Atomic Physics

Quantization of Energy

Atomic Models

Quantum Mechanics

Electric and Magnetic FieldsSummary

• A changing magnetic field can induce a current in a circuit (Faraday’s Law of Induction)

• A magnetic field is created around a current-carrying wire (Ampere’s Law)

• Electric field lines start on positive charges and end at negative charges (Coulomb’s Law / Gauss’s Law)

• Magnetic field lines always form closed loops with no beginning and no end (Gauss’s Law for magnetism)

• These unrelated observations, experiments and equations were all known by the mid-1800s, but nothing linked them together.

Maxwell’s Equations

• James Clerk Maxwell (1831-1879)– Scottish theoretical physicist & mathematician

• Maxwell’s Equations– Set of differential equations that describe the

relationship between electric and magnetic field

– Summarized all previous work of Coulomb, Ampere, Gauss, Faraday & others

Name Differential form Integral form

Gauss's law:                                                                                

Gauss's law for magnetism

:

                                                                 

Maxwell-Faraday equation(Faraday's law of induction):

                                                                                          

Ampère's circuital law

(with Maxwell's correction):

                                                                                                                                           

Maxwell’s Equations

Relax!!! You don’t need to use these.

Maxwell’s Equations

• Predicted:– a changing magnetic field would create a changing

electric field, which would, in turn, create a changing magnetic field, and so on

– existence of electromagnetic waves that move through space at the speed of light

– light is an electromagnetic wave

• Confirmed:– Heinrich Hertz in 1887– generated and detected the first E/M waves

Electromagnetic Waves

• Oscillating electric and magnetic fields

• E-field and B-field are at right angles to each other

• Propagates at a right angle to both fields (transverse wave)

Electromagnetic Waves

• EM waves can be produced most easily by an oscillating charged particle

• Frequency of oscillation determines frequency of the EM wave

• Wavelength related to frequency by:

m/s 10 x 0.3

/8

c

fc

Electromagnetic Radiation

• Energy is the ability to do work• E-fields & B-fields store energy because they

exert a force (do work) on charged particles• Electromagnetic Radiation:

– transfer of energy associated with electric and magnetic fields

– can be transferred to objects in the EM wave’s path– can be converted to other forms, such as heat– Continuous distribution of wavelengths on the

electromagnetic spectrum.

Electromagnetic Spectrum

Blackbody Radiation

• All objects emit electromagnetic radiation– Continuous distribution of wavelengths from

the infrared, visible, and UV portions of the EM spectrum

– Intensity distribution of different wavelengths varies with temperature

– At low temps: mostly infrared (invisible)– Temp increases: distribution shifts to visible &

UV– Metals glow: red > yellow > white > blue

Blackbody Radiation

• Most objects absorb some incoming radiation and reflect the rest

• Blackbody:– Ideal system that absorbs all incoming radiation– Hollow object with a small opening– Perfect absorber and perfect radiator– Emits radiation based only on its temperature

• In 1900, Max Planck (1858-1947), proposed that the walls of a blackbody contained billions of submicroscopic electric oscillators, which he called resonators. These resonators, produced the blackbody radiation.

Blackbody RadiationClassical Theory Exper. Data / Planck’s Theory

as wavelength approaches zero, the amount of energy should become infinite

as wavelength approaches zero, the amount of energy radiated also approaches zero

energy absorbed and emitted by a single resonator is continuous

energy absorbed and emitted by a single resonator occurs in certain discrete amounts

Quantization of Energy

• Planck found that the total energy of a resonator is an integer multiple of the frequency

• Because the energy of each resonator comes in discrete units, it is said to be quantized.

• Allowed energy states are called quantum states or energy levels.

• Einstein applied the concept of quantized energy to light.

• Photon: quantized unit of light energy• Photons are absorbed or given off by electrons

“jumping” from one quantum state to another.

Quantization of Energy

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The Photoelectric Effect

• Classical physics predicts:– Light waves of any frequency

should have enough energy to eject electrons if the intensity is high enough

– At low intensities, electrons should be ejected if light shines on the metal for a long enough period of time

– Increasing the intensity of the light waves should increase the kinetic energy of the photoelectrons.

– Maximum kinetic energy of a photoelectron should be determined by the light’s intensity

When light strikes a metal surface, the surface may emit electrons, called photoelectrons.

The Photoelectric Effect• Experimental evidence shows that:

– No photoelectrons emitted if the light frequency falls below a certain threshold frequency, even if the intensity is very high

– Threshold frequency, ft, depends on material

– If light frequency exceeds ft

• # of photoelectrons emitted is proportional to light intensity• Maximum kinetic energy of photoelectrons is proportional to the

frequency and is independent of the intensity• Electrons are emitted instantaneously, even at low intensities

• Classical physics could not explain the photoelectric effect … but Einstein could!

Einstein’s Explanation

• EM waves are quantized

• Think of light as a stream of particles, called photons

• Photon energy given by Planck’s equation

• When photons collide with electrons in metal, they transfer energy to electrons

hfE photon

Einstein’s Explanation

• If photon energy is greater than work function of the metal, photoelectrons are ejected

• If photon has more energy than the work function, the difference is the kinetic energy of the photoelectrons ejected from the surface

Maximum KE of Photoelectrons

frequency threshold:

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Compton Shift• American physicist Arthur

Compton (1892-1962) proposed that momentum & energy should be conserved in a collision between photons & electrons

• After a collision, scattered photon should have a lower energy, therefore a lower frequency (longer wavelength)

• In 1923, conducted experiments with X rays to demonstrate this change in wavelength, known as Compton shift.

Models of the Atom

• Thomson Model / “Plum Pudding” Model– Discovery of electron

in 1897– Negative electrons in

sphere of positive charge

Models of the Atom

• Rutherford Model / Planetary Model– 1911 experiment by

Geiger & Marsden demonstrated that practically all of atom’s mass and all positive charge must be centrally located in atom (nucleus)

– Electrons orbit nucleus like planets around Sun

Problems with theRutherford Model

• Electrons orbiting the nucleus would undergo centripetal acceleration

• Accelerating electrons would radiate EM waves• Electrons radiating EM waves would lose energy• Loss of energy would cause electron’s orbital

radius to drop• Frequency of emitted radiation would increase• Electrons would rapidly collapse into nucleus

Need a better model!

Atomic Spectra• Fill a glass tube with pure atomic gas• Apply a high voltage between electrodes• Current flows through gas & tube glows• Color depends on type of gas• Light emitted is composed of only certain wavelengths

Atomic Spectra• Emission Spectrum: diagram or graph that

indicates the wavelengths of radiant energy that a substance emits (bright lines)

• Absorption Spectrum: same thing, just for light absorbed by a substance (dark lines)

What does this have to do with atomic models?

The Bohr Model• Similar to Rutherford’s model, but only allows certain,

discrete orbits• Electrons are never found between orbits, but can

“jump” from one orbit to another• Electrons only emit radiation when they jump from an

outer orbit to an inner one• Energy of emitted photon is equal to energy decrease

of electron. This determines frequency of emitted radiation.

• Energy of emitted photon is quantized – only certain quantities are allowed. Hence, electrons undergo “quantum leaps”. (Obligatory pop culture reference)

hfEEE finalinitialphoton

Energy Levels & Emission Spectra• Lowest energy state: ground state

– Radius of this state: Bohr radius– Electrons usually here at ordinary temps

• How do electrons “jump” between states?– Absorb photon with energy (hf) exactly equal to

energy difference between ground state & excited state

– Absorbed photons account for dark lines in absorption spectrum

Energy Levels & Emission Spectra• Spontaneous emission:

– Electron in excited state jumps back to a lower energy level by emitting a photon

– Does NOT need to jump all the way back to the ground state

– Emitted photon has energy equal to energy difference between levels

– Accounts for bright lines on emission spectrum

– Jumps between different energy levels correspond to various spectral lines

The Bohr Model

Successes• Account for wavelengths

of all spectral lines of hydrogen

• Provides explanation for auroras

• Gave expression for radius of hydrogen atom

• Predicted energy levels of hydrogen

• Also successful when applied to hydrogen-like atoms (only one electron)

Failures

• Unsuccessful when applied to multi-electron atoms

• Did not explain why electrons do not radiate energy when in a stable orbit

• Did not explain why other orbits do not occur

• Combined classical and non-classical physics

The Dual Nature of Light

• Is light a particle or a wave?– Particle: blackbody radiation, photoelectric effect– Wave: interference, diffraction

• Which model is correct?– Both are correct, but depends on the situation– Each phenomenon exhibits only one or the other

natures of light– True nature of light is not describable in terms of

a single classical idea

The Dual Nature of LightLow Frequency Light

(Wave Nature)

• Very low energy– Difficult to detect a single

photon– Photon nature of light not

evident

• Long wavelength– Wave effects, like

diffraction and interference are easy to observe

High Frequency Light

(Photon Nature)

• Very high energy– Easy to detect single

photons– Photon nature of light is

evident

• Short wavelength– Wave effects, like

diffraction and interference are more difficult to observe

Matter Waves

• Since light can be described as either a particle or a wave, can we do the same for all objects, like atoms and people and cars?

• Louis de Broglie thought so!

• In 1924, proposed that all matter may have wave properties and particle properties

• Matter has a dual nature, just like light!

• Proposed idea of matter waves

Matter Waves

• The larger the momentum of an object, the smaller its wavelength

Matter Waves

• Frequency of matter waves can be found with Planck’s equation

Evidence for Matter Waves• 1927: Davisson & Germer, showed that electrons can

be diffracted by a single crystal of nickel• Electron diffraction is possible because the de Broglie

wavelength of an electron is approx. equal to distance between atoms (the size of the diffraction grating)

• Large-scale objects don’t demonstrate this well because large momentum generates wavelengths much smaller than any possible aperture through which the object could pass (won’t be diffracted)

Bohr Model Explained• De Broglie hypothesized that only certain

electron orbits are stable• Circumference of orbit must contain an

integral multiple of electron wavelengths• Similar to standing waves on a string

The Uncertainty Principle

• Wave nature of particles restricts the precision of our measurements

• Werner Heisenberg (1927):– It is fundamentally impossible to make

simultaneous measurements of a particle’s position and momentum with infinite accuracy

– The more we learn about a particle’s momentum, the less we know of its position, and vice versa.

The Uncertainty Principle:A Thought Experiment

• Imagine trying to measure an electron’s position and momentum with a powerful microscope

• In order to see the electron, thereby determining its location, at least one photon of light must bounce off the electron and pass through the microscope to your eye

• When the photon strikes the electron, it transfers some energy & momentum to the electron. So we are less sure of the electron’s momentum.

The Uncertainty Principle:A Thought Experiment

Schrodinger’s Wave Equation

• Erwin Schrodinger (1926) proposed a wave equation for de Broglie’s matter waves

• Each particle can be represented by a wave function , , dependent on the position of the particle and time

The Electron Cloud

• Max Born (1926) interpreted Schrodinger’s wave function to show probability of finding an electron at certain locations

• ||2 is proportional to probability of finding the electron at a certain position

• Peak probability for an electron in the ground state corresponds to Bohr radius

Quantum Mechanical Model• Electrons are not confined to particular orbital

distances as assumed in Bohr model• Electron cloud: a probability cloud

– Density at each location related to probability of finding electron at that location

– Wave function predicts geometry for energy levels (some spherical, others more complex)

– Most probable location still corresponds to Bohr radii, but impossible to determine actual location

• Mathematical picture of the atom that explains certain aspects of atomic structure that Bohr model cannot explain