sign in sign up
<<
Home , Azimuthal quantum number

Physics in three units • I: How light interacts with particles • II: Simple systems – a) ,their structure, interaction with light Quantum Physics Part II – b) Quantum wave functions and quantum tunneling • III: More complex systems – a)Bosons, Fermions, and Bose Condensates – b)Superconductivity and superfluidity – c) and

Review of What We Learned From Quantum Part I Bright Line Spectra • Black Body Radiation • Spectra of Atoms • – Bright Line, Absorption, From The Sun • Einstein’s Interpretation of these two – Mathematical Models of Results results • Wave-particle duality • Diffraction and interference • DeBroglie’s Hypothesis of matter waves • The

Black Body Radiation Photoelectric Effect • What it is – The electromagnetic radiation • Shine light on a metal surface given off by a hot object • No emitted unless the frequency – Does not agree with classical is above a critical threshold value physics • Once past the threshold, the amount of – Ultraviolet Catastrophe emitted electrons is proportional to the • Planck’s Solution 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 • Bright Line Spectra • The energy is give by: � hfE • Absorption Spectra • The total power of a beam of photons is • Spectrum from the Sun given by: E # photons P �� hf t t • The threshold effect for emitting electrons is due to the work function for the metal:

max hfKE �� �

Atomic Spectra: Key to the Structure of the Bright Line Spectra Prism A very thin gas heated in a discharge tube emits light only at characteristic frequencies. Observer Light Source

1752 – Scottish Physicist Joseph von Fraunhofer German Optician Thomas Melville • Takes a spectrum of the sun • Compares the spectrum for a pure hot gas (not a flame) to the spectrum of a black body • Sees an almost countless number of dark lines • Surprising result: Line Spectra superimposed on a continuous background.

Modern- Super High Resolution Spectrum

2 Gustav Kirchoff How do the spectra relate to the German Physicist 12 March 1824 – 17 October 1887 atoms? First to show that the absorption Anders Jonas Ångström lines from an emission spectrum lines up with the absorption lines in 13 August 1814– 21 June 1874 - Swedish an absorption spectrum. • In 1862 Angstrom studies But…By very careful work, he was spectrum of Hydrogen in able to determine that there were some spectral absorption lines on the great detail. sun had no matching lines on earth. A worldwide search began to find the Question: How can we make sense elusive element: Result – Discovery of “Helium”. of all of these lines of different frequencies of light? Helium comes from Greek word “Helios” (sun)

Johann Jakob Balmer Additional Hydrogen Swiss Mathematician 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: n1=1 n2=1 n3=1 � 11 1 � R� �� � � 22 � � � 2 n �

A constant An integer > 2 All wavelengths � 111 � 1890’s: A result waiting Using this equation, Balmer predicts additional lines agree with a R� �� � for an explanation! before they are discovered. generalized model � 2 2 � � � 1 nn 2 �

Cathode Rays J. J. Thomson British Physicist and Nobel Laureate Late 1800’s: Studies were being conducted in what happens when electricity is discharged into a rarefied 18 December 1856 – 30 August 1940 gas.

How Thomson Discovers the (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 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 27.10 Early Models of the Atom The only way to account for the large angles Rutherford did an experiment that showed that was to assume that all the positive charge was the positively charged nucleus must be contained within a tiny volume – now we know extremely small compared to the rest of the that the radius atom. He scattered alpha particles – helium of the nucleus nuclei – from a metal foil and observed the is 1/10000 that scattering angle. He found that some of the of the atom. angles were far larger than the plum-pudding model would allow.

The work was carried out by one of Hans Geiger’s graduate students, Earnest Marsden.

Gold Foil Experiment 9:07

27.10 Early Models of the Atom Rutherford’s Result Was A Total Surprise

“This is quite the most incredible Therefore, Rutherford’s event that has ever happened in my model of the atom is life. It was almost as if you fired a mostly empty space: 15” shell at a piece of tissue paper and it came back and hit you!1

1Introducing 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.

Bohr arrives in England in 1911 and initially works with J.J. Thomson. However, the two do How Do Radio Transmitters Work? not get along with each other. (The Great When he arrived he spoke An accelerating charge produces electromagnetic radiation. If almost no English. He brought the charge oscillates with a specific frequency, then the radiation Dane) a dictionary and the complete works of Charles Dickens to will have the same frequency. learn the language. Danish Physicist Classical Physics- All accelerating charges The Grandfather of Quantum Physics produce electromagnetic energy. 7 October 1885 – 18 November 1962

Alice and Bob: How Can Atoms Exist?

Niels Bohr (The Great Dane) However, Bohr Hits it Off With Rutherford.

5 Niels Bohr (The Great Dane) Physicists Can Learn From Unit Analysis 2 2 1 2 � ��mkg � � ��mkg � 1 Energy: mv � 2 �s �2 � � �ss �

1 Frequency f �s � �m� Planck’s Constant Units � :for �kghhfE � �m � �s �

�m� Angular � �kgmvrL � �m � Momentum �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 Bohr’s • In 1913 Bohr combines three ideas together. First – The line spectra formula from Balmer – The quantizing of angular momentum from Nicholson Postulate – The need to define stable orbits for Rutherford’s model

6 Principle , n 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 The angular momentum can not take on any value (as would be �� mM protonelectron �� eqq nucleus � Zeq the case for classical physics). Centripetal Force Provided By Centripetal Force Provided By The angular momentum must 2 2 2 be an integer multiple of h/2π vm mGm vm KZe planet planetsun electron F � F � FG � 2 Fc � E 2 c r r r r L �1 Solution for atoms 1 � Solution for planets � rvmL � planetplanet rvmL electronelectron L2 � 2� 2 2 2 L L � 2 r � rn � 2 n ... r � 2 2 eKZm GMm KmZe elec n � nL � Conservation of Angular Momentum Quantinization of Angular Momentum st is Kepler’s 2nd Law is Bohr’s 1 Postulate.

Adding Energy to Bohr’s Model

Bohr • Once the radius and the angular defines momentum are known, it is fairly radius of straightforward to determine the total energy of the atom depending on which each orbit orbit the electron is in. 2 • Procedure: r � � n2 n ZeKm 2 – Determine the Kinetic Energy elec – Determine the Potential Energy – Add them together

Bohr Derives the Balmer Formula � 111 � R� �� � � � 2 nn 2 � Bohr’s 2nd � 1 2 � The value for R calculated by Postulate 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 Bohr’s Formula for Energy

1. The atom is quantized by a single quantum • Overall energy levels: number “n”, which relates to the angular Z is the number of momentum of the state that the electron is in. protons in the nucleus. 2. The same number defines the energy of the 2 atom. Z En �� 6.13 eV 3. Absorption and emission of a photon can only n2 occur if the between two states is exactly equal to energy of the photon being n is a quantum absorbed or emitted. number. It can be 1, 4. The quantum number, n, defines the “shell” for 2, 3, … the electron.

Chladni Plate Vibrations 1 E �� 6.13 eV n n2

When n=∞, E=0, electron is ionized from atom.

What might Chladni patterns look More complicated structure

like in 3D? • Additional spectral lines were observed • It was proposed by 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 The patterns of the value according to are complex, but much simpler 22 llL �� 1 than these! � � �

8 Orbital Shapes- Derived Later Electron cloud, or probability distribution, for n = 2 states in hydrogen

Orbitals

Chemistry How the relates to the azimuthal quantum number • You learned about the l quantum numbers in chemistry.

l number orbital type 0 S 1 P 2 D 3 F

Overview of Magnetism How Do They Work 6:25 Magnetic Moment and Orbital Angular Momentum

Orbital Magnetic Moment

�orb � IA Charge: e 2�rv 2�rT ��� � 1 T 1 v � e Lorb 2�r �orb �� T � m 2 v

9 Still more complicated structure The strength of the depends The Zeeman Effect on magnetic field • 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 “”, 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: � � � lml

Measuring the magnetic fields of Three quantum numbers: n, l, m stars • Since the optical splitting depends on the • Bohr builds on Sommerfeld’s work and strength of the magnetic field, observation works out a bunch of details for “selection of the degree of splitting is a way to rules”. measure the magnetic field strength in • These rules showed that certain stars. transitions between states were not allowed. • We will learn more about forbidden transitions when we get to particle physics.

The fourth quantum number Example of a (The anomalous Zeeman effect) – Austrian Theoretical Physicist • When a photon is emitted or absorbed, the 25 April 1900 – 15 December 1958 l quantum number must change by ±1. In 1925, additional spectral splitting was �l � �1 observed that could not be explained.

It was an accepted fact that often theorists were terrible with experimental equipment. • The reason for this is that the photon has For some reason, Pauli had the reputation angular momentum. that by his just stepping into a laboratory he could make equipment fall apart. L 1�� A famous physicist, Otto Stern, would not photon � 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 Intrinsic

Pauli hypothesized that of the anomalous Zeeman electrons effect could be explained by a “hidden is either rotation”. This would result in a fourth “up” or quantum number, “s”, “down”. which would explain the result. 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. Why We Can Not Walk • Instead, the spin of an electron is said to Through Walls? be an intrinsic property of the electron (like its mass). Alice and Bob • We now understand that all fundamental particles have a property called spin.

Pauli Exclusion Principle Complex Atoms Two electrons can not occupy the same . 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. 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 The Periodic Table of the Elements

In order to understand the electron We can now understand the organization of the distributions in atoms, another principle is periodic table. needed. This is the Pauli exclusion principle: Electrons with the same n are in the same shell. No two electrons in an atom can occupy the Electrons with the same n and l are in the same same quantum state. subshell. The quantum state is specified by the four The exclusion principle limits the maximum quantum numbers; no two electrons can have number of electrons in each subshell to 2(2l + 1). the same set.

Review of What We Learned From Quantum Part I • Black Body Radiation • Photoelectric Effect • Einstein’s Interpretation of these two results • Wave-particle duality

The entire periodic table, all chemical properties, can be • Diffraction and interference explained by the combined work of the cast of characters that we have studied so far. • DeBroglie’s Hypothesis of However, this is not the end of the story for quantum theory. matter waves Now the story gets even stranger! • The uncertainty principle

De Broglie’s Hypothesis applied A Revolution in Quantum Thought

h h to atoms So Who is � �� correct? p mv Bohr? De Broglie? � � 2�rn For in-phase Neither? h n 2�r mv � mv � h � mv � � �n � 2�r� � � 2� � mv � 2� � h n� mvr 2� This is Bohr’s Quantum Condition!

12 Particle-Wave Duality

Bohr-Schrödinger-Heisenberg (6:21)

Schrodinger • Defines a type of that can • Challenges electron “orbits” as just being an be used to solve for many properties of an imaginary tool to visualize the atom. atom. • He treated atoms as simple oscillators in which • The Schrodinger equation is a complex he could define the momentum, p, and the partial differential equation that can be degree to which the charge, q, was displaced from equilibrium position. solved to find this wave function. • He comes up with a very abstract, complicated • Once the wave function is found, it can be algebra. used to explain all of the observed results. • It explains the observed quantum results, but offers no pictures to visualize the atom.

Imaginary Numbers i 1���

Examples of imaginary numbers: Complex Numbers � � 6.15,2.127,3,3 iiii

2 22 22 2 One of the foundations of � � ii ��� 933 � � � � ii ����� 933 quantum physics Imaginary numbers play an important role in many areas of physics.

13 Complex Numbers Visualizing Complex Numbers

A complex number is one in which part of the number Imaginary Part is real and part of the number is imaginary. � i21 2i

Example of a complex number: � i21 1i � i12

1 2

Real Part Imaginary Part Real Part

Complex numbers are sometimes used in place of Cartesian coordinates.

Complex Conjugate Visualizing Complex Numbers

Change the sign of the imaginary Imaginary Part part of the complex number. 2i Example: A �1� 2i 1i A � i12 * A �1� 2i 1 2 Real Part A* A* is the complex conjugate of A. � i12

The square of the absolute Absolute Value of A Complex Number Equivalent to the length of the vector value of a complex number described by the complex number The product of a complex number and its complex 2i conjugate is equal to the square of the absolute value of a complex number.

1i � i12 1 Example: � �� �� 2121 ii � 1 2 2 Real Part � �� �� 2121 � iiiii ������� 5414221 2 *� AAA 2 i 22 ���� 52112

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 2i � i12

1i � i12 1i

1 � 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 Principle of Schrodinger • Very few exact solutions Equation Least Time Least Action • Usually done numerically by computer • The function Ψ that you end up with is the William Rowan Hamilton (1805–1865) wave function. It varies at different places Irish Physicist in space. H � K � PE • The probability of finding the electron at a particular place in space is given by The Schrodinger equation is a complex partial * differential equation that can �P �� be solved to find the wave function.

How Physicists Use The Schrodinger Equation (1/2) Schrodinger’s Wave Function, Ψ Goal: Predict the outcome of a measurement. • The Schrodinger wave function is not directly • Solve Schrodinger’s equation for a given formula for potential energy, using calculus, and/or using computers to solve the (1882–1970) • Max Born showed that the absolute value equation. squared of the wave function is equal to the • You now have the functions, Ψ(x) and Ψ*(x) probability of finding an object at a particular location. • Pick a special operation that you can apply to the function • No more exact answers, said Born. In Ψ(x) that will give you a new function. all we get are probabilities. • Example: For position the is xΨ You multiply the function by its location to get a new function.. German-British physicist • Example: For momentum the operator is d i� You use calculus to differentiate the function dx and multiply it by some constants.

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.

A wave function which satisfies the non-relativistic • Carefully add up all of the values for (Ψ* operator Ψ). You Schrödinger equation with have to consider every valid point is space. Anything PE=0. In other words, this that is non-zero must be included in this sum. corresponds to a particle traveling freely through empty space. The real part • In reality, this summing process is done by doing an of the wave function is plotted here. integral with calculus.

• 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

Some trajectories of a particle in a All solutions to the equation box according to Newton's laws of have Ψ=0 at x=0 and at x=L. (A), and according to the Schrödinger equation of quantum mechanics (B- F). In (B-F), the horizontal axis is Lowest energy state is not E=0. position, and the vertical axis is the This is called the zero point real part (blue) and imaginary part energy. The lowest energy of a (red) of the wave function. system can never be zero.

L Expectation value for the x ��� “location” of the particle = the The states (B,C,D) are energy 2 middle of the box. eigenstates, but (E,F) are not.

4:17 Particle in A Box Visualized � p �� 0 Expectation value for the “momentum” of the particle = 0

Particle in a box solution Solution to the Particle in A Box Relationship between total energy, Uncertainty in T �� PEKE KE, and PE. But PE = 0 �x � � everywhere inside the box. Position

T � KE Inside the box �x � �

2 p Classical Relationship Between Kinetic Uncertainty in K � Energy and Momentum. This still holds. �p � � 2m Momentum

n � 22 px � ����� 2 p 32

16 Quantum Tunneling and Quantum Tunneling Radioactive Decay

Minute Physics: What is Quantum Tunneling 1:04

Desktop Physics – Quantum Tunneling 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