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**Blackbody Radiation Photoelectric Effect Wave-Particle Duality SPH4U**

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**Everything comes unglued**

The predictions of “classical physics” (Newton’s laws and Maxwell’s equations) are sometimes completely, utterly WRONG. classical physics says that an atom’s electrons should fall into the nucleus and STAY THERE. No chemistry, no biology can happen. classical physics says that toaster coils radiate an infinite amount of energy: radio waves, visible light, X-rays, gamma rays,…

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**The source of the problem**

It’s not possible, even “in theory” to know everything about a physical system. knowing the approximate position of a particle corrupts our ability to know its precise velocity (“Heisenberg uncertainty principle”) Particles exhibit wave-like properties. interference effects!

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**The scale of the problem**

Let’s say we know an object’s position to an accuracy Dx. How much does this mess up our ability to know its speed? Here’s the connection between Dx and Dv (Dp = mDv): That’s the “Heisenberg uncertainty principle.” h 6.610-34 J·s “It is physically impossible to predict simultaneously the exact position and exact momentum of a particle.”

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**Atomic scale effects Small Dx means large Dv since**

Example: an electron (m = 9.110-31 kg) in an atom is confined to a region of size x ~ 510-11 m. How is the minimum uncertainty in its velocity? Plug in, using h = 6.610-34 to find v > 1.1106 m/sec

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Example The speed of an electron (m = 9.110-31 kg) is measured to have a value of 5 x 103 m/s to an accuracy of percent. Determine the uncertainty in determining its position.

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Example The speed of an bullet (m = kg) is measured to have a value of 300 m/s to an accuracy of percent. Determine the uncertainty in determining its position.

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Example A proton has a mass of 1.67 x kg and is close to motionless as possible. What minimum uncertainty in its momentum and in its kinetic energy must it have if it is confined to a region : 1.0 mm An atom length 5.0 x 10-10m About the nucleus of length 5.0 x 10-15m

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Example A proton has a mass of 1.67 x kg and is close to motionless as possible. What minimum uncertainty in its momentum and in its kinetic energy must it have if it is confined to a region : 1.0 mm

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Example A proton has a mass of 1.67 x kg and is close to motionless as possible. What minimum uncertainty in its momentum and in its kinetic energy must it have if it is confined to a region : An atom length 5.0 x 10-10m

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Example A proton has a mass of 1.67 x kg and is close to motionless as possible. What minimum uncertainty in its momentum and in its kinetic energy must it have if it is confined to a region : About the nucleus of length 5.0 x 10-15m Notice that when we consider a particle (say a proton), that is confined to a small region, the Quantum Mechanics requires that such a particle cannot have a precise momentum (or even momentum of zero). This means that even at absolute zero, this proton must have kinetic energy. This energy is called the “zero point energy”, and there is no way to avoid this.

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**Quantum Mechanics! At very small sizes the world is VERY different!**

Energy can come in discrete packets Everything is probability; very little is absolutely certain. Particles can seem to be in two places at same time. Looking at something changes how it behaves.

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**Another Consequence of Heisenberg’s Uncertainty Principle**

A quantum particle can never be in a state of rest, as this would mean we know both its position and momentum precisely Thus, the carriage will be jiggling around the bottom of the valley forever

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Blackbody Motivation The black body is importance in thermal radiation theory and practice. The ideal black body notion is importance in studying thermal radiation and electromagnetic radiation transfer in all wavelength bands. The black body is used as a standard with which the absorption of real bodies is compared.

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Blackbody Radiation Hot objects glow (toaster coils, light bulbs, the sun). As the temperature increases the color shifts from Red to Blue. The classical physics prediction was completely wrong! (It said that an infinite amount of energy should be radiated by an object at finite temperature.) Note humans are ‘hot’ 300K so we emit light, just not much in the visible spectrum. Try infrared.

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**Maxwell Boltzmann Distributions**

In 1859 Scottish physicist James Clerk Maxwell developed his theory on the kinetic theory of gasses that explained the macroscopic properties of pressure, temperature and volume. If we accept the idea that when we heat a gas, that heating causes the molecules to move faster and thus bang into the walls of the container more frequently. O.K. James how does this kinetic theory of gasses relate temperature to pressure. I think gas consists of billions and billions of fast randomly moving molecules that bounce off each other as well as the walls of the container.

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**Maxwell Boltzmann Distributions**

Maxwell statement was a bold one. He claimed that the macroscopic properties of a gas (that was easily measured in a laboratory) could be predicted by a microscopic model that consisted of billions and billions of gas molecules. The molecules act like tiny spherical marbles. With the diameter of the marbles much smaller than the distance between them. The collisions between the molecules where elastic (no energy was lost). In between the collisions the molecules moved according to Newton’s Laws (constant speed and straight line). The initial position and velocity of each molecule was random. My last statement uniquely allowed me to apply a branch of mathematics called statistics to prove my theory was quantitatively consistent with the physical properties of gasses that where measured in the laboratories. I had to make 4 assumptions before I was comfortable with my kinetic model.

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**Maxwell Boltzmann Distributions**

602,214,179,300,000,000,000,000 You used statistics and thus used the averages. Why did you not calculate the motion of the molecules using my Laws? Man, I just couldn’t. There are just too many gas molecules, the task would be impossible. Even in a small sample called a mole, there are 6x1023 molecules. Temperature is actually the measure of microscopic mean square velocity (average velocity multiplied by itself). Maxwell’s theory is the prediction of the probable velocity distribution of the molecules. That is it gives the range of velocities.

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**Maxwell Boltzmann Distributions**

Because all atoms of an element have roughly the same mass, the kinetic energy of identical atoms is determined by velocity (KE= ½mv2) I devised an apparatus that allowed me to determine the kinetic energies and thus the velocities How did you go about measuring these average values?

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**Maxwell Boltzmann Distributions**

Fraction of molecules Kinetic energy Molecules hit disk here first Molecules hit disk last The faster moving (higher Kinetic Energy molecules) start hits the disk early at around 11 The slower moving (lower Kinetic Energy molecules) start hits the disk later at around 3 The resulting disk looks like this. If we plot the intensity of the dots on a graph we get a graph of fraction of atoms/molecules vs. kinetic energy:

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**Maxwell Boltzmann Distributions**

Fraction of molecules Kinetic energy Why is the graph skewed? This curve is characteristic of all molecules The curve is elongated due to how atoms collide, and to the units of the graph Recall all particles are in motion. An average speed will be reached. The graph is skewed because 0 is the lower limit, but theoretically there is no upper limit More than that the graph is skewed because the x-axis has units of energy not velocity. velocity Same data, different axes.. v=1, KE=1 v=2, KE=4 v=3, KE=9

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**Maxwell Boltzmann Distributions**

Molecules with “most probable speed” Distribution of Kinetic Energy at temperature = T1 No. of molecules with K.E. Ea rxn occurs! Area under curve total no. of molecules

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**Maxwell Boltzmann Distributions**

Ludwig Boltzmann, building upon the work of Maxwell formalized the Theorem of the Equipartition of Energy. That is: the average kinetic energy in the translational motion of a molecule should equal the average kinetic energy in its rotational motion. I also looked at what happens to the particles when the energy of the system decreases. This gave a new interpretation of the Second Law to Thermodynamics. When a system reaches thermal equilibrium, all the energy will be equally shared among all degrees of freedom.

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**The Second Law of Thermodynamics?**

Maxwell Boltzmann Distributions The Second Law of Thermodynamics? When a system starts to fall apart, the atoms in the system become more disordered and the entropy increases. This disorder can be measured as the probability of that particular system. That is, it is defined as the number of ways the system can be assembled from its collection of atoms. k – Known as Boltzmann’s constant (1.38x10-23 J/K) W – probability that a particular arrangement will occur Boltzmann created the field of Statistical Mechanics, a tool that where the properties of macroscopic bodies are predicted by the statistical behaviour of their microscopic parts.

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**Black Body Radiation Max Planck**

When an object is heated it releases energy in the form of a broad spectrum of electromagnetic waves. A black body is an ideal body which allows the whole of the incident radiation to pass into itself (without reflecting the energy) and absorbs within itself this whole incident radiation. This propety is valid for radiation corresponding to all wavelengths and to all angels of incidence. Therefore, the black body is an ideal absorber and emitter of radaition. The blackbody will then radiate at a wavelength that is related to its absolute temperature. One should picture a hot oven with an open door emitting radiation into its cooler surroundings or, if the surroundings are hotter, one pictures a cool oven with an open door taking in radiation from its surroundings. It is the open oven door, which is meant to look black—and hence absorbs all colours or frequencies. This gives rise to the term black body.

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Black Body Radiation We noticed that the dominant wavelength (the highest peak on the curve) shifts to a lower wavelength (or higher frequency) when the temperature increases. When measurements were made of the radiation leaving the opening of the oven, it was discovered that the intensity (or brightness) of the radiation leaving the oven corresponded to the Wavelength of the radiation.

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Black Body Radiation This means that in ideal conditions, the radiation depends only on the temperature. So any substance (metal, glass, coal, you, me) that is at a temperature of 4227 oC (4500K) will glow orange-yellow in colour

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Black Body Radiation Fraction of molecules Kinetic energy Hey, Max! Did you notice the shape of your graph looks very similar to the shape of my graph Then, maybe we can interpret the electromagnetic waves jostling around inside the oven statistically like the way you did with the jostling molecules inside a container.

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**Black Body Radiation Predicted by Theory**

The theory predicted that the amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. But the predicted continual increase in radiated energy with frequency (dubbed the "ultraviolet catastrophe") did not happen. Observed in Nature When my colleagues tried this they came up with an intensity equation, that agreed well at the large wavelengths (low frequencies) , but at shorter wavelengths the intensity went off the scale so that it predicted an infinite intensity at the ultraviolet or higher region of the spectrum. Nature knew better!

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**The Rayleigh-Jeans model certainly did not match the data.**

Black Body Radiation Yes, the model says if you opened the oven to look inside you would be vapourized buy a UV blast. You are right, the amount of radiation predicted by the theory would be unlimited, and as the temperature rose the intensity would even increase. “Is it hot in here?” What went wrong? The Rayleigh-Jeans model applied the statistical model to waves, rather than the particles that were used in Maxwell’s model. You can fit a infinite number of waves at higher and higher frequency into a container. This lead to the model being used incorrectly. The Rayleigh-Jeans model certainly did not match the data.

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Black Body Radiation Planck introduced the idea of electric oscillators inside the cavity along with the notion that all possible frequencies being present. This blackbody problem intrigued me. I looked at the data collected by numerous scientists. I came to the conclusion that what ever the proper formula is, it should only contain temperature and radiation frequency (or wavelength), plus a universal constant or two. I also expected the average frequency to increase when the temperature of the oven increased as the hotter walls caused these electric oscillators to vibrate faster and faster until thermal equilibrium was achieved. Planck compared the data with another theory predicted by his friend Wien (which better matched the data) and tried to consolidated them both.

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**Success Eureka I Did it Black Body Radiation Then what happened**

I looked at Wien’s formula and included the requirement of the inclusion of all frequencies. Eureka

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**Energy is Discontinuous**

Black Body Radiation c1 and c2 where constants chosen by Planck to make the equation fit the experiments. Energy is Discontinuous To calculate the probabilities at various arrangements, Planck divided the energy of the oscillators into finite chunks. So the total energy was written as E=ne (where e is arbitrarily small amount of energy) in which e=hf where h is some small constant and f is the frequency. Planck ended up rejecting a classical approach and followed Boltzmann's statistical approach . The graph fits the experimental data exactly. I don’t know what it means, but boy am I happy. That’s it. The small energy packets DID NOT GO TO ZERO, They came in quantum chunks given by e=hf But, why doe this work?

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**Black Body Radiation n=6 n=5 n=4 E=nhf n=3 n=2 n=1 1 2 3 4 5 6 7 8 9**

10 11 Frequency We can now see why the classical theory failed in the high frequency region of the black body curve. Here the energy quanta are so large that only a few energy modes are excited, thus the radiation emitted drops to zero and no ultraviolet catastrophe occurs. There are more modes available at lower frequencies to absorb or emit the correct energy packet. This way we don’t turn into a charcoal briquette when looking at an open fire. I had no reason whatsoever to propose this, but I accepted it. It is not possible for an absorber to absorb or emit energy in a continuous range, but rather it must absorb or emit in discontinuous chunk called Quanta

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**Blackbody Radiation: First evidence for Q.M.**

Max Planck found he could explain these curves if he assumed that electromagnetic energy was radiated in discrete chunks, rather than continuously. The “quanta” of electromagnetic energy is called the photon. Energy carried by a single photon is E = hf = hc/ Planck’s constant: h = X Joule sec Note humans are ‘hot’ 300K so we emit light, just not much in the visible spectrum. Try infrared. E = nhf, n=1, 2, 3, 4

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**Questions A series of light bulbs are glowing red, yellow, and blue.**

Which bulb emits photons with the most energy? The least energy? Blue! Lowest wavelength is highest energy. E = hf = hc/l Red! Highest wavelength is lowest energy. Which is hotter? (1) stove burner glowing red (2) stove burner glowing orange Hotter stove emits higher-energy photons (shorter wavelength = orange)

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**Colored Light Which coloured bulb’s filament is hottest? 1) Red**

Visible Light Which coloured bulb’s filament is hottest? 1) Red 2) Green 3) Blue 4) Same max Coloured bulbs are identical on the inside – the glass is tinted to absorb all of the light, except the color you see.

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Photon A red and green laser are each rated at 2.5mW. Which one produces more photons/second? 1) Red 2) Green 3) Same Red light has less energy/photon so if they both have the same total energy, red has to have more photons!

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**Wein‘s Law Wein Displacement Law**

- It tells us as we heat an object up, its color changes from red to orange to white hot. - You can use this to calculate the temperature of stars. The surface temperature of the Sun is 5778 K, this temperature corresponds to a peak emission = 502 nm = about 5000 Å.

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**Wien’s Displacement Law (nice to know)**

To calculate the peak wavelength produced at any particular temperature, use Wien’s Displacement Law: T · peak = *10-2 m·K temperature in Kelvin!

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**The Wave – Particle Duality**

OR

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Light Waves Until about 1900, the classical wave theory of light described most observed phenomenon. Light waves: Characterized by: Amplitude (A) Frequency (n) Wavelength (l) Energy of wave is a A2

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**Waves or Particles ? Physical Objects:**

Ball, Car, cow, or point like objects called particles. They can be located at a location at a given time. They can be at rest, moving or accelerating. Falling Ball Ground level

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**Waves or Particles ? Common types of waves:**

Ripples, surf, ocean waves, sound waves, radio waves. Need to see crests and troughs to define them. Waves are oscillations in space and time. Direction of travel, velocity Up-down oscillations Wavelength ,frequency, velocity and amplitude defines waves

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**Particles and Waves: Basic difference in behaviour**

When particles collide they cannot pass through each other ! They can bounce or they can shatter

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**Waves and Particles Basic difference:**

Waves can pass through each other ! As they pass through each other they can enhance or cancel each other Later they regain their original form !

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**And then there was a problem…**

In the early 20th century, several effects were observed which could not be understood using the wave theory of light. Two of the more influential observations were: 1) The Photo-Electric Effect 2) The Compton Effect

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Photoelectric Effect Electrons are attracted to the (positively charged) nucleus by the electrical force In metals, the outermost electrons are not tightly bound, and can be easily “liberated” from the shackles of its atom. It just takes sufficient energy… Classically, we increase the energy of an EM wave by increasing the intensity (e.g. brightness) A = amplitude of the wave. Energy a A2 But this doesn’t work ??

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PhotoElectric Effect An alternate view is that light is acting like a particle The light particle must have sufficient energy to “free” the electron from the atom. Increasing the Intensity is simply increasing the number of light particles, but its NOT increasing the energy of each one! Increasing the Intensity does diddly-squat! However, if the energy of these “light particle” is related to their frequency, this would explain why higher frequency light can knock the electrons out of their atoms, but low frequency light cannot…

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Nobel Trivia For which work did Einstein receive the Nobel Prize? 1) Special Relativity E = mc2 2) General Relativity Gravity bends Light 3) Photoelectric Effect Photons 4) Einstein didn’t receive a Nobel prize.

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Photoelectric Effect Light shining on a metal can “knock” electrons out of atoms. Light must provide energy to overcome Coulomb attraction of electron to nucleus

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The Apparatus When the emission of photoelectrons from the cathode occurs, they travel across the vacuum tube toward the anode, due to the applied potential. Even when the variable potential is dropped to zero, the current does not drop to zero, because the kinetic energy of the electrons is still adequate enough to allow some to cross the gas (thus creating a current). If we make the variable source of electrical potential negative then this has the effect of reducing the electron flow. If the anode is made more negative, relative to the cathode, a potential difference, the cutoff potential, V0, is reached when the electrons are all turned back. The cutoff potential corresponds to the maximum kinetic energy of the photoelectrons. They do not have the KE to make it across the gap.

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**Classical physics prediction**

Electrons can be emitted regardless of the incident frequency, though it will take longer time for smaller incident wave amplitude. There should be a time delay between the wave illumination and the emission of electrons. The higher the wave intensity, the higher electron energy, and thus the higher the stopping voltage.

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**Modern physics explanation**

The electromagnetic wave consists of many lumped energy particles called photons. The energy of each individual photon is given by the Joule

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**Modern physics explanation**

If N is the total number of photons incident during time interval T, then the total incident optical energy in Joules is: The incident energy per second (power) is given by: n=N/T is the number of incident photons per second. Watt = J/Sec.

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**Modern physics explanation**

Interaction (absorption / emission) between the electromagnetic wave and matter occurs through annihilation/creation of a quantized energy (photon). In the photoelectric effect, each single absorbed photon gives its total energy (hf) to one single electron. This energy is used by the electron to: Overcome the attraction force of the material. Gain kinetic energy when freed from the material.

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**Modern physics explanation**

Work function (): It is the minimum energy required by an electron to be free from the attraction force of the metal ions. Some of the electrons may need more energy than the work function to be freed. Total Energy +ve Zero The most energetic electrons in the material -ve

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**Modern physics explanation**

Total Energy +ve hf Zero The most energetic electrons in the material -ve

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**Modern physics explanation**

The most energetic electron outside the material hf Total Energy +ve hf Zero -ve

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**Modern physics explanation**

The electrons that need only the work function to be freed, will have the greatest kinetic energy outside the metal. The electrons requiring higher energy to be freed, will have lower kinetic energy.

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**Modern physics explanation**

Thus, there is a minimum required photon energy (hfo) to overcome the work function of the material (note f0 is called the cutoff frequency). If the incident photon energy is less than the work function, the electron will not be freed from the surface, and no photoelectric effect will be observed. = No photoelectric current If hf< If f< fo

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**Modern physics explanation**

The most energetic electrons are stopped by the reverse biased stopping potential Vo.

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**Modern physics explanation**

Slope = h/e The stopping potential doesn’t depend on the incident light intensity. The stopping potential depends on the incident frequency.

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**Photoelectric Equation**

Since the cutoff potential is related to the maximum kinetic energy with which the photoelectrons are emitted: for a photoelectron of charge e and kinetic energy Ek, and retarding potential V0. Then we have (loss is KE = gain in PE) : Ek=eV0. Ephoton(hf)=Φ+Ek (Φ, the work function, is energy with which the electron is bound to the surface, Ek is the kinetic energy of the ejected photoelectron) Ek=hf-Φ : This tells us that if f is small such that hf=Φ, no electrons will be ejected.

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Threshold Frequency Photoelectrons are emitted from the photoelectric surface when the incident light is above a certain frequency f0, called the threshold frequency. Above the threshold frequency, the more intense the light, the greater the current of photoelectrons

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Threshold frequency The intensity (brightness) of the light has no effect on the threshold frequency. No matter how intense the incident light, if it is below the threshold frequency, not a single photoelectron is emitted.

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**Photoelectric Effect Summary**

Each metal has “Work Function” (Φ) which is the minimum energy needed to free electron from atom. Light comes in packets called Photons E = h f h=6.626 X Joule sec Maximum kinetic energy of released electrons K.E. = hf – Φ Photoelectrons are emitted from the photoelectric surface when the incident light is above a certain frequency f0, called the threshold frequency.

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**Photoelectric Effect (Summary)**

“Classical” Method Vary wavelength, fixed amplitude electrons emitted ? What if we try this ? Increase energy by increasing amplitude electrons emitted ? No No No Yes, with low KE No If light was really a wave, it was thought that if one shined light of a fixed wavelength on a metal surface and varied the intensity (made it brighter and hence classically, a more energetic wave), eventually, electrons should be emitted from the surface. Yes, with high KE No No electrons were emitted until the frequency of the light exceeded a critical frequency, at which point electrons were emitted from the surface! (Recall: small l large n) Another symbol for frequency

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**Photo-Electric Effect (Summary)**

In this “quantum-mechanical” picture, the energy of the light particle (photon) must overcome the binding energy (work function, Φ) of the electron to the nucleus. If the energy of the photon does exceed the binding energy, the electron is emitted with a KE = Ephoton – Ebinding. The energy of the photon is given by E=hn, where the constant h = 6.6x10-34 [J s] is Planck’s constant. “Light particle” Before Collision After Collision

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Summary If light is under your control: You can set the frequency (wavelength, colour) and intensity. Your apparatus can count any ejected electrons. You create a higher potential relative to the metal plate, then the ejected electrons will be pulled into the collector and forced into the ammeter circuit. If you are interested in the energy of the ejected electrons, you would make the potential of the collector for and more negative with respect to the surface and eventually you will reach a voltage level where the ejected electrons can no longer reach the collector. This potential is called the Stopping potential, Vo. The maximum kinetic energy of the ejected electrons will then be: By the definition of the eV, the Stopping Potential expressed in volts will have the same numerical value as the electron energy expressed in eV. That is a Stopping Potential of 2.7 V implies a maximum electron energy of 2.7 eV

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Summary How does this explain the photoelectric effect? For our metal with 2.7 eV work function, then a single photon would need an energy of 2.7 eV to eject an electron. If you used red light (650 nm), then the photons in the beam would have energy 1eV=1.60x10-19J These photons will be absorbed, but they do not have enough energy to eject electrons.

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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 J sec. It NEVER changes. All lines on this type of graph will be parallel, only differing in their y-axis intercept (-f) and their x-axis intercept (the threshold frequency). The threshold frequency 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 = hf – Φ hf =Φ Ephoton =Φ Note that red light has such a low frequency (energy) that it will never eject photoelectrons - that is, the energy of a red photon is less than the work function of the metal. Energy (eV) Slope= Planck’s constant, h Curve for material 1 Curve for material 2 fo (material 1) Φ (material 1) fo (material 2) Frequency (Hz) Φ (material 2)

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Review If suitable light is allowed to fall on plate 'P', it will give out photo electrons as shown in the figure. The photo electrons are attracted by the collector 'C' connected to the +ve terminal of a battery. The glass tube is evacuated. When the collector 'C' is kept at +ve potential, the photo electrons are attracted by it and a current flows in the circuit which is indicated by the galvanometer. The negative potential of the plate 'C' at which the photo electric current becomes zero is called Stopping Potential or cut-off potential. Stopping potential is that value of retarding potential difference between two plates which is just sufficient to halt the most energetic photo electrons emitted. It is denoted by "Vo" The Minimum amount of energy which is necessary to start photo electric emission is called Work Function. If the amount of energy of incident radiation is less than the work function of metal, no photo electrons are emitted. It is denoted by Φ. Work function of a material is given by Φ=hf0. It is a property of material. Different materials have different values of work function. Threshold frequency is defined as the minimum frequency of incident light which can cause photo electric emission i.e. this frequency is just able to eject electrons with out giving them additional energy. It is denoted by f0.

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Question What happens to the rate electrons are emitted when increase the brightness? more photons/sec so more electrons are emitted. Rate goes up. What happens to max kinetic energy when increase brightness? no change: each photon carries the same energy as long as we don’t change the color of the light

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**Photoelectric Effect: Light Frequency**

What happens to rate electrons are emitted when increase the frequency of the light? as long the number of photons/sec doesn’t change, the rate won’t change. What happens to max kinetic energy when increase the frequency of the light? each photon carries more energy, so each electron receives more energy.

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**Question Which drawing of the atom is more correct?**

This is a drawing of an electron’s p-orbital probability distribution. At which location is the electron most likely to exist? 1 2 3

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Question You observe that for a certain metal surface illuminated with decreasing wavelengths of light, electrons are first ejected when the light has a wavelength of 550 nm. Determine the work function for the material. Determine the Threshold Potential when light of wavelength 400 nm is incident on the surface

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Question You observe that for a certain metal surface illuminated with decreasing wavelengths of light, electrons are first ejected when the light has a wavelength of 550 nm. Determine the work function for the material. It is quicker is we use hc=1240eV nm

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Question You observe that for a certain metal surface illuminated with decreasing wavelengths of light, electrons are first ejected when the light has a wavelength of 550 nm. Determine the Threshold Potential when light of wavelength 400 nm is incident on the surface

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Question Suppose you find that the electric potential needed to shut down a photoelectric current is 3 volts. What is the maximum kinetic energy of the photoelectrons. The given potential is the stopping potential V0 This is the maximum kinetic energy of the photoelectron

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Question If the work function of the material is known to be 2eV, what is the cut-off frequency of the photons for this material. The cutt-off frequency is the frequency above which electrons can be freed from the material. That is, the frequency of radiation whose energy is equal to the work function or

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**So is light a wave or a particle ?**

On macroscopic scales, we can treat a large number of photons as a wave. When dealing with subatomic phenomenon, we are often dealing with a single photon, or a few. In this case, you cannot use the wave description of light. It doesn’t work !

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**Is Light a Wave or a Particle?**

Electric and Magnetic fields act like waves Superposition, Interference and Diffraction Particle Photons Collision with electrons in photo-electric effect Both Particle and Wave !

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**Are Electrons Particles or Waves?**

Particles, definitely particles. You can “see them”. You can “bounce” things off them. You can put them on an electroscope. How would know if electron was a wave? Look for interference!

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**Young’s Double Slit w/ electron**

2 slits-separated by d Source of monoenergetic electrons Go to physics 2000 web site for JAVA version L Screen a distance L from slits

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Electrons are Waves? Electrons produce interference pattern just like light waves. Need electrons to go through both slits. What if we send 1 electron at a time? Does a single electron go through both slits?

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**Electrons are Particles**

If we shine a bright light, we can ‘see’ which hole the electron goes through. (1) Both Slits (2) Only 1 Slit But now the interference is gone!

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**Electrons are Particles and Waves!**

Depending on the experiment electron can behave like wave (interference) particle (localized mass and charge) If we don’t look, electron goes through both slits. If we do look it chooses 1.

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**Electrons are Particles and Waves!**

Depending on the experiment electron can behave like wave (interference) particle (localized mass and charge) If we don’t look, electron goes through both slits. If we do look it chooses 1. I’m not kidding it’s true!

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Schroedinger’s Cat Place cat in box with some poison. If we don’t look at the cat it will be both dead and alive! Here Kitty, Kitty! Poison

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Momentum of a Photon Compton found that the conservation of momentum did hold for X-ray scattering collisions at an angle (Compton effect)

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**Incident X-ray wavelength Scattered X-ray wavelength**

The Compton Effect In 1924, A. H. Compton performed an experiment where X-rays impinged on matter, and he measured the scattered radiation. Louis de Broglie M A T E R Incident X-ray wavelength l1 l2 > l1 Scattered X-ray wavelength l2 e Electron comes flying out Problem: According to the wave picture of light, the incident X-ray gives up energy to the electron, and emerges with a lower energy (ie., the amplitude is lower), but must have l2=l1.

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**Quantum Picture to the Rescue**

If we treat the X-ray as a particle with zero mass, and momentum p = E / c, everything works ! Incident X-ray p1 = h / l1 e Electron initially at rest l2 > l1 Scattered X-ray p2 = h / l2 e pe The relationship that p=E/c is obtained from the Energy-momentum formula which Einstein derived (Special Relativity). He showed that for particles which are moving close to the speed of light, they have a total energy given by: E2 = sqrt( p2c2 + m2c4), Where p = momentum, c=speed of light, and m=mass of the particle. For m=0 (like a photon), we get E=pc, or, by rearranging, p=E/c. This explanation is beyond what I expect of you, but I provided it just in case you’re interested in where this relation p=E/c came from. Compton found that if the photon was treated like a particle with mometum p=E/c, he could fully account for the energy & momentum (direction also) of the scattered electron and photon! Just as if 2 billiard balls colliding!

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**Interpretation of Compton Effect**

“Light particle” l1 l2 Before Collision After Collision The Compton Effect describes collisions of light with electrons perfectly if we treat light as a particle with: p = h/l and E = hn = hc/l = (h/l)c = pc Let the incident photon energy be referred to as E1=hc/l1. Let the emergent photon energy be referred to as E2=hc/l2. The fact that the emergent photon has a larger wavelength than the incident implies that E2 < E1. In other words, it’s as if the photon transferred some of its energy to something else, and emerged with a lower energy. In fact it did. It struck an electron and gave up some of its energy. Therefore, the photon’s energy must be lower after the collision with the electron. Since the wavelength is inversely proportional to the energy, this means the photon’s wavelength is larger after the collision. The difference in energy between the incident and emergent photon is carried away by the struck electron.

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**Compton Scattering (nice to know)**

Compton assumed the photons acted like other particles in collisions Energy and momentum were conserved The shift in wavelength is Compton wavelength

96
**DeBroglie’s Relation p = h / l**

The smaller the wavelength the larger the photon’s momentum! The energy of a photon is simply related to the momentum by: E = pc (or, p = E / c ) The wavelength is related to the momentum by: l = h/p The photon has momentum, and its momentum is given by simply p = h / l .

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**Quantum Summary Particles act as waves and waves act as particles**

Physics is NOT deterministic Observations affect the experiment

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**Four Quantum Paradoxes**

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**Paradox 1 (non-locality): Einstein’s Bubble**

Situation: A photon is emitted from an isotropic source.

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**Paradox 1 (non-locality): Einstein’s Bubble**

Situation: A photon is emitted from an isotropic source. Its spherical wave function Y expands like an inflating bubble.

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**Paradox 1 (non-locality): Einstein’s Bubble**

Situation: A photon is emitted from an isotropic source. Its spherical wave function Y expands like an inflating bubble. Question (Albert Einstein): If a photon is detected at Detector A, how does the photon’s wave function Y at the location of Detectors B & C know that it should vanish?

102
**Paradox 1 (non-locality): Einstein’s Bubble**

It is as if one throws a beer bottle into Lake Ontario. It disappears, and its quantum ripples spread all over the Atlantic. Then in Copenhagen, the beer bottle suddenly jumps onto the dock, and the ripples disappear everywhere else. That’s what quantum mechanics says happens to electrons and photons when they move from place to place.

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**Paradox 2 (Y collapse): Schrödinger’s Cat**

Experiment: A cat is placed in a sealed box containing a device that has a 50% chance of killing the cat. Question 1: What is the wave function of the cat just before the box is opened? When does the wave function collapse? Question 2: If we observe Schrödinger, what is his wave function during the experiment? When does it collapse? The question is, when and how does the wave function collapse. What event collapses it? How does the collapse spread to remote locations?

104
**Paradox 3 (wave vs. particle): Wheeler’s Delayed Choice**

A source emits one photon. Its wave function passes through slits 1 and 2, making interference beyond the slits. The observer can choose to either: (a) measure the interference pattern at plane s1, requiring that the photon travels through both slits. or (b) measure at plane s2 which slit image it appears in, indicating that it has passed only through slit 2. * * * The observer waits until after the photon has passed the slits to decide which measurement to do.

105
**Paradox 3 (wave vs. particle): Wheeler’s Delayed Choice**

Thus, the photon does not decide if it is a particle or a wave until after it passes the slits, even though a particle must pass through only one slit and a wave must pass through both slits. Apparently the measurement choice determines whether the photon is a particle or a wave retroactively!

106
**Paradox 4 (non-locality): EPR Experiments Malus and Furry**

An EPR (einstein Poldalsky Rosen) Experiment measures the correlated polarizations of a pair of entangled photons, obeying Malus’ Law [P(qrel) = Cos2qrel]

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**Paradox 4 (non-locality): EPR Experiments Malus and Furry**

An EPR Experiment measures the correlated polarizations of a pair of entangled photons, obeying Malus’ Law [P(qrel) = Cos2qrel] The measurement gives the same result as if both filters were in the same arm.

108
**Paradox 4 (non-locality): EPR Experiments Malus and Furry**

An EPR Experiment measures the correlated polarizations of a pair of entangled photons, obeying Malus’ Law [P(qrel) = Cos2qrel] The measurement gives the same result as if both filters were in the same arm. Furry proposed to place both photons in the same random polarization state. This gives a different and weaker correlation.

109
**Paradox 4 (non-locality): EPR Experiments Malus and Furry**

Apparently, the measurement on the right side of the apparatus causes (in some sense of the word cause) the photon on the left side to be in the same quantum mechanical state, and this does not happen until well after they have left the source. This EPR “influence across space time” works even if the measurements are light years apart. Could that be used for FTL signaling? Sorry, SF fans, the answer is No!

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**Four Interpretations of Quantum Mechanics**

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**The Collapse Interpretation**

Scientists who subscribe to the Collapse interpretation make a choice. They believe that when you accept the electron’s wave nature, you must give up on the electron’s particle nature. In this interpretation, the electron leaves the source as a particle that is governed by one set of laws, but then “expands” into a spread-out wave as it passes through the slits. The electron is now governed by new laws. However, before we can measure this wavy, spread-out quantum electron it “collapses” back into a particle and arrives at only one of the many possible places on the screen. The consequence of choosing the Collapse interpretation line of thinking is that you must accept that an electron physically changes from particle to wave and back again. These two realities, including the laws that describe them, alternate uncontrollably

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**The Pilot Wave Interpretation**

The Pilot Wave interpretation avoids this unexplained collapse altogether. Scientists who subscribe to this interpretation choose to believe that the electron always exists as a classical particle and is only ever governed by one kind of physical law, for both the familiar classical as well as quantum phenomena. However, to account for the electron’s wave behaviour this description requires the introduction of an invisible guiding wave. In this interpretation, wave-particle duality is explained by assuming that electrons are real particles all of the time, and are guided by an invisible wave. The electron’s wave nature is attributed to this abstract wave, called a Pilot Wave, which tells the electron how to move. To obtain the interference pattern in the double-slit experiment, this wave must be everywhere and know about everything in the universe, including what conditions will exist in the future. For example, it knows if one or two slits are open, or if a detector is hiding behind the slits. The Pilot Wave interpretation embodies all of the quantum behaviour, including all the interactions between classical objects like the electron, the two-slit barrier, and the measuring devices. In contrast to the Collapse interpretation where the collapsing electron wave was considered real, in the Pilot Wave interpretation the wave is an abstract mathematical tool. This interpretation has a consequence. The Pilot Wave interpretation, which was invented to deal with an electron as a real physical object, suffers the fate of being permanently beyond detection

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**The Many-Worlds Interpretation**

Supporters of the Many Worlds interpretation, similar to the Pilot Wave idea, choose to accept that electrons are classical particles. Then they go even further, demanding that all elements of the theory must correspond to real objects—unlike the collapsing electron or the Pilot Wave. Supporters insist on only measurable, physical objects within the world. This world is constantly splitting into many copies of itself. When electrons demonstrate wave behaviour they exist in a superposition of many different states. To Many Worlds supporters, who maintain the idea of an electron as a classical particle, a parallel universe must exist for each of the electron’s possible states. When the electron reaches the slits, it has to choose which slit to go through. At that moment, the entire universe splits into two. In one universe, the electron passes through the left slit as a real particle. In the other universe it passes through the right slit as a real particle. The consequence of accepting the Many Worlds interpretation, with many quantum particles constantly facing similar choices, is the requirement that our universe must be constantly splitting into an almost infinite number of parallel universes, each having its own copy of every one of us

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**The Copenhagen Interpretation**

Advocates of the Copenhagen interpretation choose to limit their discussion directly to the experiment and to the measurements on physical objects. Questions are restricted to what can be seen and to what we actually do. They try to think about experiments in a very honest way, without invoking extra theoretical ideas like the on-off switching of the Collapse idea, or the guidance supplied by the invisible Pilot Wave, or the proposed splitting into Many Worlds. It is tempting to come up with mental pictures about what is happening that go beyond the results of an experiment, and to try to interpret what is happening by means of those hidden theoretical mechanisms. The previous interpretations attributed the mysterious wave–particle duality to imaginative mathematics. In the Copenhagen interpretation much of this mystery is attributed to what happens when an experimenter enters the lab and interacts with the quantum mechanical system. With the Copenhagen perspective, the mathematics only deals with the experimenter’s information about measurement interactions with the quantum mechanical system. The consequence of accepting the Copenhagen interpretation is a fundamental restriction on how much you can read into experimental results. We know that electrons are particles when they are fired from the source, and we know that they are particles when they hit the screen. What happens to electrons in the middle, what they are “doing”, or what they really “are” is not possible to know. In the Copenhagen interpretation these are unfounded questions. We may call an electron a wave or a particle, but ultimately those names are no more than suitable models.

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Blackbody Radiation Photoelectric Effect Wave-Particle Duality sections 30-1 – 30-4 Physics 1161: Lecture 28.

Blackbody Radiation Photoelectric Effect Wave-Particle Duality sections 30-1 – 30-4 Physics 1161: Lecture 28.

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