Planck s Quantum Hypothesis Blackbody Radiation

Transcription

1 Planck s Quantum Hypothesis Blackbody Radiation The spectrum of blackbody radiation has been measured(next slide); it is found that the frequency of peak intensity increases linearly with temperature. A black body is an ideal object that absorbs all incident radiation on it. The body will radiate energy (emission spectrum), and have a specific continuous spectrum related to its equilibrium temperature. Real bodies emit similar spectra.

2 This figure shows blackbody radiation curves for three different temperatures. Note that frequency increases to the left. Wien s Law states: λ P T = 2.90 x 10 3 m K λ P = peak wavelength (m) T = absolute temperature (K) It is an inverse relationship between the wavelength and the temperature.

3 This spectrum could not be reproduced using 19 th -century physics. A solution was proposed by Max Planck in 1900: The energy of atomic oscillations within atoms cannot have an arbitrary value; it is related to the frequency: The constant h is now called Planck s constant.

4 Planck found the value of his constant by fitting blackbody curves: Planck s proposal was that the energy of an oscillation had to be an integral multiple of hf. This is called the quantization of energy.

5 Photon Theory of Light Einstein suggested that, given the success of Planck s theory, light must be emitted in small energy packets: These tiny packets, or particles, are called photons.

6 Example problem What is the energy of a photon of blue light ( =450 nm) in electron volts? λ

7 E = hf = h c λ Solution: E= (6.63x10-34 ) x (3.00x10 8 ) (4.5x10-7 ) = 4.4x10-19 J = 2.7 ev

8 The photoelectric effect: If light strikes a metal, electrons are emitted. The effect does not occur if the frequency of the light is too low; the kinetic energy of the electrons increases with frequency. Photoelectric Effect

9 If light is a wave, classical wave theory predicts: 1. Number of electrons and their energy should increase with intensity 2. Frequency would not matter *These assumptions proved to be incorrect.

10 If light is particles, theory predicts: Increasing intensity increases number of electrons but not energy Above a minimum energy required to break atomic bond, kinetic energy will increase linearly with frequency There is a cutoff frequency below which no electrons will be emitted, regardless of intensity

11 The particle theory assumes that an electron absorbs a single photon. Plotting the kinetic energy vs. frequency: This shows clear agreement with the photon theory, and not with wave theory.

12 When the photon strikes an electron in the metal it takes a certain amount of energy to free the ejected photoelectron. Energy conservation gives: hf = E Kmax + W 0 hf = initial photon energy E kmas = maximum kinetic energy of photoelectron W 0 = Work Function. The amount of energy required to liberate the electron from metal surface

13 Example problem: What is the maximum kinetic energy and speed of an electron ejected from a Sodium surface whose work function is 2.28 ev illuminated by light of wavelength 410 nm?

14 Solution: E = h c λ = 4.85x10-19 J hf = E Kmax + W 0 E kmax = E W 0 E kmax = 4.85x x10-19 =1.202 x10-19 J E kmax = 1 2 mv 2 v= 5.1x10 5 m/s

15 The Stopping Potential The Stopping potential is the minimum potential difference (V 0 ) required to stop the photelectrons emitted in the photoelectric effect. The relationship is just energy conservation: qv 0 = 1 2 mv 2

16 Momentum of a Photon Clearly, a photon must travel at the speed of light. Looking at the relativistic equation for momentum, it is clear that this can only happen if its rest mass is zero. We already know that the energy is hf; we can put this in the relativistic energy-momentum relation and find the momentum: p = h λ

17 Collision of a photon and a free electron ( Compton Effect )

18 Compton Effect Compton did experiments in which he scattered X-rays from different materials. He found that the scattered X-rays had a slightly longer wavelength than the incident ones, and that the wavelength depended on the scattering angle( φ ). The relationship is derived by using conservation of energy and momentum.

19 This is another effect that is correctly predicted by the photon model and not by the wave model.

20 Photon Interactions Photons passing through matter can undergo the following interactions: 1. Photoelectric effect: photon is completely absorbed, electron is ejected 2. Photon may be totally absorbed by electron, but not have enough energy to eject it; the electron moves into an excited state 3. The photon can scatter from an atom and lose some energy 4. The photon can produce an electron-positron pair.

21 Pair Production In pair production, energy, electric charge, and momentum must all be conserved. Energy will be conserved through the mass and kinetic energy of the electron and positron; their opposite charges conserve charge; and the interaction must take place in the electromagnetic field of a nucleus, which can contribute momentum.

22 Wave-Particle Duality; the Principle of Complementarity We have phenomena such as diffraction and interference that show that light is a wave, and phenomena such as the photoelectric effect and the Compton effect that show that it is a particle. Which is it? This question has no answer; we must accept the dual wave-particle nature of light.

23 The principle of complementarity states that both the wave and particle aspects of light are fundamental to its nature. Indeed, waves and particles are just our interpretations of how light behaves.

24 Wave Nature of Matter Louis De Broglie (1924 PhD thesis) proposed that just as light sometimes behaves as a particle, matter sometimes behaves like a wave.the wavelength of a particle of matter is: This wavelength is extraordinarily small. He was awarded a nobel prize for his PhD Thesis (1929)

25 The Davisson-Germer Experiment (1927)clearly showed that matter could have wave-like properties They scattered electrons of a crystal and an interference pattern was observed. When the De Broglie wavelength of the electrons was calculated it agreed with the interference pattern

26 Early Models of the Atom It was known that atoms were electrically neutral, but that they could become charged, implying that there were positive and negative charges and that some of them could be removed One popular atomic model was the plumpudding model proposed by J.J. Thomson

27 This model had the atom consisting of a bulk positive charge, with negative electrons buried throughout.

28 Rutherford did an experiment (1909) that showed that the positively charged nucleus must be extremely small compared to the rest of the atom. He scattered alpha particles helium nuclei from a metal foil and observed the scattering angle. He found that some of the angles were far larger than the plum-pudding model would allow.

29 The only way to account for the large angles was to assume that all the positive charge was contained within a tiny volume now we know that the radius of the nucleus is 1/10000 that of the atom.

30 Therefore, Rutherford s model of the atom is mostly empty space: The planetary model

31 Atomic Spectra: Key to the Structure of the Atom A very thin gas heated in a discharge tube emits light only at characteristic frequencies.

32 An atomic spectrum is a line spectrum only certain frequencies appear. If white light passes through such a gas, it absorbs at those same frequencies (absorption versus emission lines).

33 The wavelengths of photons emitted from hydrogen have a regular pattern: This is called the Balmer series. R is the Rydberg constant:

34 Other series include the Lyman series: And the Paschen series:

35 A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.

36 Photon Emission Bohr proposed that the possible energy states for atomic electrons were quantized only certain values were possible. Then the spectrum could be explained as transitions from one level to another.

37 Atomic Absorption A hydrogen atom can absorb a photon and cause the electron to move up to an excited energy state. Again the change in the energy states would equal the energy of the photon. The Bohr model does not work well for atoms With larger nuclei. It was not till Schrödinger came along (1926) and developed a more complete theory of quantum mechanics

38 Atomic Energy States in Bohr Model of Hydrogen If the angular momentum is quantized, the energy of the atom s stationary states should also be quantized E n = 13.6eV n 2 n is the quantum number of the energy state. The ground state is when n=1

39 With this relationship of the Bohr model of the hydrogen atom all the atomic emission lines of hydrogen can be predicted. Summary of Bohr s Assumptions for Hydrogen model 1. The electron moves in circular orbits about the proton under the influence of the Coulomb force of attraction. 2. Only certain electron orbits are stable. These are orbits that hydrogen does not emit energy in the form of radiation. 3. Radiation is emitted by hydrogen when the electron "jumps" from a more energetic initial state to a lower state. 4. The size of the allowed electron orbits is determined by the electron's orbital angular momentum.

40 Example: What is the wavelength of light emitted from A hydrogen atom when its electron drops from n=3 to the ground state (n=1)?

41 Solution: The energy of the emitted photon is equal to the difference of its final and initial energy states: ΔE = E f E i = = ev = 1.93x10-18 J E = h c λ, λ = hc E =1.03x10-7 m

42 Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct. More modern Quantum theory describes an electron probability distribution; this figure shows the distribution for the ground state of hydrogen:

43 The Heisenberg Uncertainty Principle Quantum mechanics tells us there are limits to measurement not because of the limits of our instruments, but inherently. This is due to the wave-particle duality, and to interaction between the observing equipment and the object being observed.

44 Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of its momentum to the electron.

45 The uncertainty in the momentum of the electron is taken to be the momentum of the photon it could transfer anywhere from none to all of its momentum. In addition, the position can only be measured to about one wavelength of the photon.

46 Combining, we find the combination of uncertainties: This is called the Heisenberg uncertainty principle. It tells us that the position and momentum cannot simultaneously be measured with precision.

47 This relation can also be written as a relation between the uncertainty in time and the uncertainty in energy: This says that if an energy state only lasts for a limited time, its energy will be uncertain. It also says that conservation of energy can be violated if the time is short enough.

48 Philosophic Implications; Probability versus Determinism The world of Newtonian mechanics is a deterministic one. If you know the forces on an object and its initial velocity, you can predict where it will go in theory. This is not as simple as it seems, as in the case of the three body problem, for which there is no analytical solution. Quantum mechanics is very different as predictions are all based on probabilities.