Lecture 14

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

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What defines the granular nature of our Universe? h λ - = = 2 34 6.626 10 kgm s deBroglie h mv mv λ λ -24 -10 -35 for an atom, 10 f ~1 or a person 60 0 ~ 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?

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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 = = - & & 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… εβ εβ - - 647 48 678 QM treatment classical treatment e e ( 29 β - 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 β - 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 π π = 1 2 1 2 vib,class 2 q . 2 osc kT const mkT k π - = � � � � � � 6 1 2 2 we use 1 2 bx e dx b 0 π = 1 2 . 2 osc m const kT k ν = vib,class q . kT const πν = osc where we used 1 2 m k ν = ° vib,QM q kT h = 1 const h

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( 29 + + = 2 2 2 Translation kinetic energy: , 2 x y z p p p p q m H ( 29 β - 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 . x y z p p p mkT x y z x y z const e dp dp dp dq dq dq - 6 447 4 48 Volume g - - - = 2 2 2 2 2 2 trans,class q .
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