Lecture 14 - What defines the granular nature of our...

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Unformatted text preview: What defines the granular nature of our Universe? h - = = 2 34 6.626 10 kgm s deBroglie h mv mv -24-10-35 for an atom, 10 f ~1 or a person 60 ~ 10 ~ ~ Kgm s Kgm s m m mv mv Our classical behavior vs the atomic quantum characteristics are a consequence of the absolute size of h So far, we solved the Q.M. problem and then count all states to get the partition function Q. While trying to count states, we invoke the classical limit and h Can we obtain Q assuming classical behavior for the H ? What do you expect to be different between the 2 answers? So far, we solved the Q.M. problem and then count all states to get the partition function Q. While trying to count states, we invoke the classical limit and h Can we obtain Q assuming classical behavior for the H ? We can obtain the states energies by solving the classical Hamiltonian equations h& = = - & & where j=1,2,...3N j j q p q j j H H p We count the states by analogy to the quantum treatment -- 647 48 678 QM treatment classical treatment e e ( 29 - class q e dpdq p,q H ( 29 = + 2 2 1 conjugated pair p,q 1 Harmonic Oscillator: , 2 2 osc p p q k q m H ( 29 - vib,class , q p q e dp dq- H- + = 2 2 1 2 2 osc p k q mkT kT e dp dq--- 2 2 1 2 2 vib,class q osc p k q mkT kT e dp e dq-- ( 29 = 1 2 1 2 vib,class 2 q . 2 osc kT const mkT k - = 6 1 2 2 we use 1 2 bx e dx b = 1 2 . 2 osc m const kT k = vib,class q . kT const = osc where we used 1 2 m k = & vib,QM q kT h = 1 const h ( 29 + + = 2 2 2 Translation kinetic energy: , 2 x y z p p p p q m H ( 29 - trans,class 3 conjugated variables pairs , q x y z x y z p q e dp dp dp dq dq dq- H + +- = 2 2 2 2 trans,class- q ....
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Lecture 14 - What defines the granular nature of our...

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