Wavefunctions, One Particle r and r r s are the variables. ˆ(ˆ r, ˆ p, ˆ Hamiltonian H r r r s ) n is a state index and could Wavefunction ψ n ( r r, r s ) have several parts. For an e − in hydrogen ψ = ψ n,l,m l ,m s ( r r, r s ) ˆ(ˆ r, ˆ p, ˆ H r r r s ) ψ n ( r r, r s ) = E n ψ n ( r r, r s ) 8.044 L 17 B1
ψ n ( r r,r s ) often factors into space and spin parts. ψ space r ) ψ spin ψ n ( r r,r s ) = ( r ↔↔ ( r s ) ↔ n n ∞ − αx 2 / 2 H n ( ψ space ( x ) ↓ e α x ) H.O. in 1 dimension n ir k · r r ψ space ( r r ) ↓ e free particle in 3 dimensions n 8.044 L 17 B2
ψ spin ↔↔ ( r s ) n Spin is an angular momentum so for a given value of the magnitude S there are 2 S + 1 values of m S . For the case of S = 1 / 2 the eigenfunctions of the z component of r s are φ 1 / 2 ( r s ) and φ − 1 / 2 ( r s ) s ) = ¯ h φ 1 / 2 ( r ˆ z φ 1 / 2 ( r S s ) 2 ¯ h ˆ z φ − 1 / 2 ( r s ) = − φ − 1 / 2 ( r s ) S 2 8.044 L 17 B3
ψ spin ˆ z . For ↔↔ ( r s ) is not necessarily an eigenfunction of S n example one might have 1 φ − 1 / 2 ( r 1 ψ spin ↔↔ ( r s ) = ∞ φ 1 / 2 ( r s ) + s ) ∞ n 2 2 In some cases ψ n ( r s ) may not factor into space and r, r spin parts. For example one may find ψ n ( x, r s ) = f ( x ) φ 1 / 2 ( r s ) + g ( x ) φ − 1 / 2 ( r s ) 8.044 L 17 B4
Many Distinguishable Particles, Same Potential, No Interaction Lump space and spin variables together r r 1 ,r s 1 � 1 r r 2 ,r s 2 � 2 etc. ˆ(1 , 2 , · · · N ) = ˆ 0 (1) ˆ 0 (2) ˆ 0 ( N ) + + H H H · · · H In this expression the single particle Hamltonians all have the same functional form but each has arguments for a different particle. 8.044 L 17 B5
The same set of single particle energy eigenstates is available to every particle, but each may be in a dif- ferent one of them. The energy eigenfunctions of the system can be represented as products of the single particle energy eigenfunctions. ψ { n } (1 , 2 , · · · N ) = ψ n 1 (1) ψ n 2 (2) · · · ψ n N ( N ) { n } � { n 1 , n 2 , · · · n N } . There are N #s, but each n i could have an infinite range. ˆ(1 , 2 , · · · N ) ψ { n } (1 , 2 , · · · N ) = H E { n } ψ { n } (1 , 2 , · · · N ) 8.044 L 17 B6
� Many Distinguishable Particles, Same Potential, Pairwise Interaction N ˆ 0 ( i ) + 1 ˆ(1 , 2 , · · · N ) = ˆ int ( i, j ) H � H � H 2 i =1 i = j The ψ { n } (1 , 2 , · · · N ) are no longer energy eigenfunc- tions; however, they could form a very useful basis set for the expansion of the true energy eigenfuncti o ns. 8.044 L 17 B7
Indistinguishable Particles ˆ P ij f ( · · · i · · · j · · · ) � f ( · · · j · · · i · · · ) ij ) 2 = I ˆ ˆ ˆ ( P → eigenvalues of P ij are + 1 , − 1 It is possible to construct many-particle wavefunctions which are symmetric or anti-symmetric under this in- terchange of two particles. ij ψ (+) ψ (+) P ij ψ ( − ) − ψ ( − ) ˆ ˆ = = P 8.044 L 17 B8
Identical ⇒ no physical operation distinguishes be- tween particle i and particle j . Mathematically, this ˆ means that for all physical operators O ˆ , P ˆ [ O ij ] = 0 ˆ must also be eigenfunctions of ⇒ eigenfunctions of O ˆ P ij . (+) ⇒ energy eigenfunctions ψ E must be either ψ E or ( − ) . ψ E 8.044 L 17 B9
⇒ states differing only by the interchange of the spa- tial and spin coordinates of two particles are the same state. Relativistic quantum mechanics requires (+) integer spin ↔ ψ E [ Bosons ] ( − ) half-integer spin ↔ ψ E [ Fermions ] 8.044 L 17 B10
Composite Pa rticles • Composite Fermions and Composite Bosons • Count the number of sign changes as all the con- stituents are interchanged • Well defined statistics (F-D or B-E) as long as the internal degrees of freedom are not excited 8.044 L 17 B 1 1
The constitutents of nuclei and atoms are e, p & n . Each has S = 1 / 2. N even ⇒ even # of exchanges. ψ � (+) ψ ⇒ B-E also N even ⇒ integer spin N odd ⇒ odd # of exchanges. ψ � ( − ) ψ ⇒ F-D also N odd ⇒ half-integer spin 8.044 L 17 B 1 2
Particle Nuclear Spin Electrons Statistics H (H 1 ) 1 1 B-E 2 D (H 2 ) 1 1 F-D T (H 3 ) 1 1 B-E 2 He 3 1 2 F-D 2 He 4 0 2 B-E Li 6 1 3 F-D Li 7 3 3 B-E 2 H 2 0 or 1 2 B-E x 2 integer () × 2 B-E 8.044 L 17 B 1 3
Let α ( r s ) , β ( r s ) , · · · be single particle w avefunctions. r,r r,r A product many-particle wavefunction, α (1) β (2), does not work. Instead, use a sum of all possible permutations: Ψ (+) 1 = ∞ ( α (1) β (2) + α (2) β (1)) 2 2 Ψ (+) ∞ 1 ∞ 1 = � ( α (1) β (2) γ (3) · · · ) N permutations N ! α n α ! � 8.044 L 17 B 1 4
The antisymmetric version results in a familiar form, a determinant. Ψ ( − ) 1 = ∞ ( α (1) β (2) − α (2) β (1)) 2 2 � states � � α (1) β (1) γ (1) · · · � � � � � α (2) β (2) γ (2) · · · � Ψ ( − ) ∞ 1 � � = � particles � � N N ! α (3) β (3) γ (3) · · · � � � � . . . � . . . � . . . � � � � 8.044 L 17 B 1 5
• Ψ ( − ) = 0 if 2 states are the same since 2 columns N are equal: Pauli Principle. • Ψ ( − ) = 0 if 2 particles have the same r r and r s since N 2 rows are equal. • Specification: indicate which s.p. ψ s are used. { n α , n β , n γ , · · ·} An ⇒ # of entries, each ranging from 0 to N but with α n α = N . � 8.044 L 17 B 1 6
| 1 , 0 , 1 , 1 , 0 , 0 , · · · � Fermi-Dirac | 2 , 0 , 1 , 3 , 6 , 1 , · · ·� Bose-Einstein � ↔ E α n α = E Prime indicates n α = N � α α Example Atomic configurations (1 S ) 2 (2 S ) 2 (2 P ) 6 � Ne (1 S ) 2 (2 S ) 2 (2 P ) 6 (3 S ) 1 � Na (1 S ) 1 (2 S ) 1 � He* 8.044 L 17 B 1 7
Statistical Mechanics Try Canonical Ensemble − E ( state ) /kT Z ( N, V, T ) = � e states ↔ − E ( { n α } ) /kT = � e { n α } � � ↔ − E α n α /kT = � � e α { n α } This can not be carried out. One can not interchange the � over occupation numbers and the � over states because the occupation numbers are not independent ( n α = N ). � 8.044 L 17 B 1 8
T=0 LOWEST POSSIBLE TOTAL ENERGY BOSE: ALL N PARTICLES IN LOWEST ε SINGLE PARTICLE STATE n α ( ε ) N δ ( ε ) ε FERMI: LOWEST N SINGLE PARTICLE STATES EACH USED ONCE ε < ε F , ε F CALLED THE FERMI ENERGY n α ( ε ) 1 ε F ε 8.044 L17B19
MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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