Chapter 28Quantum Physics(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

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-

Photon Model Explanation of the Photoelectric Effect

Dependence of photoelectron kinetic energy on light intensity

KEmax is independent of light intensityKE depends on the light frequency and the work functionThe intensity will change the number of photoelectrons being emitted, but not the energy of an individual electron

Time interval between incidence of light and ejection of the photoelectron

Each photon can have enough energy to eject an electron immediately

Photon Model Explanation of the Photoelectric Effect, cont

Dependence of ejection of electrons on light frequency

There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electronWithout enough energy, an electron cannot be ejected, regardless of the light intensity

Photon Model Explanation of the Photoelectric Effect, cont

Dependence of photoelectron kinetic energy on light frequency

Since KEmax = hf – ϕAs the frequency increases, the kinetic energy will increase

Once the energy of the work function is exceeded

There is a linear relationship between the kinetic energy and the frequency

The cutoff frequency and wavelength

Rewriting hc

Photoelectric Effect Features, Summary

The experimental results contradict all four classical predictions

Einstein extended Planck’s concept of quantization to electromagnetic waves

All electromagnetic radiation can be considered a stream of quanta, now called photons

A photon of incident light gives all its energy hf to a single electron in the metal

Photons and Waves Revisited Some experiments are best

explained by the photon model

Some are best explained by the wave model

We must accept both models and admit that the true nature of light is not describable in terms of any single classical model

Light has a dual nature in that it exhibits both wave and particle characteristics

The particle model and the wave model of light complement each other

Louis de Broglie

1892 – 1987

Originally studied history

Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons

Pronounced “de broy”

Wave Properties of Particles

Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties

The de Broglie wavelength of a particle is

Frequency of a Particle

In an analogy with photons, de Broglie postulated that particles would also have a frequency associated with them

These equations present the dual nature of matter:• particle nature, E and p• wave nature, f and λ

Particle / Wave Duality Summary

The two equations can be rewritten as:

Defining the angular frequency ω and the wave number k:

Relativistic or Non-relativistic

Use Non-relativistic Version in HW

Make sure your KE is in J!

ExampleFind the wavelength of a non-relativistic electron traveling with KE = 20eV.

Electron Diffraction, Set-Up

Electron Diffraction, Experiment

Parallel beams of mono-energetic electrons are incident on a double slit

The slit widths are small compared to the electron wavelength

An electron detector is positioned far from the slits at a distance much greater than the slit separation

Electron Diffraction, contIf the detector collects electrons for a long enough time, a typical wave interference pattern is produced

This is distinct evidence that electrons are interfering, a wave-like behavior

The interference pattern becomes clearer as the number of electrons reaching the screen increases

Electron Diffraction ExplainedAn electron interacts with both slits

simultaneously

If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern

It is impossible to determine which slit the electron goes through

In effect, the electron goes through both slits

The wave components of the electron are present at both slits at the same time

Werner Heisenberg1901 – 1976

Developed matrix mechanics

Uncertainty Principle

Noble Prize in 1932

The Uncertainty Principle, Introduction

In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty

Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy

The inescapable uncertainties do not arise from imperfections in practical measuring instruments

The uncertainties arise from the quantum nature of matter

Heisenberg’s Uncertainty Principle (1D)

You cannot tell the position and the momentum of a particle simultaneously.

Example

Another Uncertainty Principle

Another Uncertainty Principle can be expressed in terms of energy and time:

A particle that have a short life-time Δt will have large uncertainty with its energy ΔE.

Erwin Schrödinger

1887 – 1961

Best known as one of the creators of quantum mechanics

His approach was shown to be equivalent to Heisenberg’s

Schrödinger EquationThe (time-independent) Schrödinger equation of a particle of mass m in a potential energy well V(x) is given by:

The complex function ψ is called the wave function. It determines the probability of all experimental outcomes.

Wave function and probability

Potential Energy for a Particle in a Box

The picture on the right represents the potential energy V(x) of a “box” (or a square well). A particle inside the box cannot go beyond 0 and L because of the infinitely high energy.Solution to the Schrödinger equation gives:

Graphical Representations for a Particle in a Box

Energy is quantized.