Unformatted text preview: 4/11/11 PHYS 360 Quantum Mechanics Mon Apr 11, 2011 Lecture 34: Is there really a force between fermions? Is there a force law? Chapter 5: IdenOcal ParOcles Two
par(cle systems The wave funcOon: Ψ r1 , r2 , t ( ) The wave equaOon: HW #10 due Monday, Apr 18: Ch 5, #5, 7, 9, 13, 14 The Hamiltonian: 1 i ∂Ψ = HΨ ∂t
2 2 2 2 ∇− ∇ + V r1 , r2 , t 2 m1 1 2 m2 2 H =− ( )
2 The probability “density”: If the potenOal is independent of Ome, apply separaOon of variables and solve for Ome
dependent eigenfuncOon as: Ψ r1 , r2 , t ( ) 2 d 3r1d 3r2 Ψ n r1 , r2 , t = ψ n r1 , r2 e ( ) ( ) − iEnt / This is the probability that parOcle 1 is found in volume element d3r1 and parOcle 2 is found in the volume element d3r2 at the Ome t. The normalizaOon condiOon is: −
2 …where E is the total energy of the system (eigenstate). The corresponding Ome
independent equaOon is: 2 2 ∇ 2ψ r , r − ∇ 2ψ r , r + V r1 , r2 ψ n r1 , r2 = Enψ n r1 , r2 2m 1 1 n 1 2 2m 2 2 n 1 2 ( ) ( ) ( )( ) ( ) ∫ Ψ (r , r , t ) 1 2 d 3r1d 3r2 = 1
3 4 Bosons and Fermions Put parOcle 1 in state a and parOcle 2 in state b: ψ + r1 , r2 = A ⎡ψ a ( r1 )ψ b ( r2 ) + ψ b ( r1 )ψ a ( r2 ) ⎤ ⎣ ⎦ ψ + r2 , r1 = A ⎡ψ a ( r2 )ψ b ( r1 ) + ψ b ( r2 )ψ a ( r1 ) ⎤ ⎣ ⎦
+ 1 2 + 2 1 ψ r1 , r2 = ψ a ( r1 )ψ b ( r2 )
Problem: this is not actually possible if you cannot tell the diﬀerence between the two parOcles. For indisOnguishable parOcles we must write: ( ) () () ψ (r , r ) = ψ (r , r ) ψ ± r1 , r2 = A ⎡ψ a ( r1 )ψ b ( r2 ) ± ψ b ( r1 )ψ a ( r2 ) ⎤ ⎣ ⎦ ( ) ψ − r1 , r2 = A ⎡ψ a ( r1 )ψ b ( r2 ) − ψ b ( r1 )ψ a ( r2 ) ⎤ ⎣ ⎦ ψ − r2 , r1 = A ⎡ψ a ( r2 )ψ b ( r1 ) − ψ b ( r2 )ψ a ( r1 ) ⎤ ⎣ ⎦
− 1 2 − 2 1 () () ψ ( r , r ) = −ψ ( r , r ) 5 6 1 4/11/11 ψ ± r1 , r2 = A ⎡ψ a ( r1 )ψ b ( r2 ) ± ψ b ( r1 )ψ a ( r2 ) ⎤ ⎣ ⎦
This version doesn’t assume you can tell them apart. But which sign do we choose? Plus sign = bosons, minus sign = fermions ⎧all particles with integer spin are bosons, and ⎨ ⎩all particles with half integer spin are fermions ( ) Does the Pauli Exclusion Principle mean that the two electrons in the ground state of helium are NOT in the same state? Suddenly, we have the Pauli Exclusion Principle: ψ − r1 , r2 = A ⎡ψ a ( r1 )ψ a ( r2 ) − ψ a ( r1 )ψ a ( r2 ) ⎤ = 0 ⎣ ⎦ ( ) Two fermions cannot be in the same state! 7 8 Does the Pauli Exclusion Principle mean that the two electrons in the ground state of helium are NOT in the same state? Exchange Forces For simplicity lets work things out in just one dimension. We’ll consider a set of one
parOcle wave funcOons that are orthogonal and normalized. If we have two dis(nguishable parOcles the combined wave funcOon is: ψ x1 , x2 = ψ a ( x1 )ψ b ( x2 ) YES. They have the same spaOal wave funcOons, but diﬀerent spin states. (Singlet – one spin up, the other spin down.) The Pauli Exclusion Principle applies to the enOre wave funcOon, including spin. ( ) (You can think of these one
parOcle states as eigenfuncOons of the inﬁnite square well, for example.) ψrχs () ()
9 Now consider indis(nguishable parOcles…. 10 For bosons: Case 1: Dis(nguishable par(cles ψ + x1 , x2 =
For fermions: ( ) 1 2 ⎡ψ a x1 ψ b x2 + ψ b x1 ψ a x2 ⎤ ⎣ ⎦ () () () () (x − x )
1 2 2 d = x2 a + x2 b −2 x a x b Case 2: Indis(nguishable par(cles ψ − x1 , x2 = ( ) 1 2 ⎡ψ a x1 ψ b ( x2 ) − ψ b x1 ψ a x2 ⎤ ⎣ ⎦ () () () (x − x )
1 2 2 ± = x2 a + x2 b −2 x a x b 2 x 2 ab Now, for all three cases let’s calculate the expectaOon value of the square of the distance beteen the two parOcles: ...where x ab ≡ ∫ xψ a x ψ b x dx () * () (x − x )
1 2 2 2 = x12 + x2 − 2 x1 x2
11 This last term is called an overlap integral. It is zero unless there is some overlap in the wave funcOons. 12 2 4/11/11 DisOnguishable (x − x )
1 2 2 d = x2 a + x2 b −2 x a x b ≡ ( Δx )
2 ab 2 d Bosons Fermions (x − x )
1 2 2 + = x2 a + x2 b −2 x a x b −2 x x b +2 x (x − x )
1 2 2 − = x2 a + x2 b −2 x 2 ab a Bosons ( Δx ) ( Δx ) 2 + = ( Δx ) ( Δx ) 2 d −2 x 2 ab Note: Even though electrons are fermions, the spaOal part can be either symmetric or anOsymmetric (because of the spin degree of freedom). Fermions 2 − = 2 d +2 x 2 ab 13 14 1. Why is it hard to squeeze atoms closer together in solids? 2. What keeps the Sun from condensing into a solid? 3. What is a neutron star? 4. How can a neutron star become a black hole? 15 3 ...
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 Spring '11
 DURBIN,STEPHEN
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