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A variational principle on certain operators in general and the density matrix in particular
Volume29A, number 2
PHYSICS LETTERS
A VARIATIONAL IN GENERAL AND
7 April 1969
PRINCIPLE ON CERTAIN OPERATORS THE DENSITY MATRIX IN PARTICULAR* L. L. VAN ZANDT
Department of Pkysics, Purdue University, Lafayette, India Received 26 February 1969
Approximate solutions of first order operator equations of motion may be obtained variationaly. We present a variational principle which, applied to the density matrix reduces to Frenkel's principle in some special cases.
A p p r o x i m a t e s o l u t i o n s to the S c h r o e d i n g e r equation for c o m p l e x s y s t e m s m a y be o b t a i n e d f r o m v a r i a t i o n a l p r i n c i p l e s of v a r i o u s s o r t s . One such p r i n c i p l e due to F r e n k e l [1] m a y be u s e d to d e r i v e the " t i m e dependent H a r t r e e - F o c k " a p p r o x i m a t i o n [2] f o r f e r m i o n p l a s m a s a s w e l l as more complicated self-consistent field app r o x i m a t i o n s [3]. In t h e r m o d y n a m i c c a l c u l a t i o n s [4] o r in s o m e s p e c i a l s i t u a t i o n s , [5] w o r k i n g d i r e c t l y with wave functions, a s F r e n k e l ' s p r i n c i p l e r e q u i r e s , m a y be a w k w a r d r e l a t i v e to a c a l c u l a t i o n in t e r m s of the s t a t i s t i c a l d e n s i t y o p e r a t o r . We p r e s e n t h e r e a v a r i a t i o n a l p r i n c i p l e , s i m i l a r in s t r u c t u r e to F r e n k e l ' s a p p l y i n g to the d e n s i t y m a t r i x o r any o p e r a t o r with a f i r s t o r d e r equation of motion [5]. By way of c o n c r e t e e x a m p l e , let
i t ~plat = [H, p]
(1)
be the equation of m o t i o n whose s o l u t i o n s we seek. At s o m e s t a r t i n g i n s t a n t , t = 0, p m a y be c h o s e n a r b i t r a r i l y without l o s s of g e n e r a l i t y (although the a p p r o x i m a t e s o l u t i o n s w i l l be m o r e v a l u a b l e the m o r e p r o b a b l e the s t a t e of the s y s t e m d e s c r i b e d by p(0) s u b j e c t to the conditions that p+ = p and the e i g e n v a l u e s of p a r e nonnegative. A s h o r t t i m e l a t e r , P = p(O) + (aplat) At
(2)
Now a p l a t determined from eq. (1) may be too c o m p l e x to be useful. We m a y t h e r e f o r e a p p r o x i m a t e ap/a t b y s o m e s i m p l e r o p e r a t o r 2. The g r e a t e r the c o m p l e x i t y of the a s s u m e d f o r m of 2, the m o r e p r e c i s e m a y be the r e s u l t i n g a p p r o x i m a t i o n s ; the l e s s e r the c o m p l e x i t y the m o r e l i k e l y that the r e s u l t i n g e x p r e s s i o n s w i l l * Supported by ARPA
a c t u a l l y l e a d to a c o m p u t a b l e r e s u l t . F o r m i n g I = Wr {(~p/at-2) + (~p/~t- ~ } ,
(3)
we s e e that I i s c l e a r l y n o n - n e g a t i v e and v a n i s h e s only when 2 and bp/at a r e i d e n t i c a l . If in a c t u a l l y c o n s t r u c t i n g a p o s s i b l e 2 we include s o m e u n d e t e r m i n e d p a r a m e t e r s , bi, then the " b e s t " ~- out of the p a r a m e t r i z e d s e t m a y be obtained by v a r y i n g I, 6 I = 0 =-2Re T r {62 + ( a p / / ~ t - 2 ) }
(4)
in which the v a r i a t i o n s of 2 + a r e c a r r i e d out by v a r y i n g the b i. To be p h y s i c a l l y meaningful, p m u s t be H e r m i t i a n ; if 2 i s not, we r e p l a c e it by ½(E+E +) and the bi m a y a l w a y s be c o n s i d e r e d r e a l , hence the i m a g i n a r y p a r t of the t r a c e in eq. (4) i s m e a n i n g l e s s and we m a y d r o p the r e s t r i c t i o n to the r e a l p a r t . To m a k e eq. (4) useful, in s e l e c t i n g a s t a r t i n g p(O), we a s s u m e a p a r a m e t r i c dependence on t i m e and s e t 2 equal to the d e r i v a t e of p with r e s p e c t to t h i s e x p l i c i t t i m e dependence. 0 = T r {[~bi ( ~ e x p l i e i t ] [ ( ~ t d y n a m i c a l - ( ~ e x p l i c i ~ } or finally
ap(t) 1 _ ~t)]} 0 = T r {[~-~-~T-][~-li[H,p(t)]
(5)
If we c o n s i d e r a p u r e s t a t e , p2 = p this m a y be shown$ to be equivalent to F r e n k e l ' s p r i n c i p l e :
(6) SThis demonstration is part of a larger work to be presented elsewhere. 55
Volume 29A, number 2
PHYSICS LETTERS
Eq. (5) a p p l i e s to s i t u a t i o n s where p2 ¢ p and to operators where
8A/Ot ¢ [A,B]
2.
(7) 3. 4.
and t h e r e f o r e may be of m o r e g e n e r a l utility.
5.
References 1. J. Frenkel, Wave mechanics, Advanced general
FLUCTUATION
ROUNDING OF IN T H E T H R E E
7 April 1969
theory (Clarendon Press, Oxford, England, 1934) p. 253. A. D. Mclachlan and M. A. Ball, Rev. Mod. Phys. 36 (1964) 844. L.L. Van Zandt, Phys. Rev. 172 (1968)372. for example, W. Kohn and J. M. Luttinger, Phys. Rev. 108 (1957) 590. for example, R.B. Thomas jr., Phys. Rev. 171 {1968) 827.
THE SUPERCONDUCTING DIMENSIONAL REGIME
TRANSITION
J. I. GITTLEMAN, R.W. COHEN and J. J. HANAK RCA Laboratories, Princeton, N . J . , USA Received 25 February 1969
Excess conductivity due to superconducting fluctuations in the three dimensional regime has been observed in high resistivity granular films of aluminum and tin and varied as (T-Tc)-Z for d > ~ (T). The coefficient of proportionality exceeded the theoretical value by about 50%.
We have m e a s u r e d the rounding of the r e s i s t i v e t r a n s i t i o n of thick g r a n u a l A1 and Sn f i l m s for which the t h i c k n e s s d was g r e a t e r than the t e m p e r a t u r e dependent c o h e r e n c e length ~(T) over m o s t of the m e a s u r e d t e m p e r a t u r e range. T h e o r y [ 1] p r e d i c t s that the r e s i s t a n c e R(T) in the t h r e e d i m e n s i o n a l r e g i m e d ~>3~(T) v a r i e s a c c o r d i n g to the r e l a t i o n
Ro/R(T)
= 1 + po(~'3(T)
,
(1)
w h e r e R 0 and Po a r e the n o r m a l r e s i s t a n c e and r e s i s t i v i t y and ~ ( T ) is the e x t r a conductivity due to f l u c t u a t i o n s which is given by ~3 = = e2/32~(T)G • = ( T - T c ) / T c. Using the m i c r o scopic theory [2] to e x p r e s s ~ (T) in t e r m s of the P i p p a r d c o h e r e n c e length ~o and the e l e c t r o n ic m e a n f r e e path l, we have ~ ( T ) = 8.9 × 10"6(}o~E)-½(~2 • cm) -1 f
(2)
The e s s e n t i a l d i f f e r e n c e s between cr3(T) for the t h r e e d i m e n s i o n a l case eq. (2) and the e x t r a conductivity ag(T) for the two d i m e n s i o n a l r e g i m e < ~ T ) a r e : ( a ) a3 dep,en.ds on the m a t e r i a l [1, 3] d ~ p a r a m e t e r s ~o and l while •cr2 is independent ofJ t the c h a r a c t e r of the materzal, and (b) a3 ~ e - z , t w h e r e a s cr2~ c -1. We have v e r i f i e d the E-7 t e m p e r a t u r e dependence of ~a in g r a n u a l A1 and Sn. The magnitude of ~ ( T ) for A1 i s about 56
50o/o l a r g e r than that p r e d i c t e d by eq. (2), while in the case of Sn, difficulties in d e t e r m i n i n g the g e o m e t r y p r e v e n t e d a d i r e c t c o m p a r i s o n . P r e v i o u s e x p e r i m e n t a l work [4] has been confined to the two d i m e n s i o n a l r e g i m e . In o r d e r to o b s e r v e t h r e e d i m e n s i o n a l fluctuations~ s a m p l e s a r e r e q u i r e d for which [1] d > 3}(T) over the m e a s u r e d t e m p e r a t u r e range. S u p e r c o n d u c t o r s of the g r a n u a l type [5] can r e a d i l y be made to m e e t this r e q u i r e m e n t . G r a n u l a r f i l m s w e r e made by c o s p u t t e r i n g e i t h e r A1 or Sn and SiO 2 onto g l a s s s u b s t r a t e s . The t e m p e r a ture dependent r e s i s t a n c e of each s p e c i m e n was then m e a s u r e d u s i n g a dc p o t e n t i o m e t r i c method. The m e a s u r i n g c u r r e n t was l e s s than 0.3 A / c m 2 which is well below the value for which c u r r e n t dependent effects were observed. Fig. 1 is a plot of the c o n t r i b u t i o n to the cont ! ductivity due to the fluctuations Po ~ v e r s u s ¢-Y for typical s p e c i m e n s of g r a n u a l A1 and Sn. T a b l e 1 s u m m a r i z e s the p r o p e r t i e s of these specimens : Table 1 Properties of Granular A1 and Sn Vol.% Thicokness Tc Do Electron mean SiO2 (A) (OK) (~ .cm) free path~ ~) Al 27 1870 2.415 1.8x10-3 0.9 Sn 19 3160 4.590 :~ We use the valuep/ 1.6xlO-11~-cm 2 [6].
PHYSICS LETTERS
A VARIATIONAL IN GENERAL AND
7 April 1969
PRINCIPLE ON CERTAIN OPERATORS THE DENSITY MATRIX IN PARTICULAR* L. L. VAN ZANDT
Department of Pkysics, Purdue University, Lafayette, India Received 26 February 1969
Approximate solutions of first order operator equations of motion may be obtained variationaly. We present a variational principle which, applied to the density matrix reduces to Frenkel's principle in some special cases.
A p p r o x i m a t e s o l u t i o n s to the S c h r o e d i n g e r equation for c o m p l e x s y s t e m s m a y be o b t a i n e d f r o m v a r i a t i o n a l p r i n c i p l e s of v a r i o u s s o r t s . One such p r i n c i p l e due to F r e n k e l [1] m a y be u s e d to d e r i v e the " t i m e dependent H a r t r e e - F o c k " a p p r o x i m a t i o n [2] f o r f e r m i o n p l a s m a s a s w e l l as more complicated self-consistent field app r o x i m a t i o n s [3]. In t h e r m o d y n a m i c c a l c u l a t i o n s [4] o r in s o m e s p e c i a l s i t u a t i o n s , [5] w o r k i n g d i r e c t l y with wave functions, a s F r e n k e l ' s p r i n c i p l e r e q u i r e s , m a y be a w k w a r d r e l a t i v e to a c a l c u l a t i o n in t e r m s of the s t a t i s t i c a l d e n s i t y o p e r a t o r . We p r e s e n t h e r e a v a r i a t i o n a l p r i n c i p l e , s i m i l a r in s t r u c t u r e to F r e n k e l ' s a p p l y i n g to the d e n s i t y m a t r i x o r any o p e r a t o r with a f i r s t o r d e r equation of motion [5]. By way of c o n c r e t e e x a m p l e , let
i t ~plat = [H, p]
(1)
be the equation of m o t i o n whose s o l u t i o n s we seek. At s o m e s t a r t i n g i n s t a n t , t = 0, p m a y be c h o s e n a r b i t r a r i l y without l o s s of g e n e r a l i t y (although the a p p r o x i m a t e s o l u t i o n s w i l l be m o r e v a l u a b l e the m o r e p r o b a b l e the s t a t e of the s y s t e m d e s c r i b e d by p(0) s u b j e c t to the conditions that p+ = p and the e i g e n v a l u e s of p a r e nonnegative. A s h o r t t i m e l a t e r , P = p(O) + (aplat) At
(2)
Now a p l a t determined from eq. (1) may be too c o m p l e x to be useful. We m a y t h e r e f o r e a p p r o x i m a t e ap/a t b y s o m e s i m p l e r o p e r a t o r 2. The g r e a t e r the c o m p l e x i t y of the a s s u m e d f o r m of 2, the m o r e p r e c i s e m a y be the r e s u l t i n g a p p r o x i m a t i o n s ; the l e s s e r the c o m p l e x i t y the m o r e l i k e l y that the r e s u l t i n g e x p r e s s i o n s w i l l * Supported by ARPA
a c t u a l l y l e a d to a c o m p u t a b l e r e s u l t . F o r m i n g I = Wr {(~p/at-2) + (~p/~t- ~ } ,
(3)
we s e e that I i s c l e a r l y n o n - n e g a t i v e and v a n i s h e s only when 2 and bp/at a r e i d e n t i c a l . If in a c t u a l l y c o n s t r u c t i n g a p o s s i b l e 2 we include s o m e u n d e t e r m i n e d p a r a m e t e r s , bi, then the " b e s t " ~- out of the p a r a m e t r i z e d s e t m a y be obtained by v a r y i n g I, 6 I = 0 =-2Re T r {62 + ( a p / / ~ t - 2 ) }
(4)
in which the v a r i a t i o n s of 2 + a r e c a r r i e d out by v a r y i n g the b i. To be p h y s i c a l l y meaningful, p m u s t be H e r m i t i a n ; if 2 i s not, we r e p l a c e it by ½(E+E +) and the bi m a y a l w a y s be c o n s i d e r e d r e a l , hence the i m a g i n a r y p a r t of the t r a c e in eq. (4) i s m e a n i n g l e s s and we m a y d r o p the r e s t r i c t i o n to the r e a l p a r t . To m a k e eq. (4) useful, in s e l e c t i n g a s t a r t i n g p(O), we a s s u m e a p a r a m e t r i c dependence on t i m e and s e t 2 equal to the d e r i v a t e of p with r e s p e c t to t h i s e x p l i c i t t i m e dependence. 0 = T r {[~bi ( ~ e x p l i e i t ] [ ( ~ t d y n a m i c a l - ( ~ e x p l i c i ~ } or finally
ap(t) 1 _ ~t)]} 0 = T r {[~-~-~T-][~-li[H,p(t)]
(5)
If we c o n s i d e r a p u r e s t a t e , p2 = p this m a y be shown$ to be equivalent to F r e n k e l ' s p r i n c i p l e :
(6) SThis demonstration is part of a larger work to be presented elsewhere. 55
Volume 29A, number 2
PHYSICS LETTERS
Eq. (5) a p p l i e s to s i t u a t i o n s where p2 ¢ p and to operators where
8A/Ot ¢ [A,B]
2.
(7) 3. 4.
and t h e r e f o r e may be of m o r e g e n e r a l utility.
5.
References 1. J. Frenkel, Wave mechanics, Advanced general
FLUCTUATION
ROUNDING OF IN T H E T H R E E
7 April 1969
theory (Clarendon Press, Oxford, England, 1934) p. 253. A. D. Mclachlan and M. A. Ball, Rev. Mod. Phys. 36 (1964) 844. L.L. Van Zandt, Phys. Rev. 172 (1968)372. for example, W. Kohn and J. M. Luttinger, Phys. Rev. 108 (1957) 590. for example, R.B. Thomas jr., Phys. Rev. 171 {1968) 827.
THE SUPERCONDUCTING DIMENSIONAL REGIME
TRANSITION
J. I. GITTLEMAN, R.W. COHEN and J. J. HANAK RCA Laboratories, Princeton, N . J . , USA Received 25 February 1969
Excess conductivity due to superconducting fluctuations in the three dimensional regime has been observed in high resistivity granular films of aluminum and tin and varied as (T-Tc)-Z for d > ~ (T). The coefficient of proportionality exceeded the theoretical value by about 50%.
We have m e a s u r e d the rounding of the r e s i s t i v e t r a n s i t i o n of thick g r a n u a l A1 and Sn f i l m s for which the t h i c k n e s s d was g r e a t e r than the t e m p e r a t u r e dependent c o h e r e n c e length ~(T) over m o s t of the m e a s u r e d t e m p e r a t u r e range. T h e o r y [ 1] p r e d i c t s that the r e s i s t a n c e R(T) in the t h r e e d i m e n s i o n a l r e g i m e d ~>3~(T) v a r i e s a c c o r d i n g to the r e l a t i o n
Ro/R(T)
= 1 + po(~'3(T)
,
(1)
w h e r e R 0 and Po a r e the n o r m a l r e s i s t a n c e and r e s i s t i v i t y and ~ ( T ) is the e x t r a conductivity due to f l u c t u a t i o n s which is given by ~3 = = e2/32~(T)G • = ( T - T c ) / T c. Using the m i c r o scopic theory [2] to e x p r e s s ~ (T) in t e r m s of the P i p p a r d c o h e r e n c e length ~o and the e l e c t r o n ic m e a n f r e e path l, we have ~ ( T ) = 8.9 × 10"6(}o~E)-½(~2 • cm) -1 f
(2)
The e s s e n t i a l d i f f e r e n c e s between cr3(T) for the t h r e e d i m e n s i o n a l case eq. (2) and the e x t r a conductivity ag(T) for the two d i m e n s i o n a l r e g i m e < ~ T ) a r e : ( a ) a3 dep,en.ds on the m a t e r i a l [1, 3] d ~ p a r a m e t e r s ~o and l while •cr2 is independent ofJ t the c h a r a c t e r of the materzal, and (b) a3 ~ e - z , t w h e r e a s cr2~ c -1. We have v e r i f i e d the E-7 t e m p e r a t u r e dependence of ~a in g r a n u a l A1 and Sn. The magnitude of ~ ( T ) for A1 i s about 56
50o/o l a r g e r than that p r e d i c t e d by eq. (2), while in the case of Sn, difficulties in d e t e r m i n i n g the g e o m e t r y p r e v e n t e d a d i r e c t c o m p a r i s o n . P r e v i o u s e x p e r i m e n t a l work [4] has been confined to the two d i m e n s i o n a l r e g i m e . In o r d e r to o b s e r v e t h r e e d i m e n s i o n a l fluctuations~ s a m p l e s a r e r e q u i r e d for which [1] d > 3}(T) over the m e a s u r e d t e m p e r a t u r e range. S u p e r c o n d u c t o r s of the g r a n u a l type [5] can r e a d i l y be made to m e e t this r e q u i r e m e n t . G r a n u l a r f i l m s w e r e made by c o s p u t t e r i n g e i t h e r A1 or Sn and SiO 2 onto g l a s s s u b s t r a t e s . The t e m p e r a ture dependent r e s i s t a n c e of each s p e c i m e n was then m e a s u r e d u s i n g a dc p o t e n t i o m e t r i c method. The m e a s u r i n g c u r r e n t was l e s s than 0.3 A / c m 2 which is well below the value for which c u r r e n t dependent effects were observed. Fig. 1 is a plot of the c o n t r i b u t i o n to the cont ! ductivity due to the fluctuations Po ~ v e r s u s ¢-Y for typical s p e c i m e n s of g r a n u a l A1 and Sn. T a b l e 1 s u m m a r i z e s the p r o p e r t i e s of these specimens : Table 1 Properties of Granular A1 and Sn Vol.% Thicokness Tc Do Electron mean SiO2 (A) (OK) (~ .cm) free path~ ~) Al 27 1870 2.415 1.8x10-3 0.9 Sn 19 3160 4.590 :~ We use the valuep/ 1.6xlO-11~-cm 2 [6].