By a straightforward calculation we obtain in Java

Integration Quick Response Code in Java By a straightforward calculation we obtain
By a straightforward calculation we obtain
Java qr-code implementation in java
use java denso qr bar code encoding toembed qr code on java
l'I'k) =
Java barcode creator in java
using barcode printing for java control to generate, create barcode image in java applications.
VI -
Barcode scanner in java
Using Barcode decoder for Java Control to read, scan read, scan image in Java applications.
a~ atl'l'o)
QR implement on .net
use qr code 2d barcode encoder todeploy qr-codes with .net
Thus the one-phonon state is a superposition of unperturbed states in which there are any number of particles p, - p for all p, plus an additional particle of momentum k. The average number of particles that have momentum k in the perturbed ground state is
QR Code ISO/IEC18004 barcode library with .net
generate, create quick response code none on .net projects
(n k ) = ('I'olata k l'l'o) =
Control denso qr bar code data in visual basic
denso qr bar code data with
--2 1 - ak
Control code 128 code set c data in java
to insert barcode code 128 and code 128c data, size, image with java barcode sdk
(k"* 0)
Code 39 Full ASCII barcode library with java
using java todraw code 39 full ascii on web,windows application
Control universal product code version a data for java
to generate upc code and gs1 - 12 data, size, image with java barcode sdk
Therefore the total number of excited particles in the perturbed ground state is
GTIN - 13 creation for java
use java ean-13 supplement 2 integration tomake ean-13 for java
(n k ) =
Include barcode in java
using java toprint bar code with web,windows application
k*O 1 - a k
Identcode creation in java
use java identcode generation tocompose identcode for java
_k_ 2
8 = 3
Barcode implement on vb
using .net framework togenerate barcode with web,windows application
Control qr code 2d barcode data with .net
quick response code data in .net
The number of particles of zero momentum in the perturbed ground state is
Control ean13+5 size on visual basic
ean13+2 size for visual
Draw barcode code39 with visual
using visual studio .net (winforms) crystal toproduce barcode code39 with web,windows application
which shows that the approximation (13.92) is justified. It is instructive to calculate the wave functions in configuration space. In each term of the sum in (13.114), the number n == L Ik is half the total number k>O of particles with nonzero momentum. Thus we must have N ~ 2n. We rewrite (13.114) as follows:
Ean13+2 barcode library with .net
generate, create ean13+5 none in .net projects
Control gs1128 size on .net
ean/ucc 128 size on .net
L 'IE'k=n . '2""
[(-a 1 )"(-a 2 )'2 ]1/1 , /2 , )
We imagine N
~ 00
at the end of the calculation. The normalized configuration
space wave function for an unperturbed state specified by the occupation numbers {no, n 1 , } is 1 1 "Pei(PI"rl+ ... +PN'rN) (13.12_~ ( r l' ... , r NIn) no, l' . . . V N/2 N! nk!) ';:
where among the N momenta PI' ... ,PN, no are 0, n i are k I , etc. The symbol P denotes a permutation of rl> ... , rN' Hence
(r 1 , , r N l/ 1 , 12 " , , ) 1 1
= V N/ 2 VN!(N - 2n)!
where among the n vectors PI' ... , Pn' II are kl> 12 are k 2 , etc. Now consider the sum appearing in (13.121):
IXn) =
This may be rewritten as
'I' '2" ..
[(-al)"(-a 2)" '"
]1 / 1,/ 2",,)
E'k =
where, in the first line, k 1 ~ k 2 ~ . . . denotes any ordering of the momenta. In the second line each momentum independently ranges through half of momentum space excluding k = 0 (as denoted by k > 0). The configuration space representation of (13.124) is obtained by substituting (13.123) for 1/1,/2",,): 1 1 Xn(r 1 , .. ,r N ) = V N / 2 n!VN!(N - 2n)! ~P
e ik "(r,-r2)
e ik ".(r 3 -r4 )]
Let us define
f(ij) = f(r i
f(r) = - -2
ak e
ik r '
fd k
ak e
ik r '
Since ak depends only on Ikl, it follows that f(r) depends only on Irl. We can then write
Xn(r I , .. , rN ) =
V~/2 n!VN!( ~ _ 2n )! (~ ~P [f(12) ... f(34)]
in which there are n factors f(ij).
The number of distinct ways of choosing the arguments of j(ij) from the N coordinates f1> . , r N can be found by filling the n boxes in the following with N balls. The boxes are identical, each holding two balls, and the order of the two balls is irrelevant.
100100C8 00 1 00
1 2 n-l
000 ... 000
N - 2n balls left over
It is evident that there are
(N - 2n )!n!2 n
distinct ways to fill the boxes, whereas there are N! permutations of r 1,. . ,rN' Hence
Xn(r1, ,r N ) =
/(N-2n)! , NnI:[j(12) j(34)] (13.126) N.
where the sum is extended over all distinct ways of filling the "boxes." Therefore
'l'o(r1, ,r N) = Z
Xn(r1, ,r N )
= V NI2 {I + [/(12) + j(34) + ... ]
+ [j(12)j(34) + j(12)j(56) + .. , ]
+ [j(12)j(34)j(56) + ... J + ... }
It differs from the function
(13.128) only in that (13.128) contains extra terms of the type
j(12)j(13), j(12)j(13)j(34), ...
in which the same particle appears in more than one "box." These terms may be shown to belong to a higher order in the calculation than we have considered. Therefore we may take (13.128) to be the wave function to the order /a 3 ju. The function j(r) defined in (13.125) has the following asymptotic behavior:
j(r) "" - -
(r ro)
j(r) "" - 32 V
~ (~
(r ro)
Thus although (13.128) vanishes at rij = a, (13.1;7) does so only approximatel~. but its value at rij = a is of a higher order than a 3/V . The one-phonon wave function (13.116) is easily shown to have the following form in configuration space:
which verifies the result of the last section. The extension of the present results to a dilute Bose gas with attractive interaction, which has a more realistic phase diagram, has also been worked out.'