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Advances in chemical physics - Prigogine I.

Prigogine I., Rice S.A. Advances in chemical physics - Advision, 1970. - 167 p.
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ФввОМ) == Фвв(~к, t)
Хвв(к, ю) = ЗСвв(-к>ю) (35)
Another important property of the after-effect function Ф^в(к, t) can be derived. Note that
Ф^вС-к, -0 = U BJ-\
Since the trace is invariant to a cyclic permutation of the operators, it follows that
-0 №(0,= -Ф„,(к, г) (36)
Фвв( — — 0 = — Фвв(к> 0
С. The Reciprocal Relations
The operators Ak and Ёк are
4 = Е«й».е'|‘г"]+
The single particle properties {^m} and {Ёт} are Hermitian. A_k and are consequently the Hermitian conjugates of Ak, Ёк. Observables can quite generally be classified as time-even or time-odd depending on whether they do or do not change sign on time reversal. All even time derivatives of the coordinates are even under time reversal while all odd derivatives are odd under time reversal. Thus the Hamiltonian is time even, the angular momentum is time odd and the linear momentum is time odd. Time-even properties are represented by real Hermitian operators, while time-odd properties are represented by imaginary Hermitian operators.
The properties A m and Ё m have the property that
А*т = УлАт
8*m = Чв$т
where уA and у B are the time reversal signatures of the observables. Thus if A m is time-even yA = 1, whereas if it is time-odd yA = — 1. The same holds true for у B.
From the definitions of Ak and Bk it follows that
А*к = улА-к = уАА+к
Ё*к = YвВ-к = У B&+к
where A+ k and Ё+ k denote the Hermitian conjugates of Ak and Bk.
In the absence of external magnetic fields, or for that matter any external pseudovector field, the exact energy eigenstates of a system can only be degenerate with respect to the total angular momentum of the system: This source of degeneracy can be removed if we assume that the body is enclosed in a container with rigid walls. It is always possible in this case to choose the energy eigenstates to be real. Consider the matrix elements of Ak and Ёк in the energy representation in which the eigenstates are real. From the preceding relation it is seen that
(n\A*k\m) = yA(n\A+ k\m) (37)
Since the states are real
0n \A*k\ m)* = (n\Ak| m) = yA(n \A+k\ m)* = yA(m \Ak\ n) (38)
The last equality follows from the definition of the Hermitian conjugate. Consequently
(n\Ak\ m) = yA{m |Ak\ n) and similarly (39)
(n \Bk\ m) = yB(m \Bk\ n)
The operators Ak and Ёк are seen to be symmetric or antisymmetric depending on their symmetry in time.
Let us now consider the time dependence of the one-sided correlation function,
(At(t)B_k(0)} = Tr 0 Xt еШг/й Ё_к Because the trace is invariant to a cyclic permutation in the order of the operators,
<Ak(t)B-k(0)y = <Лк(0)В_к(-0>
The trace can be expanded in the complete set of energy eigenstates described above. Then
<Лк(0#-к(0)> = £ 0„(n \Ak\ *w)(m |£_k| n) eia>nmt
where conm = (E„ — Em)jh. Substitution from Eq. (39) shows that <4к(0Д-к(0)> = У а У в £ 0„(« l^-fel m)(m \Ak\ n) e<a,""^,
so that
(ii) <Лл(0^-к(0)> = УлУв<В-к(0)Ак(-ф The complex conjugate of the one-sided function is
(iii) <A(0B-*(O)>* = X p„(n |A] m)*0n\6-t\ nf e-'""-'
= £ Pn(« l4l rn)(m |i_k| n) ei<w
= Yb £ P»(n U-fel ™)(m l4l n) e,w
The second equality follows from the definition of the Hermitian conjugate and is <Bk(0)A-k(t)). The third equality follows from the reality of the states and is <^4-fc(0)l?fe(0>. Consequently
<Ak(t)B_k(0)>* = <Bk(0)A_k(t)> = yAyB<A_k(0)B+k(t)>
Analogous relations follow for the one-sided correlation function <B_k(0)Ak(t)>,
(i) (B_k(0)Ak(t)> = <B.k(-t)Ak(0))
(ii) <B_k(0)Ak(t)> = yAyB<Ak(-t)B-k(0)>
(iii) <B-k(0)A(0>* = 0)> = УаУв<В^)А-М}
Consider now the after-effect function
From the properties (i), (ii), and (iii) for (Ak(t)B_k(0)> and (В_к(0)Ак(ф it is easily seen that
(О Флв(к5 0 = — Фвл( —к, —О
00 Ф л в(к, 0 = + у л у в Ф вл(-к, 0 (40а)
(iii) Ф*лв(к> 0 = Флв(-к> 0 = УаУв®ва(К О
These relationships can be combined with the transformation properties of Флв(к, 0 under reflection summarized in Eq. (32)
(iv> ФАb(~к» 0 = ЕлeвФл в(к, 0
Conditions (i) and (ii) together give
Флв(к> 0 = — УаУвФавО^, — t) (40b)
Conditions (iii) and (iv) together give
Ф*лв(к, 0 =елБвФлв(к> 0 = УаУв®ва(К 0 (40с)
Ф^в(к) 0 = ^а ^в Ф вл(к> 0 (40d)
where ХА = Уа^а ап^ ^в = Увев аге the signatures of the properties Ak and Ёк under combined inversion and time reversal.
It should be noted that
Ф в в(к, 0 = — Ф в в(к,— О Ф*вв(к> 0 = Фвв(к) 0 (40е)
Фвв(к, 0 = Фвв( —к, t)
so that Фвв(к, t) is a real odd function of the time and a real even function of k.
From Eq. (33) it follows that
1ab(к, ю) = Ъ-л ^b1ba(к, w) (40f)
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