Early and the Wave Nature of Matter Winter 2018 Press CTRL-L to view as a slide show.
Last Time Last time we discussed: Optical systems Midterm 2
Today we will discuss: Quick of X-ray diffraction Compton scattering
An
April 27, 1900 In a speech before the Royal Institution, Lord Kelvin proposed that we basically understood all of physics, except for "two dark clouds on the hoirizon." These were The null result of the Michelson-Morley experiment The inability of classical physics to explain blackbody radiation
Need for Physics The electromagnetic radiation emitted by a heated object Emission of electrons by an illuminated metal Spectral Lines Emission of sharp spectral lines by gas atoms in an electric discharge tube
Light is emitted from hot objects in bundles (quanta) of energy
An object at any temperature emits electromagnetic radiation The spectrum of the radiation depends on the temperature and properties of the object
Graph As T increases, the total energy emitted increases As T increases, the peak of the distribution shifts to shorter wavelengths
Wien s Law Based on thermodynamics and EM Theory Gave good agreement at short wavelengths
Wien s Law But it wasn t very good at long wavelength
Rayleigh-Jeans Law Also based on thermodynamics and EM Theory Gave good agreement at long wavelength
Rayleigh-Jeans Law But it didn t agree at all at short wavelength This was called the "ultraviolet catastrophe"
Classical physics gave two laws that didn t agree... and neither agreed with all the data
Max Planck 1858-1947 Introduced a "quantum of action," h
Planck s Resolution Oscillators in matter could only emit light at discrete energies E n = nhf n is called the quantum number f is the frequency of oscillation h is Planck s constant
Planck s Constant There are many versions h = 6.626 10 34 Js (SI) h = 4.136 10 15 ev s (ev) hc = 1240 ev nm = h/2π ("hbar")
Today s Interpretation Light from atoms in the blackbody is emitted as photons from atoms Each photon of light carries energy E = hf
Conclusion 1 At very small scales, measurements of many physical quantities can only have discrete values. These include Energy of bound electrons Angular momentum Angular momentum component along one axis Spin
Light is absorbed by matter in quanta of energy called photons
Light is absorbed by matter in quanta of energy photons
Light can knock electrons off metallic surfaces The effect was first discovered by Hertz The successful explanation of the effect was given by Albert Einstein in 1905
When light strikes E, photoelectrons are emitted Electrons strike C and form a current in the circuit A voltage is placed on C. If V > 0, it attracts electrons. If V < 0, it repels electrons.
By adjusting the voltage, we can measure the kinetic energy of the photoelectrons
Current/Voltage Graph When V > 0, the current quickly reaches a saturation level No current flows for voltages less than or equal to V s, the stopping potential
Einstein s Explanation Light is emitted only in packets of energy called photons with E = hf The maximum kinetic energy of the liberated photoelectron is KE max = hf Φ where Φ is the work function of the metal The work function is the minimum energy needed to knock an elctron off the metal surface
Cutoff Frequency The cutoff frequency is related to the work function Φ E c = hf c = Φ If the energy of the photon equals the work function, there is just enough energy to remove the electron from the metal surface with no extra kinetic energy.
Cutoff Wavelength A cutoff wavelength can be defined: E c = hf c = hc λ c Wavelengths greater than λ c incident on a material with a work function Φ don t result in the emission of photoelectrons
Conclusion 2 We can think of light as particles, each having energy and momentum: E = hf p = h/λ
Consequences of Conclusion 2 Understanding light in terms photons is helpful in understanding many phenomena including effect Compton scattering
X rays can behave as waves
Diffraction of by Crystals For diffraction to occur, the spacing between the lines must be approximately equal to the wavelength of the radiation to be measured The regular array of atoms in a crystal can act as a three-dimensional grating for diffracting x rays
Schematic for X-ray Diffraction A beam of X-rays with a continuous range of wavelengths is incident on the crystal In directions where there is constructive interference from waves reflected from the layers of the crystal, the diffracted radiation is very intense The diffraction pattern is detected by photographic film
X rays can behave as particles high energy photons
Arthur Holly Compton 1892-1962 Discovered the Compton effect Worked with cosmic rays Director of the lab at U of Chicago
The Compton Compton directed a beam of x rays toward a block of graphite He found that the scattered x rays had a slightly less energy than the incident x rays The amount of energy reduction depended on the angle at which the x rays were scattered The change in wavelength is called the Compton shift
Compton assumed the photons acted like other particles in collisions Energy and momentum were conserved The shift in wavelength is λ = h (1 cos θ) m e c
and Electromagnetic Waves Light has a dual nature. It exhibits both wave and particle characteristics This applies to all electromagnetic radiation The photoelectric effect and Compton scattering offer evidence for the particle nature of light Interference and diffraction offer evidence of the wave nature of light
Wavelength What is the energy of photon that has the wavelength of 1 nm, the size of a typical atom? E = hf = hc λ = 1240eVnm 1nm = 1240eV
Work Function The work function of a given metal is 2.3 ev. What is the maximum wavelength of light (minimum photon energy) that will produce photoelectons? Φ = hf = hc λ = 2.3eV λ = hc φ = 1240eVnm 2.3eV = 539nm
A 4.4 MeV gamma ray Compton scatters through an angle of 20 (measured with respect to its incident direction). What is its energy after scattering? λ = λ λ 0 = h (1 cos θ) m e c First, we need the relation between E and λ. E = hf = hc λ λ = hc E
A 4.4 MeV gamma ray Compton scatters through an angle of 20 (measured with respect to its incident direction). What is its energy after scattering? λ = λ λ 0 = hc E hc E 0 = h (1 cos θ) m e c λ h (1 cos θ) m e c 1 E 1 = 1 (1 cos θ) E 0 m e c2 = hc E
A 4.4 MeV gamma ray Compton scatters through an angle of 20 (measured with respect to its incident direction). What is its energy after scattering? 1 E = 1 1 + (1 cos θ) E 0 0.511MeV 1 E = 1 4.4MeV + 1 0.511MeV (1 cos 20 ) E = 2.90MeV
de Broglie and Matter Waves Electrons as well as photons share in wave-particle duality
and Electromagnetic Waves Light has a dual nature. It exhibits both wave and particle characteristics The photoelectric effect and Compton scattering offer evidence for the particle nature of light Interference and diffraction offer evidence of the wave nature of light
Photon Momentum If a photon has energy, it also has momentum. Relativity tells us p = h λ
Photon Momentum If a photon has energy, it also has momentum. Relativity tells us p = h λ
Photon Momentum If a photon has energy, it also has momentum. Relativity tells us p = h λ
Wave Properties of Particles In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties
de Broglie Wavelength and Frequency Furthermore, he postulated that both light and matter obeyed the relationships: E = hf p = h λ
The Davisson-Germer Experiment Davisson and Germer reasoned that if x rays diffracted from crystals, electron waves should diffract as well Electron diffraction experiments showed that λ = h/p as de Broglie predicted
A guess about how matter waves should behave
Measurement Theory Physics had always assumed that position, velocity, etc., could be measured accurately In the early 20 th century, physicists worried that at atomic levels, measurements would change systems They looked for mathematics that gave different results when two quantities were measured in different order If you measure A then B, you get different results than when you measure B then A
Measurement Theory Erwin Schrödinger suggested using derivatives to represent measurement of physical quantities: d dx (xf (x)) x d dx f (x) This led to "wave mechanics"
Measurement Theory Werner Heisenberg suggested using matrices to represent measurement of physical quantities: [ ] [ ] a b e f c d g h This led to "matrix mechanics" [ ] [ ] e f a b g h c d
Measurement Theory Wave mechanics and matrix mechanics were later shown to be identical They came to be known as "quantum mechanics"
The Wave Function In 1926 Schrödinger proposed a wave equation that describes the manner in which matter waves change in space and time Schrödinger s wave equation is a key element in quantum mechanics Schrödinger s wave equation is generally solved for the wave function, ψ
The Foundations of Postulate 1: We can represent a system (like an electron in an atom) by a function called a wave function.
The Wave Function Has both wave and particle characteristics built into it. The wave function of a particle moving in one dimension in a box with very hard walls: ψ(x, t) = A sin 2πx λ e i 2πf t wave function wavelength: p = h/λ frequency: E = hf
The Wave Function Has both wave and particle characteristics built into it. The wave function of a particle moving in one dimension in a box with very hard walls: ψ(x, t) = A sin 2πx λ e i 2πf t
The Wave Function Has both wave and particle characteristics built into it. The wave function of a particle moving in one dimension in a box with very hard walls: ψ(x, t) = A sin 2πx λ e i 2πf t
The Wave Function Has both wave and particle characteristics built into it. The wave function of a particle moving in one dimension in a box with very hard walls: ψ(x, t) = A sin 2πx λ e i 2πf t
The Wave Function Has both wave and particle characteristics built into it. The wave function of a particle moving in one dimension in a box with very hard walls: ψ(x, t) = A sin 2πx λ e i 2πf t
The Wave Function What s with the e i 2πft? e i 2πft = cos( 2πft) + i sin(2πft) This form makes calculations more convenient, but still represents a sine (or cosine) wave.
The Wave Function Only wave functions with certain wavelengths are allowed.
The Wave Function When we square the wave function, we get the probability distribution.
The concept of a probability distribution
Wave function = Probability wave The wave function gives us the probability of finding the particle at a given location. Wave functions add like waves they interfere and diffract
The Foundations of Postulate 2: We can represent things we can measure (energy, momentum, spin) with operators. An operator is anything that does something to something else. The operators in quantum mechanics do something to wave functions.
Two Operators Momentum operator Constant slope of ( ) Position operator position ( )
The Foundations of Postulate 3: Sometimes an operator satisfies the equation: operator (ψ) = number ψ If this is true, the number is the value you obtain when you measure the quantity indicated by the operator.
Huh???? Postulate 3: If momentum operator (ψ) = 3 ψ the momentum is 3 and ψ is the special function that describes a particle with momentum = 3
So what? Energy = kinetic energy +potential energy = p 2 /(2m) + P.E. If (momentum operator) 2 ψ+(potential energy) ψ = E ψ 2m Then ψ is the special wave function that describes a particle with energy E.
Schrödinger s 2 2m 2 ψ( r) + V ( r)ψ( r) = Eψ( r) Constructed by Erwin Schrödinger in 1926 It describes atoms very accurately! It was a guess it had no right to work and we don t really understand it yet...
Understanding and dealing with matter waves
The Wave Function So, what does the wave function mean? That was the subject of much debate in the 1920s! We settled on the "Copenhagen Interpretation," proposed by Neils Bohr and his associates.
The Wave Function The value of ψ 2 at some location at a given time is the probability density (the probability per unit volume) of finding the particle at that location and at that time
The Wave Function The value of ψ 2 at some location at a given time is the probability density (the probability per unit volume) of finding the particle at that location and at that time
The Wave Function Atomic orbitals are the wave functions for electrons in atoms. http://www.geo.arizona.edu/xtal/geos306/d-orbitals.gif
Conclusion 3 We can describe all we know about a particle in terms of the wave function. The absolute square of the wave function is the probability density. The measurement process can be described by "operators"
The Heisenberg One consequence of dealing with matter waves
The When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements Classical mechanics allows for measurements with arbitrarily small uncertainties
The and Waves Two special wave functions: All other wave functions are somewhere in between.
The mechanics predicts that a barrier to measurements with ultimately small uncertainties does exist In 1927 Heisenberg introduced the uncertainty principle If a measurement of position of a particle is made with precision x and a simultaneous measurement of linear momentum is made with precision p x, then the product of the two uncertainties can never be smaller than h/4π
The Mathematically, x p x h 4π It is physically impossible to measure simultaneously the exact position and the exact linear momentum of a particle Another form of the principle deals with energy and time: E t h 4π
Applied to an Electron View the electron as a particle Its position and velocity cannot both be known precisely at the same time Its energy can be uncertain for a period given by E = h/(4π t)
Conclusion 4 Measuring a quantity changes the wave function. Whenever the measurement of one quantity affects another quantity, the two quantities can not be measured accurately together. In these cases, the quantities satisfy an uncertainty principle relationship.
An application of quantum mechanics
Consider a positively charged plate with a small hole in it.
A proton is moving in the x direction along the axis of the hole.
We can make a graph of the potential energy of the proton vs position
We can make a graph of the potential energy of the proton vs position
Let the total energy of the proton be E.
The kinetic energy is the difference between E and U.
As the proton approaches the plate, it slows down. After if passes through the hole, it speeds up again.u
If the total energy is less than U, the electron can never get through the hole...
... unless the electron is a wave! Then a small wave is found on the far side of the plate.
Scanning Microscope (STM) A conducting probe with a sharp tip is brought near the surface Electrons can tunnel across the work function potential energy
Scanning Microscope By applying a voltage between the surface and the tip, the electrons can be made to tunnel preferentially from surface to tip The larger the distance to the tip, the more difficult it is for electrons to tunnel through the barrier. The STM allows measurements of the height of surface features within 0.001 nm
STM Result, Example This is a "quantum corral" of 48 iron atoms on a copper surface The diameter of the ring is 143 nm Obtained with a low temperature STM
A Thought Experiment Shoot many individual electrons toward two narrow slits (really at a crystal lattice). What do you see on the screen behind? An interference pattern is seen, but can one electron go through both slits? Try an experiment - put a pick-up coil around one slit What happens? The interference pattern goes away - but the experiment changes the system.
The Bottom Line The electron is observed at creation and detection. Between these times, the electron seems to do all possible things the electron could do. The probability of the different outcomes is the same as if the electron were a wave that experienced interference.