AP Physics Chapter 27Quantum Physics

Chapter 27: Quantum Physics

27.1 Quantization: Planck’s Hypothesis27.2 Quanta of Light: Photons and the

Photoelectric Effect27.3 Quantum “Particles”: The Compton Effect27.4 The Bohr Theory of the Hydrogen Atom27.5 Omitted

Homework for Chapter 27• Read Chapter 27

• HW 27.A: p.861-862: 16, 18, 19-27.

• HW 27.B: p.863- : 42, 43, 52, 54-58, 61, 63, 64.

27.1: Quantization: Planck’s Hypothesis

I A M C ONV INC ED THA T G OD

DOE S N OT PLA Y D ICE

Some History on the Atomic Nucleus…

J.J. Thomson Model: After discovering the electron in 1897, Sir J.J. Thomson proposed a model of an atom in 1904. This model later came to be known by different names such as the plum-pudding and watermelon models.

In this model the pudding was the positive charge of the atom and electrons were embedded in it like plums. The total positive charge was equal in magnitude to the total negative charge of the electrons. Hence the atom was a neutral particle.

Rutherford’s Model:

In 1911, Ernest Rutherford performed an experiment to observe the scattering of alpha particles by a thin gold foil. (Alpha particles consist of two protons and two neutrons). Based on the plum pudding model, Rutherford expected very little scattering because of the large momentum for alpha particles. He was surprised to observe that some alpha particles scattered through large angles and in fact some of them had back scattered. This was completely inconceivable on the basis of the plum-pudding model.

This remarkable experimental result let Rutherford to revise the atomic model. He could explain the result of his alpha scattering experiment by the nuclear model.According to the nuclear model the positive charge of the atom and most of its mass is concentrated in a very small volume at the center of the atom. This part of the atom came to be called the nucleus of the atom. The electrons revolve around the nucleus in orbits similar to the planets going around the sun.

This model has since been further refined but the basic idea of a tiny atomic nucleus at the center of outer electrons still holds true.

The Electromagnetic Spectrum

Visible Light Spectrum

Max Planck (1858-1947) – A German physicist; considered to be the founder of quantum theory, and thus one of the most important physicists of the twentieth century. Planck was awarded the Nobel Prize in Physics in 1918.

One of the problems scientists had at the end of the nineteenth century was how to explain thermal radiation.

thermal radiation – the continuous spectra of radiation emitted by hot objects. Maximum intensity shifts to shorter wavelengths (higher frequencies) with increasing temperature.

blackbody – an ideal system that absorbs and emits all radiation that falls on it.

A blackbody can be approximated by a small hole leading to an interior cavity in a block of material.

Intensity vs. Wavelength Curves for the Thermal Radiation from an Idealized Blackbody at Different Temperatures

• The location of maximum intensity shifts to shorter wavelengths with increasing temperature.

• The wavelength shift obeys Wein’s displacement law:

maxT = 2.90 x 10-3 m·K

where max is the wavelength of radiation (in meters) at which maximum intensity occurs and T is the temperature of the body (in kelvins).

Example 27.1: What is the most intense color of light emitted by a giant star of surface temperature 4400 K? What is the color of the star?

Classical theory predicts the intensity of thermal radiation is inversely related to the emitted wavelength.

I 1 4

Thus the intensity of the radiation would become infinitely large as the wavelength approaches zero. This was known as the “ultraviolet catastrophe”. In contrast, Plank’s quantum theory predicts the observed radiation distribution.

Max Planck successfully explained the spectrum of blackbody radiation by proposing a radical hypothesis. According to Planck’s hypothesis, the energy of the oscillating atoms emitting the radiation have only discrete, or particular, amounts of energy rather than a continuous distribution of energies. The energy is

E = hf

where E is the energy h is Planck’s constant (6.63 x 10-34 J·s) f is the frequency of the oscillation

On Gold Sheet

• According to Planck’s hypothesis energy is quantized, or occurs in only discrete amounts. A more specific way to represent his hypothesis is

En = n(hf) for n = 1,2,3,…

•The smallest possible amount of energy occurs when n= 1. E1 = hf.

• All other permitted values of energy are integral multiples of hf.

• The quantity hf is called a quantum of energy.

Blackbody Radiation Applet

http://www.mhhe.com/physsci/astronomy/applets/Blackbody/frame.html

Check for Understanding

1.Which scientist is credited with the discovery of the electron?

a) Albert Einsteinb) Count Rutherfordc) Robert Milikand) Max Plancke) J.J. Thomson

Answer: e

2. Which scientist is credited with the discovery of the atomic nucleus?

a) Albert Einsteinb) Count Rutherfordc) Robert Milikand) Max Plancke) J.J. Thomson

Answer: b

Check for Understanding3.

27.2: Quanta of Light: Photons and the

Photoelectric Effect

DO NOT TE LL GOD HO W T O R

UN THE UN IVE RSE

photon – a particle of light

• Photons have no mass, but they can transfer energy to or from electrons.

Summary of Subatomic Particles

• The mass of a proton and neutron equals one atomic mass unit, or amu.

• The electron-volt (eV) is a useful unit of energy for subatomic particles. One eV is equal to the amount of energy needed to change the potential of an electron by one volt.

1 eV = 1.6 x 10-19 J

Name Mass Charge

Proton 1.67 x 10-27 kg = 1 amu Positive

Neutron 1.67 x 10-27 kg = 1 amu Zero

Electron 9.11 x 10-31 kg Negative

Photon 0 Zero

The Photoelectric Effect

Towards the end of the 19the century, it had been experimentally observed that when ultraviolet light was shone on a negatively-charged electroscope, the charged leaves fell closer together; the electroscope discharged. This was the beginnings of the path to understanding what we now call the photoelectric effect.

When light shines on any metal surface, the surface can release electrons. If light were composed of waves, then eventually any wavelength of light should be able to build up enough energy to knock an electron free. However, scientists had discovered that only certain wavelengths worked with each metal and that electrons were either emitted instantaneously, or never emitted. They had also noticed that shorter wavelengths worked better than longer wavelengths.

The equation for the photoelectric effect was first explained by Albert Einstein in 1905.

On Gold Sheet

Some observations …..

•This equation is actually just a restatement of conservation of energy.

•The intensity of the light source affected the number of photoelectrons ejected from the surface since higher intensities permit more photons to strike the surface.

•The frequency of the light source affected the kinetic energy of each photoelectron.

•Since each photon can be absorbed by only ONE photoelectron (that is, there is a one-to-one correspondence), the energy of the photons directly affects the kinetic energy of the released photoelectrons.

hf

The Experiment

• The electrons with the maximum KE can be stopped from completing their journey across the photoelectric tub if there is a stopping potential set up to impede their progress. The formula that relates the KE of these photoelectrons to this stopping potential is

KEmax = UE = qVstopping or eVo

where Vstopping (Vo) is the stopping potential q (e) is the magnitude of the charge on an electron, 1.6 x 10-19 coulombs

• This formula is based on the fact that work is done on charged particles when they cross through an electric field.

• The work done (qV) equals the change in each electron’s KE.

• Incident light on the photoelectric material in a photocell causes the emission of electrons, and a current flows in the circuit.

• The voltage applied to the tube can be changed by means of a variable resistor.

• As the plots of current vs. voltage for the two intensities of monochromatic light show, the current is constant as the voltage increased. However, for negative voltages (by reversal of the battery polarity), the current goes to zero at a particular stopping voltage, which is independent of intensity.

• As would be expected classically, the current is proportional to the intensity of the incident light – the greater the intensity, the more energy there is to free additional electrons.

• The minimum energy needed to free the electrons from the material is called the work function (o).

• According to energy conservation, hf = Kmax + o , that is, the energy of the absorbed photon goes into the work of freeing the electron, and the rest is carried off by that emitted electron as kinetic energy.

• The threshold or cutoff frequency (fo ) is the lowest frequency, or longest wavelength, that permits photoelectrons to be ejected from the surface. At this frequency the photoelectrons have no extra KE (KE = 0) resulting in

0 = hfo - o

hfo = o or Ephoton = o

fo = o

h

Often the photoelectric equation is illustrated on a graph of KE vs. frequency. On this graph, the slope ALWAYS equals Planck’s constant, 6.63 x 10-34 J·s.

All the lines on this type of graph will be parallel, only differing in their y-axis intercept (-) and their x-axis intercept (the threshold frequency).

(think: y = mx+b)

f1 f2 f3

Photoelectric Effect Characteristics (Table 27.1 in textbook)

Characteristic Predicted by wave theory?

1.The photocurrent is proportional to the yesintensity of the light.

2. The maximum KE of the emitted electrons nois dependent on the frequency of the lightbut not on its intensity.

3.No photoemission occurs for light with a frequency nobelow a certain cutoff frequency fo regardless ofits intensity.

4. A photocurrent is observed immediately when the nolight frequency is greater than fo even if the lightintensity is extremely low.

• Albert Einstein received the Nobel Prize for Physics in 1921 for his discovery of the Law of the Photoelectric Effect.

• His work ended the controversy as to whether light had particle properties.

• By invoking the quantum nature of light he was able to explain experimental results that his predecessors could not explain with just the wave model of light. Einstein’s official portrait after

receiving his Nobel Prize in 1921.

Problem-Solving Hint

• Start with the formula E = hf

• Recall from wave theory that the frequency of a wave is related to the wavelength by the formula v = f

• For light, the velocity is c, 3 x 108 m/s, so we can instead write c = f

• This means we can rewrite the equation for the energy of a photon to read

E = hc where hc = 1.24 x 103 eV·nm (on your blue sheet) • This is helpful because typically the wavelength in nm is given in a problem rather than frequency.

•These formulas tell us that a photon with high frequency, and therefore with a small wavelength, is higher in energy than a photon with low frequency and long wavelength. So, gamma rays, for example, are a lot higher energy than radio waves because gamma rays have a higher frequency.

Example 27.2: What is the photon energy of visible light having wavelength 632.8 nm?

Example 27.3: A metal has a work function of 4.5 eV. Find the maximum kinetic energy of the emitted photoelectrons if the wavelength of light falling on the metal is a) 300 nmb) 250 nm

Example 27.4: When light of wavelength 350 nm is incident on a metal surface, the stopping potential of the photoelectrons is measured to be 0.500 V.a)What is the work function of the metal?b)What is the threshold frequency of the metal?c)What is the maximum kinetic energy of the photoelectrons?

Summary

• Thermal radiation, typically produced by hot objects, has a continuous spectrum.

• A blackbody is an ideal system that absorbs and emits all radiation that falls on it.

• Wein’s displacement law states that the wavelength of maximum intensity for radiation from a blackbody is inversely related to its temperature.

• Classical theory states that the wavelength of maximum intensity for radiation from a blackbody is inversely related to its temperature.

• Planck’s constant (h) is the fundamental proportionality constant between energy and frequency of thermal oscillators as well as frequency of a light wave and energy of the corresponding photons.

hf hf hf hfoKE

f

fo fo

Check for Understanding

1.A blackbodya) absorbs all radiation incident on itb) re-emits all radiation incident on itc) emits thermal radiation in a continuous spectrumd) all of these

Answer: d

2. The ultraviolet catastrophe is a consequence of a) Planck’s Theoryb) Classical Theoryc) Einstein’s Theoryd) Rutherford’s Theory

Answer: b

Check for Understanding

3. Which is a true statement about the photoelectric effect?a) Energy in the form of light can cause an atom to eject one of its

electrons.b) The frequency of light must be above a certain value for the ejection to

occur.c) An ejected electron has a KE of zero if the energy of the photon is equal

to the work function.d) all of these

Answer: d

4. A photocurrent is observed whena) the light frequency is above the threshold frequencyb) the energy of the photons is greater than the work functionc) the light frequency is below the threshold frequencyd) both a and b

Answer: d

Check for Understanding5.

Check for Understanding6.

Homework for Chapter 27.1-2

• HW 27.A: p.861-862: 16, 18, 19-27.

27.3: Quantum “Particles”: The Compton Effect

Thomson

Millikan

Rutherford

Bohr

• In 1923, American physicist Arthur H. Compton (1892-1962) explained a phenomenon he observed in the scattering of X-rays from a graphite block by considering the radiation to be composed of quanta.

• His explanation of the observed effect provided additional convincing evidence that, at least in certain types of experiments, light, and electromagnetic radiation in general, is composed of quanta, or “particles” of energy called photons.

• When X-rays of a single wavelength were scattered by the electrons in metal foil, the incident wavelength is increased in the scattered X-rays.

• The wavelength shift grew as the scattering angle increased. The nature of the scattering material did not contribute to the effect.

• This phenomenon came to be known as the Compton effect.

• Compton theorized that an X-ray photon colliding with and electron was like billiard balls in an elastic collision. He reasoned that the incident photon would transfer some energy and momentum to the electron.

• After the collision, the energy and frequency of the scattered photon should be decreased (E=hf) and its wavelength increased ( = c/f).

• He applied the principles of conservation of energy and momentum to develop the formula for the Compton effect:

= 1 - o = C (1- cos)

where o is the wavelength of the incident photon 1 is the wavelength of the scattered photon C is the Compton wavelength of the electron is the scattering angle

Compton wavelength C = h = 2.43 x 10-12 m = 2.43 x 10-3 nmof an electron mec

where h is Planck’s constant m is the mass of an electron

c is the speed of light

• Since the Compton shift is very small, it is only significant for X-ray and gamma- ray scattering where the wavelengths are on the order of C.

• Compton’s equation correctly predicted the observed wavelength shift, and Compton was awarded a Nobel Prize in 1927.

• Einstein’s and Compton’s successes in explaining electromagnetic phenomena in terms of quanta left scientists with two apparently competing theories of electromagnetic radiation.

• Classically, the radiation is pictured as a continuous wave, and this theory satisfactorily explains such wave-related phenomena as interference and diffraction.

• Conversely, quantum theory was necessary to explain the photoelectric and Compton effects correctly.

• These two theories gave rise to a description this is called the dual nature of light. That is, light apparently behaves sometimes as a wave and at other times asphotons or “particles”.

Example 27.5: X-rays of wavelength of 0.200 nm are scattered by a metal. The wavelength shift is observed to be 1.50 x 10-12 m at a certain scattering angle measured relative to the incoming X-ray.a)What is the scattering angle?b)What is the maximum shift possible for the Compton effect?

Summary

• The Compton effect is the wavelength increase of light scattered by electrons or other charged particles.

• The dual nature of light means that light must be thought of as having both particle and wave natures.

Check for Understanding

1.The Compton effect was first observed by usinga) visible lightb) infrared radiationc) ultraviolet lightd) X-rays

Answer: d

2. The wavelength shift for Compton scattering is a maximum whena) the photon scattering angle is 90°b) the electron scattering angle is 90°c) the shift is equal to the Compton wavelengthd) the incident photon is backscattered

Answer: d

Check for Understanding

3. A photon can undergo Compton scattering from a molecule as well as from a free electron. How does the maximum wavelength shift for Compton scattering from a molecule as a unit compare to that from a free electron?

) max increases for the moleculeb) max decreases for the moleculec) max doesn’t change

Answer: b. The Compton wavelength is inversely proportional to the mass of the scattering particle. max = 2C = 2h 1

mc m

4. In Compton scattering, why does the scattered photon always have a longer wavelength than the incident photon?

Answer: From energy conservation, the scattered photon has less energy after scattering because the free electron receives part of the incident energy. Since the energy of a photon is proportional to the frequency of the light or inversely proportional to wavelength, the scattered photon always has a longer wavelength.

27.4: The Bohr Theory of the Hydrogen Atom

Answer: When the light waves strike the transparent material, a chain of absorptions and reemissions occur through the material. The time delay between each absorption and reemission produces an average speed of light less than 3 x 108 meters per second.

•In the 1800s, much experimental work was done with gas discharge tubes. (neon, hydrogen, mercury, etc. vapor)

• Normally, light from an incandescent source (such as a light bulb) exhibits a continuous spectrum.

Spectra

a.

b.

c.

• When a gas is excited by heat or electricity and the light it emits is separated into its component wavelengths by a prism or diffraction grating, the result is a bright-line, or emission, spectrum, such as these of a) barium and b) calcium. Each atom or molecule emits a characteristic pattern of discrete wavelengths.

• When a continuous spectrum consisting of all wavelengths is passed through a cool gas, a series of dark lines is observed. Each line represents a “missing” wavelength – a particular wavelength the gas has absorbed. The wavelengths absorbed by any substance are the same ones it emits when excited. This absorption spectrum of the Sun shows several prominent absorption lines; the gases of the solar atmosphere produce it.

emission spectrum – or bright–line spectrum; a series of bright lines indicating which wavelengths are being emitted.

• A discrete line spectrum is characteristic of the individual atoms or molecules of a particular material.• Emission lines can be used to identify a material with a spectroscope.

absorption spectrum – a series of dark lines superimposed on a continuous spectrum.

• If white light passes through a relatively cool gas, certain frequencies or wavelengths are missing, or absorbed.

• Absorption and emission lines for a gas occur at the same frequencies.

• Hydrogen was under study because it was the simplest atom with one proton and one electron, and had a relatively simple visible spectrum.

• The spectral lines of hydrogen in the visible region is called the Balmer series.

The spectral lines of hydrogen.

• In 1913 an explanation of the spectral lines was given in A Theory of the Hydrogen Atom by Danish physicist Niels Bohr (1885-1962).

Bohr’s Postulates:

1. The hydrogen electron orbits the nuclear proton in a circular orbit (analogous to planets orbiting the Sun).

2. The angular momentum of the electron is quantized in integral multiples of

Planck’s constant, h. L = nh , n=1,2,3,… 2

3. The electron does not radiate energy when it is in certain discrete circular orbits.

4. The electron radiates or absorbs energy only when it makes a transition to

another orbit. hf = Ef - Ei

http://micro.magnet.fsu.edu/primer/java/fluorescence/exciteemit/index.html

Electron Excitation and Emission Simulation

En

rn = 0.0529 n2 nm for n= 1,2,3,…

where En is the energy level of the electronn an integer (quantum number)rn is the radius of the electron orbit

• The Bohr theory predicts that the hydrogen electron can occupy only certain orbits that having discrete radii. Each allowed orbit has a corresponding energy level. The lowest energy level (n=1) is the ground state; those above (n>1) are excited states.

• The orbits are shown at the left, with orbital radius plotted against the 1/r electrical potential of the proton. The electron in the ground state is deepest in the potential well, analogous to the gravitational potential well.

• An electron generally does not remain in an excited state for long. It decays, or makes a transition to a lower energy level, in a short time. The time an electron

spends in an excited state is called the lifetime of the excited state.

• If an electron makes a downward transition for ni to nf state, a photon is released and its energy is equal to the energy difference between the final and initial states:

E = Ef – Ei = 13.6 1 – 1 eV nf

2 ni2

• The wavelength of the photon is then

= hc = 1.24 x 103 eV·nm E E

Hydrogen Spectrum

• For transitions with nf = 1, the spectrum series is called the Lyman series (all ultraviolet).• For transitions with nf = 2, the spectrum series is called the Balmer series (visible if ni = 3,4,5, and 6).• For transitions with nf = 3, the spectrum series is called the Paschen series (infrared).

• In fluorescence, an electron that has been excited by absorbing a photon returns to the ground state in two or more steps. At each step a photon is emitted each with less energy (longer wavelength) than the absorbed light.

Some mineral are fluorescent.

A fluorescent butterfly.

Fluorescent pigment.

Example 27.6: What are the orbital radius and total energy of an electron for a hydrogen atom ina)the ground state andb)the second excited state?

Example 27.7: The electron of a hydrogen atom makes a transition from the fourth excited state to the first excited state. What are the energy and wavelength of the emitted photon?

Summary

• The Bohr theory of the hydrogen atom treats the electron as a classical particle in circular orbit around the proton, held in orbit by the electric attraction force.

• The angular momentum of the electron is quantized in integral numbers of Planck’s constant, h.

•The ground state of a hydrogen atom is the state of lowest energy for the atomic electron (in the smallest orbit, n=1). Excited states have greater energies.

Check for Understanding

1.In his theory of the hydrogen atom, Bohr postulated the quantization ofa) energyb) centripetal accelerationc) lightd) angular momentum

Answer: d

2. An excited hydrogen atom emits light when its electron a) makes a transition to a lower energy levelb) is excited to a higher energy levelc) is in the ground state

Answer: a

Check for Understanding

En

Bohr was able to arrive at a formula for the energy of a single electron atom (hydrogen or a helium ion) in the nth orbit:

Check for Understanding

Homework for Chapter 27.3-4

• HW 27.B: p.863- : 42, 43, 52, 54-58, 61, 63, 64.

Chapter 27 Formulas