1

Quantum Physics Part II

Quantum Physics in three units• I: How light interacts with particles

• II: Simple systems– 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

Bright Line Spectra

• Spectra of Atoms– Bright Line, Absorption, From The Sun

– Mathematical Models of Results

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– If the energy of light come in

quanta – small packets = hf then he can explain the 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

2

Einstein’s Interpretation

• Light comes in quantized bundles called photons

• The energy is give by:

• The total power of a beam of photons is given by:

• The threshold effect for emitting electrons is due to the work function for the metal:

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New Discoveries

• Bright Line Spectra

• Absorption Spectra

• Spectrum from the Sun

Bright Line Spectra

Light Source

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.

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 FraunhoferGerman 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

3

Gustav KirchoffGerman 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”.

Helium comes from Greek word “Helios” (sun)

How do the spectra relate to the atoms?

• In 1862 Angstrom studies spectrum of Hydrogen in great detail.

Anders Jonas Ångström

13 August 1814– 21 June 1874 - Swedish

Question: How can we make sense of all of these lines of different frequencies of light?

Johann Jakob Balmer Swiss Mathematician

May 1, 1825 – March 12, 1898

Balmer finds that he can represent the frequencies of all observed lines from Hydrogen by a simple formula:

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A constantAn integer > 2

Using this equation, Balmer predicts additional lines before they are discovered.

Additional Hydrogen Wavelengths

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All wavelengths agree with a generalized model

n1=1 n2=1 n3=1

1890’s: A result waiting for an explanation!

Cathode RaysLate 1800’s: Studies were being conducted in what happens when electricity is discharged into a rarefied gas.

J. J. Thomson

18 December 1856 – 30 August 1940

British Physicist and Nobel Laureate

How Thomson Discovers the Electron (11:08)

(This is the same guy who discovered Thomson Scattering)

Hypothesis: The Plum Pudding Model of the Atom:

Thomson Discovers Electron (2:53)

4

ROLLING WITH RUTHERFORDLab:

Ernest Rutherford

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.

30 August 1871 – 19 October 1937

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 AtomThe 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 Quantum Theory, J.P. McEvoy, Oscar Zarate

27.10 Early Models of the Atom

Therefore, Rutherford’s model of the atom is mostly empty space:

5

Rutherford’s Model Had A Lot of Critics

But the biggest problem with his model was yet to be explained.

Bohr Model of Atom

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.

Alice and Bob: How Can Atoms Exist?

Niels Bohr (The Great

Dane)

7 October 1885 – 18 November 1962

Danish Physicist

The Grandfather of Quantum Physics

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.

However, Bohr Hits it Off With

Rutherford.

Niels Bohr (The Great Dane)

6

Niels Bohr (The Great Dane) Physicists Can Learn From Unit Analysis

Energy: � �� �� �

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Angular Momentum

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Planck’s Constant has units of Angular Momentum! Is this just a coincidence???

J. J. Nicholson 1912

• 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.

Angular Momentum

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

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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

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Planets: Gravity Atom: Electromagnetic

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Conservation of Angular Momentum is Kepler’s 2nd Law

Quantinization of Angular Momentum is Bohr’s 1st Postulate.

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Adding Energy to Bohr’s Model

• 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’s 2nd

Postulate

Bohr Derives the Balmer Formula

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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.

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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:

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Z is the number of protons in the nucleus.

n is a quantum number. It can be 1, 2, 3, …

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

Chladni Plate Vibrations

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 lquantum 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

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9

Electron cloud, or probability distribution, for n = 2 states in hydrogen

Orbitals

Orbital Shapes- Derived Later

Chemistry• You learned about the l quantum numbers

in chemistry.

l number orbital type0 S1 P2 D3 F

How the Periodic Table relates to the azimuthal quantum number

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

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Still more complicated structureThe Zeeman Effect

• 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 mquantum number according to:

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The strength of the Zeeman effect depends on magnetic field

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.

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.

Example of a selection rule• When a photon is emitted or absorbed, the

l quantum number must change by ±1.

• The reason for this is that the photon has angular momentum.

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Other, “forbidden,” transitions also occur but with much lower probability.

The fourth quantum number

25 April 1900 – 15 December 1958

Wolfgang Pauli – Austrian Theoretical Physicist

(The anomalous Zeeman effect)

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.

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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.

Why We Can Not Walk Through Walls?

Alice and Bob

Pauli Exclusion PrincipleTwo 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.

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 Zprotons in its nucleus. Z is called the atomic number.

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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).

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!

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

De Broglie’s Hypothesis applied to atoms

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This is Bohr’s Quantum Condition!

A Revolution in Quantum Thought

So Who is correct?

Bohr?

De Broglie?

Neither?

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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 complexpartial 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.

Complex Numbers

One of the foundations of quantum physics

Imaginary Numbers

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Imaginary numbers play an important role in many areas of physics.

Examples of imaginary numbers:

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Complex NumbersA complex number is one in which part of the number is real and part of the number is imaginary.

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Real Part Imaginary Part

Visualizing Complex Numbers

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Real Part

Imaginary Part

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Complex numbers are sometimes used in place of Cartesian coordinates.

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Complex ConjugateChange the sign of the imaginary part of the complex number.

Example:

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A* is the complex conjugate of A.

Visualizing Complex Numbers

Real Part

Imaginary Part

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Absolute Value of A Complex NumberEquivalent to the length of the vector described by the complex number

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The square of the absolute value of a complex numberThe product of a complex number and its complex conjugate is equal to the square of the absolute value of a complex number.

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The phase of a complex number

Real Part

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The phase of a complex number

Real Part

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Changing the phase of a complex number does not change the magnitude of the complex number

Origin of the Schrodinger Equation

Principle of Least Time

Principle of Least Action

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Emily Noether

William Rowan Hamilton (1805–1865) Irish Physicist

Schrodinger Equation

The Schrodinger equationis a complex partial differential equation that can be solved to find the wave function.

Solving The Schrodinger Equation

• 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

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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.

German-Britishphysicist

Max Born (1882–1970)

How Physicists Use The Schrodinger Equation (1/2)

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• 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 use calculus to differentiate the function and multiply it by some constants.

You multiply the function by its location to get a new function..

Goal: Predict the outcome of a measurement.

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How Physicists Use The Schrodinger Equation (2/2)

• 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.

• The result of this process is a real number that represents the observable that you will try to measure.

Wave function for a moving particle

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.

Particle in a Box

Particle in A Box Visualized

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 (B-F). 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.

The states (B,C,D) are energy eigenstates, but (E,F) are not.

One of the few exact solutions to Schrodinger’s Equation.

4:17

Solution to the Particle in a BoxAll solutions to the equation have Ψ=0 at x=0 and at x=L.

Lowest energy state is not E=0. This is called the zero point energy. The lowest energy of a system can never be zero.

2

Lx ��� Expectation value for the

“location” of the particle = the middle of the box.

0��� p Expectation value for the “momentum” of the particle = 0

Particle in a box solution

Classical Relationship Between Kinetic Energy and Momentum. This still holds.

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Solution to the Particle in A BoxUncertainty in Position

Uncertainty in Momentum

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Quantum Tunneling

Minute Physics: What is Quantum Tunneling

Desktop Physics – Quantum Tunneling

Touch Screens and Quantum Tunneling (6:26)

Quantum Tunneling through a finite barrier (0:26)

2:57

1:04

Quantum Tunneling and Radioactive Decay

Radioactivity 4:17

Simple Harmonic Oscillator Simple Harmonic Oscillator

Simple Harmonic Oscillator Simple Harmonic Oscillator