Transcript lecture35
Chapter 27 UV Catastrophe Wien’s Law Schrodinger’s Cat I ( , T ) 8 k T 4 This expression is known as the Rayleigh-Jeans law. Exactly how this was derived is NOT important. However, we do note that the intensity of the radiation we observe at a given wavelength and temperature is proportional to the temperature and inversely related to the wavelength. The fact that the radiation is proportional to temperature makes perfect sense. The hotter the blackbody is, the more intense will be the light that is emitted. It’s the second bit that leads to a problem... I ( , T ) 8 k T 4 What happens to the intensity of the radiation as 0? I So the classical theory would predict if we open the door to a heated oven, we would be exposing ourselves to an unhealthy does of x-rays and high energy radiation! Of course, we know that this is not the case, or humans would never have cooked anything! is the name by which this great blunder of classical physics came to be known. Classical physics simply could not predict the observed spectrum of blackbody radiation. Blackbody Radiation Intensity Classical theory Observations Wavelength Observations of the intensity of blackbody radiation as a function of temperature revealed the following: Blackbody Observations Intensity max max max Increasing Temperature Wavelength An empirical formula was derived to describe the relationship between the wavelength at which the object radiates most strongly (max) and the temperature. max T 0.2898 10 mK 2 where the wavelength is measured in m and the temperature is measured in degrees Kelvin. At what wavelength is the intensity of the Sun’s output a maximum? (The surface temperature of the Sun is about 5000 K.) max T 0.2898 10 mK 2 max GREEN 2 2 0.2898 10 mK 0.2898 10 mK T 5000 K max 580nm Right in the middle of the “visible” spectrum! In 1900, Max Planck found that he could explain the observations of blackbody radiation at all wavelengths using the following empirical function: 8 hc I ( , T ) 5 hc / kT (e 1) 8 hc I ( , T ) 5 hc / kT (e 1) The constant h is known as Planck’s constant h 6.626 10 34 Js In trying to connect this formula with a more classical framework, Planck came up with the following explanation... Consider each electron in the blackbody as its own miniature oscillator (like a mass on a spring). These oscillators lose energy when they radiate (that is, give off light). The observations of blackbody radiation can be explained if you permit these oscillators to only occupy certain energy states: E nhf n where n is an integer known as the quantum number and f is the frequency of the oscillator. This hypothesis suggests that a physical system cannot attain all values of energy... En nhf …rather, energy is quantized -- that is to say, it comes in discreet chunks! As you ascend a flight of stairs, your potential energy increases. For this analogy quantum physics would suggest that you can only be found on the landings and are never found “between the stairs.” Of course, when you travel up a flight of stairs, you seem to move through all the space between the landings…nevertheless, the quantum theory did seem to at least explain the observations of the blackbody radiation. Planck himself even had trouble believing that energy was really quantized, however! Once you accept that energy indeed does come in discrete quantities, the world of classical physics begins to turn upside-down! Where things aren’t always what they seem…In fact, things often aren’t UNTIL they’re seen…. For example, take the famous paradox of I’m going to change the story a little bit so as not to offend the cat-lovers among you! Hey! I’m not sure I like this idea... Here’s the problem…A couple of pilgrims are getting ready to cook up their turkey in the microwave oven. Before they put the turkey in the oven, however, they’ve got to kill it first. But neither of them can stand the sight of blood, so they put the turkey inside their special turkey-killing device... Inside the box is a vial of poison that is released only when a sensor inside the box is triggered by the emission of a photon from a particular atom. The chances of this atom radiating are 50/50 each hour. After the poison has been released, the turkey will be dead and ready for cooking. What our Pilgrims want to know is: To find out, we must open the box and look. If we find the turkey dead, we won’t know at which exact moment he died, but we will at least be sure he’s dead before we cook him... But what about BEFORE we open the box to see, is the turkey alive or is he dead? According to QUANTUM PHYSICS, This makes absolutely NO sense classically. Afterall, how can something be both alive and dead at the same time? In the quantum world, the existence of the turkey is governed by what is known as a “wave function.” The wave function simply relates the probability of the turkey being alive or dead at any given instant in time. It doesn’t tell you whether or not the turkey actually IS alive, just the odds... In the world of probabilities, both the dead and alive outcomes exist simultaneously. It is only when our Pilgrims open the box to look and see that what was a probability becomes a reality…i.e., the turkey is either alive or dead. In the quantum world, however, the turkey is both alive and dead until the moment the Pilgrims open the box! Just when we thought classical physics (that is, everything you’ve learned in physics class so far) explained everything pretty well... Blackbody radiation kinda threw a wrench into the works! As you might expect, blackbody radiation was not alone... First discovered by Hertz in 1887 (as a side note in an experiment which demonstrated the wave nature of light as hypothesized by Maxwell), the photoelectric effect could only be explained by a particle model of light! A + V_ V A C evacuated chamber With monochromatic light of an appropriate frequency incident upon the photocathode, a current is detected in the ammeter. The existence of a current indicates that when the light strikes the plate, charges are being released! A + V_ V A C evacuated chamber The freed electrons will gain kinetic energy from the collision with the photons. We can determine what the maximum value of this kinetic energy is by varying the voltage supplied to this circuit. In particular, if we reverse the polarity of the battery, we will eventually stop the current entirely... In 1900, Lenard studied this system in more detail and made the following observations. Current Current High intensity Applied Voltage Applied Voltage 10 8 6 4 2 0 -2 Vo -4 -6 Low intensity Current High intensity 10 8 6 4 2 0 -2 -4 -6 Low intensity Applied Voltage 1) The magnitude of the current depends upon the intensity of the light source. 2) As the applied voltage increases, the current increases until the point at which all freed electrons are being captured by the anode. Current High intensity 10 8 6 4 2 0 -2 -4 -6 Low intensity Applied Voltage 3) A current will flow for voltages greater than the stopping potential. 4) The existence of a current depends only upon the applied voltage and the frequency of the incident light, NOT the intensity of that light. stopping potential If you now were to plot the stopping potential as a function of the frequency of the incident radiation... frequency fc cutoff frequency Recall, the potential energy of a charge in an electric field is simply qV. In this case, the stopping potential represents the maximum potential difference the electron can cross. So, KEmax = e Vo These observations contradicted the classical theory, which suggested that the current should exist for any frequency of light, so long as the intensity of the light was strong enough...