Chapter 38Photons, Electrons, and Atoms(About quantization of light, energy and the early foundation of quantum mechanics)

BlackbodyBlackbody: A “perfect” absorber. For example, a hole in a cavity.It turns out a blackbody must also emit radiation, so a blackbody is not really “black”. The radiation from a blackbody depends only on the temperature of the cavity.

Blackbody RadiationThe radiation from a wide variety of sources can be approximated as blackbody radiation:Coal, sun, human body (infrared)As mentioned such radiation depends only on the temperature of the object, and is sometimes refer to as the thermal radiation.

Material IndependenceIt is observed that as an object gets hotter, the predominant wavelength of the radiation emitted by the object decreases (hence the frequency increases). Example:

As temperature increases:Infrared Red Yellow White

This is true regardless of the material that made up the blackbody. Objects in a furnace all glow red with the furnace walls regardless of their size, shape or materials.

Temperature Dependence

The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases:

Conflict with classical physics

Ultraviolet catastrophe

Max Planck and Planck’s constant (1900)Proposed energy on the cavity wall:

h becomes known as the Planck’s constant:

All quantum calculations involves h. Sometimes it is more convenient to use:

The idea behindPlanck’s equation means it is now more difficult (or energy costly) to excite a mode of higher frequency. As a result less high frequency (low wavelength) radiations are produced, preventing ultraviolet catastrophe. Classical, the cost of a high frequency mode is the same as that of a low frequency mode.

Quantization of EnergyThe energy emitted or absorbed by the energy transition of the cavity wall is therefore given by:

The cavity cannot emit half of hf. Energy in the radiation only exists in packages (quanta) of hf.

But why hf?

Even Planck himself could not give a more fundamental reason why the equation E=hf makes sense, except that it appeared to describe blackbody radiation perfectly. Planck continues to try to find a “better” explanation.Today physicists generally accept this equation as an observed fact of nature. Its introduction is regarded as the beginning of quantum mechanics.

Photoelectric EffectWhen light shines on certain metals, electrons are sometimes released. The emitted electrons are sometimes referred to as photoelectrons.

We can measure the energy of the photoelectrons using the setup below:

The Setup

When the external potential ξ is connected as shown, it helps the electrons to flow, generating a non-zero current when photoelectrons are produced.

Reversing the potentialNow the external potential ξ is reversed. It actually resists the flow of the electrons. When the potential is big enough, it can even stop the current completely. This is the stopping potential Vs.

The stopping potential and the number of photoelectrons

Such an experiment measures the stopping potential Vs, the external potential required to stop the flow of current completely.From Vs one can deduce the maximum KE of the photoelectrons emitted by the metal, because by conservation of energy:

By studying the KE and Ne of the photoelectrons, further contradictions with classical physics were found.

On the other hand, the current gives a measurement of the rate of electrons released. Roughly speaking, one can say:

Photoelectric Effect, Results

The maximum current increases as the intensity of the incident light increases

When applied voltage is equal to or more negative than Vs, the current is zero

Photoelectric Effect Feature 1

Dependence of ejection of electrons on light frequency

Classical PredictionElectrons should be ejected at any frequency as long as the light intensity is high enough

Experimental ResultNo electrons are emitted if the incident light falls below some cutoff frequency, fc, regardless of intensityThe cutoff frequency is characteristic of the material being illuminatedNo electrons are ejected below the cutoff frequency

Photoelectric Effect Feature 2

Dependence of photoelectron kinetic energy on light frequency

Classical PredictionThere should be no relationship between the frequency of the light and the electric kinetic energyThe kinetic energy should be related to the intensity of the light

Experimental ResultThe maximum kinetic energy of the photoelectrons increases with increasing light frequency

Photoelectric Effect Feature 3

Dependence of photoelectron kinetic energy on light intensity

Classical PredictionElectrons should absorb energy continually from the electromagnetic wavesAs the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy

Experimental ResultThe maximum kinetic energy is independent of light intensityThe current goes to zero at the same negative voltage for all intensity curves

Summary

Action KE Ne

Increase intensity No effects Increase

Increase frequency Increase Increase

Observation when f >fc :

When f <fc no photoeletrons are released, independent of intensity.The cutoff frequency fc depends on the metal.

Action KE Ne

Increase intensity Increase Increase

Increase frequency No effects No effects

Classical prediction for all f :

Frequency Dependence and Cutoff Frequency

The lines show the linear relationship between KEmax and f

The slope of each line is h

The absolute value of the y-intercept is the work function

The x-intercept is the cutoff frequency

This is the frequency below which no photoelectrons are emitted

Some Work Function Values

Einstein’s Explanation• Energy in light comes in packages (photons). Each

photon carries energy E=hf. You cannot get half a photon or 1/3 of a photon.• The intensity of light is related to the number of

photons present, but not to the frequency.• Electrons are bind to the metal, so for an electron to

escape, it needs to absorb a certain threshold amount of energy ϕ, called the work function. Each metal has a different value for ϕ. The stronger the binding to the metal, the larger is ϕ.

The PictureThe picture:An electron absorbs energy hf from the radiation, spends ϕ to escape from the metal, leaving only hf - ϕ as the KE:

This explains why the slope of each line is h.

Increase f Increase KEmax

Increase intensity Increase number of e-

The cutoff frequency and wavelength

Rewriting hc

Early Models of the Atom – Rutherford

RutherfordPlanetary modelBased on results of thin foil experimentsPositive charge is concentrated in the center of the atom, called the nucleusElectrons orbit the nucleus like planets orbit the sun

The trouble with the atom

Maxwell’s equations says that all accelerating charge must radiate. As electrons orbits the nucleus it must also radiates continuously, hence losing energy.

Result: The electron theoretically should spiral into the nucleus in a very short time (10-8s)… and we should all be dead.

The Bohr Theory of Hydrogen

In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory

His model includes both classical and non-classical ideas

He applied Planck’s ideas of quantized energy levels to orbiting electrons

In this model, the electrons are generally confined to stable, non-radiating orbits called stationary states

Energy levelsBohr’s work lead to the prediction of the existence of energy levels inside atoms. The energy of an electron when measured must lie in one of the levels, it can never possess energy between two levels. In other words, the energy between the levels are forbidden.In particular, it predicts the existence of the ground state (the lowest energy level). No energy level lies below the ground state.This prevents the decay of the electron orbit because it cannot drop below the ground state.For hydrogen:

Bohr’s Hydrogen

TerminologyGround state: n =1First excited state: n =2Second excited state: n =3

Ionization energy: The energy required to free an electron = E∞-E1

For hydrogen, the ionization energy is: E∞-E1 = 0 - (-13.6eV) = 13.6eV

Energy transition of electron

Emitting a photonFind the frequency of the photon emitted when an electron drops from n=5 to n=2.

Find the wavelength and frequency for the following transitions (n):

λ f

Single electron ions other than hydrogen

Atomic spectrum

LaserLight Amplification by the Stimulated Emission of Radiation.Based on three processes:

a) Absorptionb) Spontaneous emissionc) Stimulated emission

Stimulated EmissionA photon of frequency f passes and it triggers an excited electron to fall to the lower level.

Same energy, same phase, polarization, direction.

Summary

Population Inversion

When there are more excited atoms than atoms at ground state, it is said to be population inverted. This can only happen when the system is not in thermal equilibrium.

He-Ne LaserSelection rules forbid He 2s level from decaying via radiation, so a population inversion is created. It can decay via collision with Ne, hence creating a population inversion in Ne between the 5s and 3p levels. The decay from 5s to 3p is the laser beam.