(<I>a' <I>a) ~ in Java

Attach qr codes in Java (<I>a' <I>a) ~
(<I>a' <I>a) ~
Java qr-codes creation on java
using barcode printing for java control to generate, create qr image in java applications.
3N r N!
Barcode barcode library for java
use java barcode encoding toembed barcode with java
7~<5p<5Q
Barcode barcode library on java
Using Barcode reader for Java Control to read, scan read, scan image in Java applications.
[u Pal (1) U Qal (1)] ... [u Pa) N) U Qa) N)]
Control qrcode size for .net c#
qr code 2d barcode size in visual c#.net
For fermions we must have P
Control denso qr bar code image with .net
use asp.net website qr-codes printing toprint qr with .net
Q, because the a i are all distinct. Therefore
VS .NET Crystal qr-code maker on .net
use .net crystal denso qr bar code integrating toincoporate quick response code in .net
(<I>a' <I>a)
Deploy denso qr bar code with visual basic.net
generate, create qrcode none in visual basic.net projects
(fermions)
(A.17)
Control ean/ucc 128 image for java
generate, create none on java projects
For bosons the a i are not necessarily all distinct. Suppose that among a 1, ., aN there are n a having the value a. Then (bosons)
Control upc a data in java
upc-a supplement 5 data with java
(A.I8)
Control pdf-417 2d barcode image in java
using barcode printer for java control to generate, create barcode pdf417 image in java applications.
which is not necessarily unity.
Control pdf 417 data with java
to paint pdf417 and pdf417 data, size, image with java barcode sdk
N-BODY SYSTEM OF IDENTICAL PARTICLES
Incoporate british royal mail 4-state customer barcode in java
using java toinsert royalmail4scc for asp.net web,windows application
The integer n a is called the occupation number of the single-particle level a. Obviously we have the conditions
2d Matrix Barcode barcode library on .net
generate, create 2d barcode none in .net projects
L:na=N
recognize data matrix barcode for none
Using Barcode Control SDK for None Control to generate, create, read, scan barcode image in None applications.
n a = 0,1, ... , N n a = 0,1
Control gs1 - 13 data for .net
ean13 data for .net
(boson) (fermion)
Control code 128 code set a size in word
barcode code 128 size on microsoft word
(A.19)
SSRS pdf 417 writer with .net
using sql server reporting service todraw pdf417 2d barcode in asp.net web,windows application
Instead of {a 1 , .. , an} we can equally well label the wave function by the occupation numbers {no, n 1 , ... }. Thus we also introduce the notation <I>n' defined by (A.20) <I>n = <I>a where n stands for the set {no, n b . .. }. If we wish, we may use in place of <I>a the wave function <I>'=
Control ean / ucc - 13 image with .net
using .net winforms tocompose ean13 for asp.net web,windows application
JI)(n))
Control pdf 417 size with office word
to display pdf417 and pdf-417 2d barcode data, size, image with word documents barcode sdk
(A.21)
which is normalized to unity for both bosons and fermions. It is, however, neither necessary nor convenient to do this. The reason is as follows. Suppose we are to calculate the trace of an operator. We may write Tr (2 =
(<I>~, (2<I>~)
(A.22)
where the sum extends over all distinct sets {a 1, ... , aN}' A convenient way to calculate this sum is to sum over each a i independently and to take into account the fact that a permutation of the a i must not be counted as a new term in the sum. That is, we write Tr(2 By (A.21) we have (A.23) Thus it is actually more convenient to use the wave functions <I>a as defined in (A.14). Free-Particle Wave Functions A useful choice of ua(r) is the single-particle wave function for a free particle of momentum p. The quantum number a is now explicitly p, and we have
up(r)
ai' .. ,a N
TI (n))
(<I>~, (2<I>~)
eipor/fr
(A.24)
SPECIAL TOPICS IN STATISTICAL MECHANICS
The allowed values of P are determined by the periodic boundary conditions
up(r + nL)
up(r)
(A.25)
where nand L are defined in (A.ll). This implies that the allowed values of pare
(A.26) L These values form a cubic lattice in momentum space with the lattice constant 27T iii L, which approaches 0 as V ~ 00. In this limit a volume element of size d 3p in momentum space contains (Vlh 3 ) d 3p lattice points. Thus as V ~ 00 a sum over P may be replaced by an integration over P in the following manner: V L ~ 3 d 3p (A.27)
= -p
27Tlin
The functions defined by (A.24) obviously form an orthogonal set. The completeness of this set follows from the fact that any function can be Fourier analyzed. The N-body wave functions built up from up(r) according to (A.14) are the N-body free-particle wave functions. They are denoted by <Pp(I, ... , N) and are eigenfunctions of the kinetic-energy operator K: 1 K<Pp(I, ... , N) = 2m (pi + ... +p~)<Pp(I, ... , N) (A.28) where PI" .. , PN are the momenta of the N-single-particle wave functions contained in <Pp.
Example of Calculation: System of Bosons
We calculate (<P a, Q<P a) for a system of bosons:
(<p a, Q<P a)
* r <Pa I>ij<P a = ~N(N - 1)
r <Pa 12<Pa *v
where the second equality is obtained by renaming the integration variables r I , , r N in an appropriate fashion in each term of the sum Lv ij . Using (A.14) we have
(<p a, Q<Pa)
N(N - 1) NI2
LLf d Q
r [uPaJI) ... uPa)N)]v12[uQa ,(l) uQa)N)]
N(N - 1) N I2 L L(Pal> Pa21v1QaI, Qa 2>( ~Pa3,Qa3 ... ~paN,QaJ
. P Q
(A.29)
where (A.30)
N-BODY SYSTEM OF IDENTICAL PARTICLES
In (A.29) only two terms in the sum the conditions (a) or (b):
are nonzero, namely the terms satisfying
Hence
Qa 1 = Pal' Qa 1 = Pa2'
Qa 2 = Pa2' Qa 2 = Pal'
Qa= PaJ J Qa j
, N) , N) (A.31)
(j = 3,
(<I>a' [2<1>a) =
N(N - 1)
TI (n,.!) a
+ (Pal' Pa2lulPa2, Pal ) (A.32)
X L(Pa1' Pa 2lulPa 1, P(2)
As P ranges through all the N! permutations of the set {a 1, ... , aN}' the pair {Pal' Pa 2 } takes on all possible pairs of values {a, 13} chosen from the set { a 1, ... , aN}' Suppose the occupation numbers for the single-particle states a,13 are respectively n a' n p. Then the number of ways in which the pair { a, 13} can be chosen from the set {a1> ... , aN} is
(a*13) (a=13)
(1 - ~aP)nanp + t~apna(na - 1)
Furthermore, there are (N - 2)! permutations that affect only the quantum numbers { a3' ... , aN} and thus leave { a1, a2} unchanged. Changing the label of <I> a to occupation numbers we obtain
(<I>~,[2<1>~) = t N (N-1)
(N - 2)!