v(t) = E. v(t) E~ and E, is the projection on o~r Especially, if dim ~, = 1, we have (lb)

R E M A R K O N T H E D E C A Y O F A M I X E D S T A T E P. EXNER

Faculty of Mathematics and Physics, Charles University, Prague*)

The initial decay rate of a mixed state is discussed; is shown to be zero for finite energy states.

In the previous paper [1 ] we ~nvestigated a general scheme for description of unstable systems.

One of our results concerned the initial decay rate. Generalizing the earlier results of HORWITZ and MARCHAND (see [2] and other references contained in [11) we proved there, that the initial decay rate of any pure state of the unstable system equals to zero, if this state is so called finite energy state.

Here we shall be interested in the same problem, assuming now the state of the decaying system to be in general mixed. Such assumption seems to be reasonable: firstly, from the point of view of an experiment it is too restrictive to treat only pure states of unstable systems. In particular, mixed states are generally considered in the recent studies about the influence of measuring devices on the time evolution of the unstable system [3 ]. Moreover, for this kind of problems the behaviour of the unstable system immediately after its preparation, especially the initial decay rate, is of great importance (see also [4, 5]).

We shall prove in this paper that for finite energy states the initial decay rate is equal to zero, which generalizes the result contained in [1].

W e d e a l w i t h t h e d e s c r i p t i o n o f u n s t a b l e s y s t e m s c o n s i d e r e d i n Ref. [1]. L e t us r e m i n d t h a t we u n d e r s t o o d t h e r e as unstable a n y p h y s i c a l s y s t e m w h i c h o b e y s t h e f o l l o w i n g c o n d i t i o n s :

(i) o n e c a n a s c r i b e to it a state H i l b e r t space ~ , , (ii) o~r is a p r o p e r s u b s p a c e o f a H i l b e r t space ~4 ~,

(iii) a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n U(t) o f o n e - p a r a m e t e r t r a n s l a t i o n g r o u p is r e a l i z e d o n o~ a n d oY(,, is n o t a n i n v a r i a n t s u b s p a c e o f U(t) o n ~ f i f t > 0.

H e r e o~ m e a n s t h e state H i l b e r t space o f t h e " w h o l e " s y s t e m , i.e. it c o n t a i n s also v e c t o r s c o r r e s p o n d i n g to d e c a y p r o d u c t s etc. F u r t h e r U ( t ) = exp ( - i H t ) is t h e e v o l u t i o n o p e r a t o r , a n d H is t h e r e f o r e t h e t o t a l H a m i l t o n i a n .

F o r a n y 0 ~ J r ' , , F'[I = a, we define t h e decay law ( n o n - d e c a y p r o b a b i l i t y ) o f t h e u n s t a b l e s y s t e m p r e p a r e d at t = 0 i n t h e state ~p as t h e f u n c t i o n

( t a ) P~(t) = IlV(t) ~[I z

w h e r e

v(t) = E. v(t) E~

a n d E , is t h e p r o j e c t i o n o n o~r E s p e c i a l l y , if d i m ~ , = 1, we h a v e

( l b ) Po(t) = Pv,(t) - 1(r U(t) ~)l 2 .

*) Mysllkova 7, Praha 1, Czechoslovakia.

976 Czech. ]. Phys. B 26 [1975]

P. Exner: R e m a r k on the decay o f a m i x e d states

The initial state need not be pure but can be a mixture; then we have to describe it by a density matrix Q, Ran Q c s~,. In this case we define the

decay law

by (2)

Pe(t)= Tr{Q(t)Eu} = Tr{U(t)oU+(t)E.} = Tr{V+(t) V(t)Q}

(the notion of trace is everywhere related to $~"); this definition generalizes naturally (la).

The evolution operator U(t) and the Hamiltonian H can be expressed in the form of spectral decomposition

f ~ 1 7 6 e - u ` f ?

(3)

v(t)

^{= n =} ;t d e , , ( ; t )

(see e.g. [6]). N o w we shall generalize the concept of finite energy state used in [1]:

a density matrix 0 is said to describe a finite energy state if the quantity

(4) <H>~ = f-~oo;t d/t~ /~0(;t) = Tr {0 Et~(;t)} ,

is finite. In order to show consistency o f this definition and its connection to that of finite energy pure state we shall prove the following two assertions:

Propositon 1: The function/~0(') defined by (4) determines a measure on R.

Proof:

It is sufficient that the function #0(') is non-decreasing and continuous on the right (see e.g. [7]). As any compact operator the density matrix Q has a pure point spectrum (for properties of density matrices see e.g. [8]) so that we can write it in the form

(5) = = 1 ,

k k

where

Wk > 0

for all k and E k is one-dimensional projection containing the normalized eigenvector q~k o f p in its range. For any ;t e E the spectral projection En(;t ) is bounded so that #Q(2) is defined and

(6) #o(2) = Ewk/tk(2), /~k(;t) = (~Pk, Eu(2) cp~).

The function Pc(.) is non-decreasing and continuous on the right for all k;/t0(. ) is therefore obviously non-decreasing. Let us assume dim Ran Q = oo; otherwise also the continuity can be seen trivially.

Consider some ;to c •, then to any e > 0 there exists an integer

no

such that for all n > no it holds

<

k = n + l

Czech, ]. Phys. B 26 [1976] 977

P. Exner: Remark on the decay of a mixed states

and on the other hand, to any such n one can find positive numbers

5k, k = 1, 2 ... n,

so that for all 2, 20 < 2 < 20 +

5k,

we have

(**) 0 _< ~ k ( 2 ) - pk(20) < -~e.

Let us define 6 = min

54,

then combining (*) and (**) we obtain for all 2,

l<k<=n

20 < 2 < 2 o + 5 :

o _< o(2) - . o ( 2 o ) _-< wk( k(2) - k(2o)) +

k = l

k = n + l k = l k = n + l

i.e. po(.) is continuous on the right at an (arbitrary) point 2o e R.

The present definition of the finite energy mixed state is essentially mathematical.

Naturally, one has to ask what it means physically. It is reasonable to expect the

"energy-finiteness" of a mixed state to be connected with the analogous property of pure states contained in this mixture.

Proposition2:

Let 0 be a finite energy state. Then any q~k "contained in the mixture", i.e. such to which corresponds non-zero

Wk,

describes also a finite energy state.

Proof:

In order to prove this assertion we have to say what the convergence of integral (4) means. Let us denote by H e the operators

(3a)

H+ = f ] 2 d E . ( 2 ) , H_ = f ~ o 2 d E n ( 2 ) '

so that H = H+ + H _ and

(3b) Inl = H § - H _ .

The integral (4) is understood to be convergent if both < H + ) o and < H _ ) e converge (are finite), which is equivalent to finiteness of

<1/%.

We denote for convenience

<H>k= f ] Ad#k(l)

(see (6)), and analogously

<He) k,

<IH[)k. The absolute convergence of the series (6) (we consider the more complicated case dim Ran 0 = ~ ) implies that for any measurable set M c ~ the relation holds

pQ(M) =

EWk

Pk(M),

k

978 czech. J. Phys. B 28 [1978]

P . E x n e r : R e m a r k o n t h e d e c a y o f a m i x e d s t a t e s

and consequently

(7a)

<H+ >o = ~wk<H+ >k,

k

i.e.

(7b) <H>q = ~Wk<H> k

k

and

(7c)

<IHi>o =

k

The last relation tells us that infiniteness of any

<lnl>k

(to which corresponds

Wk

> 0) implies that Q cannot be a finite energy state.

Thus we can forrflulate an

alternative definition:

a state ~ is a finite energy state if it is a mixture of finite energy pure states tpk such that the series (7) converge.

Now we are in position to prove the announced statement concerning the initial decay rate:

Theorem: Let Q be a finite energy state. Then the initial decay rate P~(0) equals to zero.

Proof:

We express the density matrix ~ through (5) and write the decay law (2) in the form

PQ(t) = 2Wk((Oj, U(t) E k U+(t) ~oj) jk

(order of summing is irrelevant due to absolute convergence); further we obtain

PQ(t) = EWulIEk U+(t) <pill 2 = ~:Wkl(q>j, U(t)

+k)l 2 =

EWkl(<pj, V(t) <pk)l z .

j k j k j k

In analogy with the notation used in [1] we write

(8a) p , ( t ) = Ilv(t) o,,l[ , p,(t) = I(<p,,, ; these function obey the following inequalities

(8b)

O < pk(t) < P (t) < P (O) = pk(O) = 1

~ k ~ k

We express in this notation the decay law:

Pc(t) = ~',Wk Pk(t).

k

The inequalities (8b) show that there holds

(9a)

0 <= po(t) -~ s k pk(t) <= Po(t)

k

Czech. I- Phys. B 26 [1976] 979

P. Exner: Remark on the decay of a mixed states

for all t, and

(9b) pe(O)

=

Po(O)= 1.

Combining the conditions (9a,b) one easily finds (see [1]) that p~(0) = 0 implies P;(0) = o.

We are therefore interested in

p;(t)

= (d/d 0

EWk pk(t).

According to Proposition 2

k

all the q~k correspond to finite energy states, i.e. all

<[Hl>k

are finite and looking for

d (~Ok ' U(t)~Ok)= d f [

e_i;~t

dpkO~)

dt ~ oo

we can interchange the derivative with the integral and obtain

oo

(10)

d (q)k, U(t)

rpk) = - i ~,e-iXt dgk(;O 9

dt -o~

Let us prove now that for all k the derivatives

p'g(t)

are bounded and continuous functions o f t: Eqs. (8), (10)imply

(11) p'k(t)

= 2 Re

(U(t) q~k, (PR) " ~ ((Pk, U(t) (Pk) =

^d

----2 Im (U(t) q)k, (Pk) " f~_ ~'~e-iXt ^{dPk0-) 9}

One can easily see that

[p;,(t)[ < 2 141 2<[HI> ~

which proves the boundedness of

Pk(t).

Moreover, ~0 k as a finite energy state belongs to the domain of

41HI

= ~/H+ + 4 - H - . Since

U(t)is

strongly continuous, we obtain

]f;~2[ e-ixt - e-i;tt~ dPk00 <----~=+~ [(~/c~H~)(Pk, [(U(t)- U(to)] x/(~H~)(Pk)[----<

at = :t:

The function

f_~176

dpk(2) is therefore continuous, and consequently

p'k(t)

is continuous.

I f dim Ran ~ < ~ , we may now write

(12)

pk(t) = 2Wk Pk(t) .

k

9 8 0 Czech. I. P h y s . B 26 [1976]

P. Exner: Remark on the decay of a mixed states

In the case dim Ran Q = ~ it is necessary to verify whether the series on the right- hand side of (12) converges uniformly with respect to t. However, this is not difficult, since the series can be majorized as follows (see (7c), (11))

IEwk pt(t)l 2wklp' (t)l 2 = 2<1/-/1>

k k k

Using (11), (12) we obtain now

especially

p'o(t) = 2 ~Wkk Im (U(t) qgk, tpk ) . f[ooRe-i~t d/~k(2),

p~(O) = 2 •Wk Im ( H ) k = 0

k

(each ( H ) k is real), and consequently P~(0) = 0. [ ] Let us now pass to discussion of the results. Notice first that in the case o f a finite energy pure state (when the density matrix Q reduces to one-dimensional projection containing a unit vector ~ in its range), the decay law (2) gives (la) and (H>~ =

= ( H ) ~ < o% so that the proved theorem really generalizes the result deduced i n [ l ] .

If P~(0) does not exist, the initial decay rate is defined as the derivative on the right P~(+0) in the case when it exists. Our theorem says that states with non-zero initial decay rate could be found among infinite energy states1). What concerns the physical realizability of infinite energy states the same is true as in the special case o f a pure state2): infiniteness o f the integral (IH])0 alone cannot serve as a criterion for exclu- sion of the state Q as physically unrealizable. The reason is that what one actually measures is a probability

T r {0 En(A)} = J ' 9 , ~ )

that the measured quantity (especially energy) would be found in an interval A (more generally in a Borel set A on R - of. Ref. [9]). This probability is defined for any state Q and it holds j'Yo~ d/~0(2) = 1. Since any real experiment is restricted to b o u n d e d region of R (bounded scale of a measuring apparatus), convergence o f quantities o f the type (4) is a matter of our extrapolation.

1) Such states obviously exist: for example let 0 be such that d,ttk(,~ ) ⁼ (1/2n) [Fk/(,~ - - ek) 2 -{- + 88 d2 for all k, F k 3> 0, then PQ(t)= ~ w k exp(--Fkt), t>= 0, i.e. P~(+ 0 ) < 0. Notice that

k

~o is an infinite energy state, though all the integrals ~ Woo 2 d/tk(~,) converge in the sense of principal value (cf. proof of Proposition 2).

2) We are indebted to Prof. I. TODOROV for discussion about this problem which has followed publication of the paper [1].

Czech. J. Phys. B 26 [1O761 9 8 1

P. Exner: Remark on the decay o f a mixed states

L e t us p o i n t o u t t h a t the initial d e c a y rate is n o t a directly m e a s u r a b l e q u a n t i t y . A c c o r d i n g t o p o s t u l a t e s o f the q u a n t u m t h e o r y a n y r e d u c t i o n o f a state r e p r e s e n t s i t s e l f t h e i m m e d i a t e process. H o w e v e r , the d u r a t i o n o f a real m e a s u r e m e n t (which a l s o i n c l u d e s p r e p a r a t i o n o f a state) is finite, so t h a t the c o n c e p t o f two m e a s u r e m e n t s a r b i t r a r i l y closely f o l l o w i n g e a c h o t h e r h a s its limitation3).

N e v e r t h e l e s s , a n i n f o r m a t i o n a b o u t the initial d e c a y r a t e c a n be u n d e r s t o o d as a u s e f u l s t a r t i n g p o i n t for e s t i m a t i n g h o w the d e c a y law behaves n e a r the t i m e origin.

A s we h a v e m e n t i o n e d , this is i m p o r t a n t n a m e l y f o r studies o f r e p e a t e d m e a s u r e m e n t s o n u n s t a b l e system.

Received 4. 7. 1975.

References

[1] HAVLi(3EK M., EXNER P., Czech. J. Phys. B 23 (1973), 594.

[2] HORWlTZ L. P., LAVITA J., MARCHAND J.-P., J. Math. Phys. 12 (1971), 2537.

13] TWAREQUE ALI S., FONDA L., GHIRARDI G. C., N. Cim. 25.4 (1975), 134.

TWAREQUE ALI S., GHtRARDI G. C., N. Cim. 24.4 (1975), 220.

[4l EKSTEIN H., SIEGERT A. J. F., Ann. Phys. (NY) 68 (1971), 509.

BESKOW A., NILSON J., Arkiv f6r Fysik 34 (1967), 561.

FONDA L., GHIRARDI G. C., RIMINI A., WEBER T., N. Cim. 15 A (1973), 689.

[5l EXN~R P., HAVLi~EK M., Proceedings of 3rd Conference of Czechoslovak Physicists (Olo- moue, 1973), 98 (in Czech).

DEGASPERIS A., FONDA L., GHIRARDI G. C., N. Cim. 21A (1974), 471.

[6] AKHmZ~R N. I., GLAZMAN I. M., Linear Operator Theory in Hilbert Space (in Russian), Nauka, Moscow 1966.

[71 KOLMOGOROV A. N., FOMIN S. V., Elements of the Theory of Functions and Functional Analysis (in Russian), Nauka, Moscow 1972.

[8l BLANK J., EXNER P., HAVLf~EK M., Selected Topics of Mathematical Physics: Linear Opera- tors on Hilbert Space t (in Czech), SPN, Prague 1975.

[9l JAtrcrI J. M., Foundations of Quantum Mechanics, Addison-Wesley PuN. Co, London 1968.

3) We wish to thank Dr. J. FORM~NEK for discussion on this point.

982 czech. ]. Phys. B 26 [1976]