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

Blackbody Blackbody: 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 Radiation The 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 Independence It 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 behind Planck’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 Energy The 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 Effect When 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 potential Now 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 V s.

The stopping potential and the number of photoelectrons Such an experiment measures the stopping potential V s, the external potential required to stop the flow of current completely. From V s one can deduce the maximum KE of the photoelectrons emitted by the metal, because by conservation of energy: By studying the KE and N e 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 V s, the current is zero

Photoelectric Effect Feature 1 Dependence of ejection of electrons on light frequency Classical Prediction Electrons should be ejected at any frequency as long as the light intensity is high enough Experimental Result No electrons are emitted if the incident light falls below some cutoff frequency, f c, regardless of intensity The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency

Photoelectric Effect Feature 2 Dependence of photoelectron kinetic energy on light frequency Classical Prediction There should be no relationship between the frequency of the light and the electric kinetic energy The kinetic energy should be related to the intensity of the light Experimental Result The maximum kinetic energy of the photoelectrons increases with increasing light frequency

Photoelectric Effect Feature 3 Dependence of photoelectron kinetic energy on light intensity Classical Prediction Electrons should absorb energy continually from the electromagnetic waves As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy Experimental Result The maximum kinetic energy is independent of light intensity The current goes to zero at the same negative voltage for all intensity curves

Photoelectric Effect Feature 4 Time interval between incidence of light and ejection of photoelectrons Classical Prediction For very weak light, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal Experimental Result Electrons are emitted almost instantaneously, even at very low light intensities Less than s

Summary Action KENeNe Increase intensityNo effectsIncrease Increase frequencyIncrease Observation when f >f c : When f <f c no photoeletrons are released, independent of intensity. The cutoff frequency f c depends on the metal. Action KENeNe Increase intensityIncrease Increase frequencyNo effects Classical prediction for all f :

Frequency Dependence and Cutoff Frequency The lines show the linear relationship between KE max 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 Picture The 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 KE max Increase intensity  Increase number of e -

The cutoff frequency and wavelength

Photon Model Explanation of the Photoelectric Effect Dependence of photoelectron kinetic energy on light intensity KE max is independent of light intensity KE depends on the light frequency and the work function The 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 electron Without 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 KE max = hf – ϕ As 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

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

Compton Scattering The loose electrons recoils from the momentum of the photon. The classical wave theory of light failed to explain the scattering of x-rays from electrons. The results could be explained by treating the photons as point-like particles having energy hƒ and momentum hf / c.

Conservation of P and E

Early Models of the Atom – Rutherford Rutherford Planetary model Based on results of thin foil experiments Positive charge is concentrated in the center of the atom, called the nucleus Electrons 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 ( s )… 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 levels Bohr’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

Terminology Ground state: n =1 First excited state: n =2 Second excited state: n =3 Ionization energy: The energy required to free an electron = E ∞ -E 1 For hydrogen, the ionization energy is: E ∞ -E 1 = 0 - (-13.6eV) = 13.6eV

Energy transition of electron

Emitting a photon Find 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

Laser Light Amplification by the Stimulated Emission of Radiation. Based on three processes: a)Absorption b)Spontaneous emission c)Stimulated emission

Stimulated Emission A 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 Laser Selection 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.