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STATES OF A MODEL SYSTEM
the systems we are interested in has many available quantum states
- many states can have identical energy --> multiplicity (degeneracy) of a level: number of quantum states with
the same energy
- it is the number of quantum states that is important in thermal physics, not the number of energy levels!
Examples for quantum states and energy levels of several atomic systems:
(multiplicity for each energy level shown in the brackets)
1. Hydrogen (one electron + one proton)
2. Lithium (3 electrons + 3 protons + 3-4 neutrons)
3. Boron (five electrons + 5 protons + 5-6 neutrons)
4. Particle confined to a cube


2M
  2
2
2
  (nx  n y  nz )
L
2
nx, ny, nz --> quantum numbers : 1, 2, 3, …k,...
Quantum states of one particle systems --> orbitals
Binary model systems
- elementary magnets pointing up or down
- cars in a parking lot
-m magnetic moment
+m magnetic moment
- binary alloys
Occupied or type A atom
Unoccupied or type B atom
A single state of the system: 1  2 3  4 5  6 7 .....  N
All states of the system generated by: (1  1 )( 2   2 )(3  3 )......( N   N ) -->generating function
Total number of states: 2N ;
N+1 possible values of the total magnetic moment: M=Nm, (N-2)m, (N-4)m, ...-Nm
number of states >> possible values of total magnetic moments (if N>>1)
if the magnetic moments are not interacting M will determine the E total energy of the system in a
magnetic field!
( number of states >> possible energy values) --> some states have large multiplicity
Enumeration of States and the Multiplicity Function
(Let us assume N even) N : number of up spins, N: number of down spins
N   N   2s
spin excess
Multiplicity function g(N,s) of a
state with a given spin excess
s 1 / 2 N
 g ( N , s)  (1  1)
N
g ( N , s) 
 2N
s  1 / 2 N
Ex. Form of g(10,s) as a function of 2s:
Binary alloy systems:
(N-t) A atoms and (t) B
atoms on N sitesthe same
multiplicity function
g(N , t) 
N!
N!

( N  t )!t! N A! N B !
N!
N!

1
 1
 N! N!
 N  s ! N  s !
2
2

Sharpness of the Multiplicity Function
- g(N,s) is very sharply peaked around s=0;
- we want get a more analytical form of g(N,s) when N>>1 and s<<N
- we will follow the same procedure as for the random-walk problem!
- we use the Stirling approximation:
1
2
2s 2
ln[ g (n, s)]  ln(
)  N ln( 2) 
2 N
N
and after find:
N
1
ln( N !)  N ln( N )  N  ln( 2N )
2
N
N
g ( N ,0 ) 

 g ( N , s)   g ( N , s)ds   g ( N , s)ds 2
s  N
g ( N , s )  g ( N ,0 ) e
N

2s2
N
2 N
2
N

g(N,s) is a Gaussian-like distribution!
Width of the g(N,s) multiplicity function governed by
for s/N=(1/2N)1/2 the value of g is e-1 of g(N,0)
1
2N
For N>>1 the distribution gets very sharp --> strong consequences for thermodynamic systems
Problems
1. Prove that:


e x dx  
2

2. Prove the Stirling approximation:
1
ln( N !)  N ln( N )  N  ln( 2N )
2
3. Approximate in the limit of large energy values the () density of states for a particle
confined in a 3D box
 ( )  lim  
d 0
n( ,   d )
d
where n(,+d) represent the number of states with energy between  and +d.
4. Starting from the
2 N 
g ( N , s) 
2 e
N
2s2
N
multiplicity function for a binary model
system, approximate the number of possible states of the system, when N=100 and and s is
between 0 and +10.
Extra problem
1. (**) Using the entropy formula given by
Renyi calculate the entropy of a binary model
system presuming that all microstates are
equally probable.
S  k  Pi ln( Pi )
i
(In the above Renyi formula the summation is over
all possible microstates, and Pi represents the
probability of microstate i)