the quantity’s value is called an “eigenvalue” of the quantity’s operator. (Mathematical operators are distinguished by little hats.) Lecture 5 The Wave Function and Quantum Probability Physical observables are represented mathematically by operators. Example: This is Schrödinger’s Equation for the energy. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics The Wave Function The physical entity may have certain values of more than one observable, so long as they are compatible, i.e., do not have uncertainty constraints with respect to each other. In this case, the wave function may be an eigenfunction of all the compatible observables. Lecture 5 The Wave Function and Quantum Probability Quantum physics involves a “wave” description of nature. x and px are not compatible Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Quantum States Example: Hydrogen Atom Lecture 5 The Wave Function and Quantum Probability The hydrogen atom – one electron – is a simple example of a quantized entity. Quantum Numbers A complete set of compatible observables: Energy (n) Total angular momentum squared (ℓ) One (any) component of angular momentum (μ) The wave function and eigenvalues are found by solving Schrödinger’s Equation. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Quantum States Lecture 5 The Wave Function and Quantum Probability Schrödinger’s equation yields the same results for Hydrogen energies as the original Bohr model. energy Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities Lecture 5 The Wave Function and Quantum Probability Does God play dice with the universe? Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities Pwill The DBacks will win the World Series. Lecture 5 The Wave Function and Quantum Probability Pwill not The DBacks will not win the World Series. Pwill + Pwill not = 1 A priori probabilities don’t make sense scientifically unless you can test them a posteriori. A priori probability: Guess or calculated (theoretical) prediction A posteriori probability: Observation or experimental (empirical) result Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities Classical Thermodynamics Lecture 5 The Wave Function and Quantum Probability Pressure, temperature In physics, experiment (or controlled observation) is the link between a priori probabilities (theory) and a posteriori probabilities. Quantum Theory Values of observables Transition rates Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities in Quantum Physics Hydrogen Wave Function Lecture 5 The Wave Function and Quantum Probability is the probability of finding the electron at position x, y and z in the tiny volume of Wave functions are interpreted in terms of probabilities. Thus, the square of the wave function gives us a picture of the electron’s probability of being anywhere within the atom. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

High electron position probability The First 100 Years of Quantum Physics Hydrogen Quantum States Electron distributions High electron position probability Lecture 5 The Wave Function and Quantum Probability n = 1 n = 2 Wave functions show the electron’s possible positions. n = 3 Low electron position probability n = 4 n = 5 Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Hydrogen Quantum States Electron distributions Lecture 5 The Wave Function and Quantum Probability n = 1 n = 2 Wave functions show the electron’s possible positions. n = 3 n = 4 n = 5 Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Hydrogen Quantum States Electron distributions Light emission – Balmer series Lecture 5 The Wave Function and Quantum Probability n = 1 n = 5  n = 2 n = 4  n = 2 n = 3  n = 2 n = 2 Wave functions can be used to compute transition probabilities n = 3 n = 4 n = 5 Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities in Quantum Physics Quantum Transitions Lecture 5 The Wave Function and Quantum Probability Light emission by excited atoms Radioactive decay Elementary particle decay and many more Quantum rules require that ℓ change by plus or minus 1 unit. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities in Quantum Physics Quantum Transition Rates Lecture 5 The Wave Function and Quantum Probability A sample of N0 identical excited atoms Unlike more complex systems, like human beings, until they actually transform, atomic systems do not age. In a very short time Δt a small number ΔN transform, or “decay” to another state (with the emission of energy or other particles.) The remainder are exactly the same as they were before. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities in Quantum Physics Quantum Transition Rates Lecture 5 The Wave Function and Quantum Probability The number that decay in a particular time interval depends upon how many there were to begin with, the length of the time interval, and a quantum decay probability, λ, calculated from the initial and final wave functions and the force that causes the transition. The quantum decay probability, λ, is calculated from the wave functions. In the next very short time Δt, another small number transform to another state. But there were fewer to start with. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Wave Function and Quantum Probability The First 100 Years of Quantum Physics Probabilities in Quantum Physics Quantum Transition Rates Lecture 5 The Wave Function and Quantum Probability This leads to a simple formula for how many of the original sample remain after a time t. Unlike more complex systems, like human beings, until they actually transform, atomic systems do not age. t ½ original population half-life Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Kinetic Angular Momentum Spin – Intrinsic Angular Momentum The First 100 Years of Quantum Physics Spin Angular Momentum Kinetic Angular Momentum Spin – Intrinsic Angular Momentum Identical Particles Pauli Exclusion Principle Lecture 6 Spin Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin Examples of Kinetic Angular Momentum Lecture 6 Spin The First 100 Years of Quantum Physics Spin Examples of Kinetic Angular Momentum Lecture 6 Spin Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin L The Quantum Mechanics of Angular Momentum Ly Lx Lz The First 100 Years of Quantum Physics Spin The Quantum Mechanics of Angular Momentum x y z L Lx Lz Ly Lecture 6 Spin For many systems the energy, square of the total angular momentum and any one of the three angular momentum components can be chosen as a complete compatible set of observables. Lx, Ly and Lz are mutually incompatible! But L2 is compatible with all three! We choose, for example, L2 and Lz. Then --- Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin Example: Hydrogen Atom Quantum Numbers Lecture 6 Spin The First 100 Years of Quantum Physics Spin Example: Hydrogen Atom Quantum Numbers Lecture 6 Spin The number of degenerate states increases with n. n ℓ μ 1 2 ±1 3 ±2 Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin What makes a magnet? Ampere’s Law The First 100 Years of Quantum Physics Spin What makes a magnet? Lecture 6 Spin Because of its charge and angular momentum, the electron in an atom is like a magnet. S N N S In the presence of an external magnetic field, there is an additional amount of energy. Ampere’s Law If an external magnetic field pointing in the z direction is applied to the atom, an additional energy exists which is proportional to Lz. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The First 100 Years of Quantum Physics Spin Lecture 6 Spin ∞ 5 4 3 2 1 n etc. This splitting of the spectral lines in the presence of a magnetic field is called the Zeeman effect. without magnetic field with magnetic field Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin + = Weak field Zeeman effect Lecture 6 Spin The First 100 Years of Quantum Physics Spin spin-orbit effect splitting Weak field Zeeman effect Lecture 6 Spin The Zeeman effect and many other experiments suggest strongly that another source of angular momentum of quantum number ½ is also present. 1 Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College + = μ = +1 s = +1/2 M = +3/2

Spin Stern-Gerlach Experiment The First 100 Years of Quantum Physics Spin In 1925, Dutch grad students, Samuel Goudsmit and George Uhlenbeck, based on the experimental results of Otto Stern and Walther Gerlach in 1922, proposed that the electron must have an intrinsic angular momentum of magnitude 1/2. Lecture 6 Spin Particle spin is an intrinsic property that behaves like and adds to ordinary angular momentum in a quantum system. Its origins are best understood in the theory of relativity. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College Stern-Gerlach Experiment

Spin Electron spin: S = 1/2 s = 1/2 or -1/2 The First 100 Years of Quantum Physics Spin Electron spin: S = 1/2 s = 1/2 or -1/2 Lecture 6 Spin Intrinsic spin obeys the same quantum mechanical rules as ordinary angular momentum. Unlike ordinary angular momentum, spin can exist in half-integer amounts. But the same rule holds for component values: s = -S, -S+1, …., S-1, S Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin Photon spin: S = 1 s=1 or -1 (not 0) Lecture 6 Spin The First 100 Years of Quantum Physics Spin Photon spin: S = 1 s=1 or -1 (not 0) Lecture 6 Spin According to Maxwell’s Equations for electromag-netism, because it has zero mass, the photon cannot have a zero component of its spin. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Spin Fermions and Bosons Fermions Bosons The First 100 Years of Quantum Physics Spin Fermions and Bosons Fermions Bosons Lecture 6 Spin Obey Fermi-Dirac Statistics Obey Bose-Einstein Statistics “Statistics” here refers to the thermo-dynamical properties of the particular particles. Particles with Half-integer Spin Electron Proton Neutron Quark He-3 Atom Particles with Integer Spin Photon Pion Gluon Hydrogen Atom Alpha Particle Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Indistinguishable Particles The First 100 Years of Quantum Physics Indistinguishable Particles Classical indistinguishability - Easily remedied Lecture 6 Spin Classical particles are in principle distinguishable. Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Indistinguishable Particles The First 100 Years of Quantum Physics Indistinguishable Particles Identical Particles Lecture 6 Spin Distinguishable That particles are absolutely identical or indistinguishable is a consequence of the Uncertainty Principle. Indistinguishable Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Indistinguishable Particles The First 100 Years of Quantum Physics Indistinguishable Particles Maxwell-Boltzmann Statistics Distribute 9 units of energy among 6 distinguishable particles Lecture 6 Spin Most ordinary gasses and liquids obey the classical Maxwell-Boltzmann statistics Energy levels 2002 different ways to distribute 9 units of energy among 6 distinguishable particles Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Indistinguishable Particles The First 100 Years of Quantum Physics Indistinguishable Particles Bose-Einstein Statistics Distribute 9 units of energy among 6 indistinguishable Bose particles Lecture 6 Spin 1 Light obeys the Bose-Einstein statistics 26 Different ways to distribute 9 units of energy among 6 identical bosons Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Indistinguishable Particles The First 100 Years of Quantum Physics Indistinguishable Particles The Pauli Exclusion Principle Lecture 6 Spin Only one identical fermion at a time may occupy an atomic state Wolfgang Pauli Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

Indistinguishable Particles The First 100 Years of Quantum Physics Indistinguishable Particles Fermi-Dirac Statistics Lecture 6 Spin Remember: only one fermion of each spin state in any quantum state. Thus, in this example, two particles can have the same energy. 5 different ways to distribute 9 units of energy among 6 identical spin ½ fermions Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College

The Emeritus College Next: Atoms, Molecules and Nuclei The First 100 Years of Quantum Physics The Emeritus College http://www.emerituscollege.asu.edu Next: Atoms, Molecules and Nuclei rjjacob@asu.edu http://www.public.asu.edu/~rjjacob/courses.html Dr. Richard J. Jacob Professor Emeritus of Physics Arizona State University The Emeritus College