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Unformatted text preview: 4/22/11 PHYS 360 Quantum Mechanics Fri Apr 22, 2011 Lecture 39: What role does quantum mechanics play in determining temperature? Outline: 1. Thermodynamics Etotal=αNkT, where α=# of “degrees of freedom” per mode. (VibraYng atoms=3; EM waves=2) 2. Fails to explain blackbody radiaYon (&speciﬁc heat) 3. Planck hypothesis: atoms emit radiaYon in quanta: E=hν. Changes average energy per mode (Boltzmann staYsYcs), solves blackbody radiaYon problem. 4. Einstein points out that consistency then requires that the radiaYon ﬁeld itself must be quanYzed, invenYon of the photon, explains photoelectron eﬀect. 1 5. Einstein returns to vibraYon of atoms. Speciﬁc heat shows deviaYons from thermodynamics (equiparYYon). Mostly solved by Einstein model, which quanYzed vibraYons (harmonic oscillators). Leads to invenYon of the phonon. 6. QuanYzed vibraYonal modes in solids (with Einstein model) provides framework for deriving entropy and temperature from a mechanical harmonic oscillator model. Thermodynamics gives a clear predicYon for blackbody radiaYon, which is totally wrong. The problem was solved by Planck with his quantum hypothesis: The atoms emifng & absorbing radiaYon do so in discrete, quanYzed units of energy. Planck quanYzed the oscillaYng atoms, as if they had discrete energy levels En=hνn(=nhν). Compare this to what we know about the quantum harmonic oscillator. Einstein went further, and said that quanYzaYon is inherent to light itself (not just the atoms that emit the light). This is the “photon” hypothesis, that he used to explain the photoelectric eﬀect. However, Einstein also realized that quanYzing atomic oscillators could solve another problem: speciﬁc heat in solids. Heat Capacity ΔE = ΔEint + ΔEother = Q + W C= ∂Eint
C ∂T → ΔT = Q High C means Q doesn’t change temperature much. Small C means even a small Q changes temperature a lot. classical expectaYon “Speciﬁc Heat” is just the heat capacity normalized per unit atom (or equivalent). 1 4/22/11 Einstein proceeded with what is now known as the Einstein model for lafce vibraYons: Include: each atom is bound by springs allowing vibraYons in the x, y, & z direcYons (3 degrees of freedom). Exclude: The moYon of one atom does not eﬀect neighboring atoms. Special feature: vibraYonal energy levels are quanYzed. The Einstein Model of Solids • assume atoms oscillate independent of each other • each atom is comprised of 3 independent quantum harmonic oscillators: N oscillators means N/3 atoms Planck used Boltzmann staYsYcs (pure classical model) to describe the energy per oscillator: E= hν → kT for high temperatures ehν / kT − 1 → 0 for low temperatures How to improve on this model: 1. Allow coherent vibraYons, collecYve moYons of many atoms instead of just one. These are called “phonons,” in direct analogy with photons. (Employ Bloch’s theorem too…) 2. Use Bose
Einstein staYsYcs instead of Maxwell
Boltzmann Einstein started with this assumpYon, applied it to the “Einstein Model” and got good ﬁts to data: copper What does any of this have to do with temperature? Well, what is temperature? How is it deﬁned? What is the role of quantum mechanics? We will now see that: 1. By quanYzing the harmonic oscillator 2. Using the Einstein model for vibraYons in a solid 3. And applying the fundamental assumpYon of staYsYcal mechanics, we can…. 4. Deﬁne Entropy and Temperature. Coun9ng Microstates: 4 quanta, 3 oscillators Macrostate: given only by macroscopic parameters (energy, volume, etc.) Microstate: lisYng energy state of each oscillator All 15 of these microstates represent the same macrostate. 2 4/22/11 A Formula for Coun9ng Microstates Fundamental Assump9on of Sta9s9cal Mechanics The fundamental assumpYon of staYsYcal mechanics is that, over Yme, an isolated system in a given macrostate is equally likely to be found in any of its microstates. Generally, # microstates (N oscillators, q quanta) ≡Ω= (q + N − 1)! q !(N − 1)!
29% 24% 24% Thus, our system of 2 atoms is most likely to be in a microstate where energy is split up 50/50. 12% 12% atom 1 atom 2 Fundamental Assump9on of Sta9s9cal Mechanics The fundamental assumpYon of staYsYcal mechanics is that, over Yme, an isolated system in a given macrostate (total energy) is equally likely to be found in any of its microstates (microscopic distribuYon of energy). Interac9ng Blocks As quanta are transferred from one block to another, the system transiYons from one microstate to another. As the system “wanders” around, what is the most likely value of q1? 29% 24% 24% Thus, our system of 2 atoms is most likely to be in a microstate where energy is split up 50/50. 12% q1 0 1 2 3 4 … q2 = 100–q1 100 99 98 97 96 … Ω1 1 300 4.52 E+4 4.55 E+6 3.44e E+8 … Ω2 2.78 E+81 9.27 E+80 3.08 E+80 1.02 E+80 3.33 E+79 … Ω1 Ω2 2.77 E+81 2.78 E+83 1.39 E+85 4.62 E+86 1.15 E+88 … 12% atom 1 atom 2 Two Important Observa9ons 1. the most likely circumstance is that the energy is divided in proporYon to the # of oscillators in each object: lnΩ is Largest in Most Likely State most likely state has 60% of quanta in block with 60% of the oscillators. 2. as the # of oscillators get large, only one such circumstance is observed: the curve gets narrower: more oscillators more oscillators 3 4/22/11 Deﬁni9on of Entropy entropy: S ≡ k ln Ω
where Boltzmann’s constant is Approaching Equilibrium k = 1.38 ×10−23 J K energy distribu9on A energy distribu9on B • assume system starts in an energy distribuYon A far away from equilibrium (most likely state). • what happens as the system evolves? • Crucial point: consider our system to contain both blocks. The macrostate of the system is speciﬁed by the total # quanta: 100. • All microstates compaYble with qtotal=100 are equally likely, but not all energy distribu9ons (values of q1) are equally likely! Second Law of Thermodynamics THE SECOND LAW OF THERMODYNAMICS If a closed system is not in equilibrium, the most probable consequence is that the entropy of the system will increase. Entropy Increases By “most probable” we don’t mean just “slightly more probable”, but rather “so ridiculously probable that essenYally no other possibility is ever observed”. Hence, even though it is a staYsYcal statement, we call it the “2nd Law” and not the “2nd Likelihood”. Temperature As system approaches equilibrium: Small dS/dE energy High temp S ≡ k ln Ω
1 dS ≡ T dEint Big dS/dE Low temp DeﬁniYon of temperature: 1 dS ≡ T dEint If the iniYal state is not the most probable, energy is exchanged unYl the most probable distribuYon is reached. 4 ...
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This note was uploaded on 04/23/2011 for the course PHYS 360 taught by Professor Durbin,stephen during the Spring '11 term at Purdue UniversityWest Lafayette.
 Spring '11
 DURBIN,STEPHEN
 Thermodynamics, mechanics

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