User Tools

Site Tools


world:ch01

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

world:ch01 [2020/07/27 14:47]
talipovm created
world:ch01 [2020/07/28 08:14] (current)
talipovm
Line 3: Line 3:
 ===== Binary systems ===== ===== Binary systems =====
  
-Consider a collection of non-interacting particles, where each particle can be in one of two states ​quantum states. As an example, we will deal with //N// elementary magnets that could have their spins oriented up or down (another example: flipping ​coins). The magnetic moment of an individual magnet is //+m// if the magnet is pointing up and //-m// if it is pointing down. The total number of configurations is $2^N$ while the total number of possible momenta of the entire system is $N+1$ (that is, $-Nm, -(N-2)m, ..., +Nm$).+Consider a collection of non-interacting particles, where each particle can be found in one of two quantum states. As an example, we will deal with //N// elementary magnets that could have their spins oriented up or down (another example: flipping ​a coin). The magnetic moment of an individual magnet is //+m// if the magnet is pointing up and //-m// if it is pointing down. The total number of configurations is $2^N$ while the total number of possible momenta of the entire system is $N+1$ (that is, $-Nm, -(N-2)m, ..., +Nm$). 
 + 
 +The spin excess of the system is equal to: 
 + 
 +$N_{up} - N_{down} = 2s$, 
 + 
 +where $N_{up}$ and $N_{down}$ are the number of magnets with spin up and down, respectively. Spin excess is defined as $2s$ instead of $s$ for convenience. It is easy to show that: 
 + 
 +$N_{up} + N_{down} = N$ 
 + 
 +$N_{up} =N/2 + s$ 
 + 
 +$N_{down} =N/2 - s$ 
 + 
 +The number of possible states with a specific spin excess can be evaluated using a simple formula from combinatorics:​ 
 + 
 +$g(N,s) = \frac {N!} {N_{up}! N_{down}!}.$ 
 + 
 +The function //g// is called a multiplicity function. The multiplicity function forms a Gaussian distribution for //s//: 
 + 
 +$g(N,s) \approx g(N,0) exp(-2s^2/​N)$ 
 + 
 +$g(N,0) = \frac {N!} { (N/2)! (N/2)! } \approx \sqrt \frac 2 {\pi N} 2^N $ 
 + 
 +<hidden Derivation>​ 
 +We will use the Stirling approximation:​ 
 + 
 +$ ln (N!) \approx 1/2 ln (2\pi) + (N+1/2) ln N - N$ 
 + 
 +</​hidden>​ 
  
-  
  
world/ch01.txt · Last modified: 2020/07/28 08:14 by talipovm