1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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 = 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. The Emeritus College Next: Atoms, Molecules and Nuclei
The First 100 Years of
Quantum Physics
The Emeritus College
Next: Atoms, Molecules and Nuclei
Dr. Richard J. Jacob
Professor Emeritus of Physics
Arizona State University
The Emeritus College