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

)(2

2222

zyx nnnLM

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

- binary alloys

-m magnetic moment +m magnetic moment

Occupied or type A atom Unoccupied or type B atom

A single state of the system: N .....7654321

All states of the system generated by: ))......()()(( 332211 NN -->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

sNN 2 spin excess

Multiplicity function g(N,s) of a state with a given spin excess !!

!

!21

!21

!),(

NN

N

sNsN

NsNg

NNs

Ns

NsNg 2)11(),(2/1

2/1

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

!!

!

!)!(

!),(

BA NN

N

ttN

NtNg

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:

and after find:

)2ln(2

1)ln()!ln( NNNNN

N

sN

Nsng

22)2ln()

2ln(

2

1)],(ln[

N

N

s

NNg

eNgsNg

22

)0,(

)0,(),(

22

NN

N

N

Ns

dssNgdssNgsNg 2),(),(),(

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) N2

1

g(N,s) is a Gaussian-like distribution!

For N>>1 the distribution gets very sharp --> strong consequences for thermodynamic systems

Problems

1. Prove that:

dxe x2

2. Prove the Stirling approximation:

d

dn

d

),(lim)(

0

)2ln(2

1)ln()!ln( NNNNN

3. Approximate in the limit of large energy values the () density of states for a particle confined in a 3D box

where n(,+d) represent the number of states with energy between and +d.

4. Starting from the 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.

N

sN e

NsNg

22

22

),(

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.

)ln( ii

i PPkS

(In the above Renyi formula the summation is over all possible microstates, and Pi represents the probability of microstate i)