From QuantumMechanical Harmonic Oscillators to Classical Ones through Maximization of Entropy -
B. Bouiil and 0. Henri-R-au Universite de Perpignan, Avenue de Villeneuve, 66025 Perpignan Cedex, France The transition from ouantum mechanical to classical behavior is always of great interest in the exposition of quantum chemistrv. This nassaee mav " be annroached either eene r d y by aid-of funhamental conside;ations, such a s t h e correspondence principle, or more intuitively by looking a t concrete situations. The last approach has been used, for instance, in this Journal to pass from the hydrogen atomic orbitals to the Bohr orbits ( I ) . On the other hand, for the harmonic oscillator, the passage from quantum to classical behavior is generally established by looking a t great quantum numbers (2). However, there is another possibility of approach that is not used in quantum chemistry textbooks, but that ought to be most fruitful. I t deals with the properties of a population of special excited oscillators that is fully quantum mechanical in nature, but near classical in behavior (3). This special form of population may be built up in several ways. Here, we shall prefer that using the maximization of entropy subject to four constraints, by aid of the Lagrange multipliers. This will lead us to a very general state, a t a given temperature, which may reduce to that of a classical oscillator a t zero absolute temperature.
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Revlew of the Quantum Harmonlc Oscillator The hamiltonian H of an harmonic oscillator is known to be the sum of the kinetic T and potential Voperators:
The System of Harmonlc Oscillators to be Considered Now, let us look a t an ensemble of quantum harmonic oscillators a t a given temperature T, which are characterized by the same Hamiltonian 5. Next, suppose that we have, a t a eiven time t = 0, some knowledee, not only of the averwe balueof theenergyofthis populaiion ofquantum oscillaro;a, but also un the mean valne of their Q coordinate and of iw. conjugate momentum. Moreover, assume that a t time t = 0, the average value of Q is Q(0), whereas that of P is P(0). For such a situation, it is suitable to describe the system by the density operator (4): ~ ( 0=) Ib(d
(6)
where I#) is the mixed state characterizing the population of quantum oscillators in the present situation. The Four Constraints Characterlzlng the System ot Harmonic Oscillators We must observe that the present model is characterized by four constraints on the density operator which are: tr Ip(0)l = 1
(7)
Here the symbol t r {A1denotes the following operation:
T and V are respectively given by:
and V=%KQ~
with K = Mw2is the force constant, and where Q and P are, respectively, the position and its conjugate momentum, whereas M is the reduced mass of the oscillator and w its frequency. I t is customary to express the Hamiltonian 1in terms of the raising and lowering a+,a operators, which are related to the Q and P coordinates according to:
where A is any operator or product of operators and {Im))is any basis. As we may observe, the first constraint (eq 7) is the usual one ( 4 ) concernine the normalization nronertv of the densitv operator p(0). 0;the other hand, the secondconstraint (eq 8) is related to the averaee enerev value of the novulation of the quantum harmonic c&illat;;;s: it is the us;a1, i (?>1,
(a; exp [iwt] - qexp [-iwt])
Next, we may write in a general form:
(19)
with, of course, the same commutator as that given by eq 4, 1.e.:
(a; exp [iwt] + a, exp [-iwt])
a. = xo
+ iyo
m; = xo
- iyo
where xo and yo are real dimensionless parameters that, owing to expressions 16 and 17, are given by:
As a consequence of the transformations 19 and 20, eq 18 becomes: p(0) = r exp [-Ac+c]
(22)
Time Evolution ot Average Values In the Helsenberg Representation
~ ( t=)
Next, we have to find the time evolution of the average values of the physical quantities characterizing the population of the harmonic oscillators described a t time t = 0 by the density operator p(0). Thus, it may be suitable to workin the Heisenberg picture (7),where we have to take the operators at time t, weighted by the density operator at time t = 0. Then, the time evolution of the average value of the operator A(t) will be given according to: A(t) = trIp(O)A(t)l
(23)
On the other hand, because of the choice of the coordinates 19 and 20 for the expression of the density operator 22, we have to express the operator A(t) in terms of c(t) and c+(t). Time Evolutlon ot the New Normal Coordinates c(f) and
c+w First, let us look a t the time dependence of the a, a+ operators appearing in eqs 2 and 3. From the quantum dynamical equation, we have (8): -.
and a similar equation for a+(t). Then, owing to the commu468
Journal of Chemical Education
Then, eqs 31 and 32 become, by aid of the Euler relations:
(L>1, 2(xo 2Mw
cos ot
-yo sin w t ~
which give the general time evolution of the Q and P coordinates of a classicalharmonic oscillator. As we may observe, if we take yo = 0, for example, q = at,this gives the more usual time evolution of Q and P. In the following, in order to simplify the exposition, we shall focus our attention on the special situation where a. = a;. This quasi-classical behavior of eqs 33 and 34 may be underlined. As it amears. we have obtained them bv startine from quantum mechanical considerations to which we ha": added only the assum~tionof some knowledge on the 0 and P coordinates and tl;e hypothesis of maximization o> the entropy. It may be also ohserved that the Lagrange multiplier j3, which is connected to the temperature, does not app_earin eqs 33 and 34 so that the time evolution of Q(t) and P(t) is temperature independent. In other words, although the temperature has heen involved in obtaining the density operator by maximization of the entropy through the temperaturedependent j3 Lagrange multiplier, it does not appear in the classical behavior of the Q and P coordinates. As a consequence, it may be said that we can pass from quantum
behavior to classical ones through the maximization of the entropy of the quantum harmonic oscillators. The Time Evolution of Klnetlc and Potential Energies
Now, let us look at the time evolution as given by eq 23 of the kinetic and potential operators T and V ,weighted by the density operator 22 and evolving according to eq 24. The calculation given in Appendix A leads to: m exp [-Am]
+ [I + (4)+ ~'(t))']
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where we have neglected the zero-point energy because of its smallness with respect to the time-dependent terms. Moreov_er,the average Hamiltonian as given by the sum of ?(t) and T(t) is
fl = '/&Q2(o)
with n(t) = % exp [-iwt]
age values of V(t) and R t ) , as given by eqs 43 and 44 reduce to: V(t) % K&O) cos2w t (45)
(36)
Moreover, accordingto the result (A2) giving r, the first term appearing in eq 35 takes the form (9):
(47)
Equations 45 and 46 are indeed examples of the classical behavior of an harmonic oscillator which may be made in connection with the time evolution of the Q and P coordinates as given by eqs 40 and 41. Equations 40,41,45,46, and 47 for T = 0 K are examples of what is named a coherent state (lo),when the zero-point energy is neglected. The Denslty Operator as Minimizing the Uncertainty Relation
where we have usedin the last step the well-known properties of series. Here m may be recognized as the mean occupation number of the harmonic oscillator at a given temperature. As a consequence, in the special case where ao = a& eq 35 becomes:
It may be of interest to consider further the operator that allows such interesting physical situations. What may be, for instance, the uncertainty relation for states as given by eq 22? The dispersion on Q is given by: AQ = dltr lpQ(t)l12
- ltr l~8~(t)ll
(48)
whereas that o n P is given by a similar relation. Next, we may observe that [tr (pQ(t))]2is the square of eq 31. On thepther hand, because of the relation between the operators V ( t )and Q(t) as given by eq 43, we have:
In a similar way, we obtain for fit): Then, performing the calculation of AV(t) and that of A n t ) by a similar way, we find: Next, we may note that according to eqs 31 and 32, we have, respectively, for a 0 = a;: a t ) = Q(0) ms wt
(40)
where we have let in eq 40:
Conclusion
Then, using the last equation giving Q(O), we obtain for the potential energy: '/&Q2(0) = h w d
(42)
On the other hand, owing to the relation 15, and recognizing in the fl Lagrange parameter, the usual k T term, where T is the temperature and k the Boltzmann constant, we may write eqs 38 and 39 in a more pictorial form:
Tit) =
4
That is the minimum allowed by the Heisenberg principle: a result that may be made in connection with the quasi-classical property of the system.
+ 1/2KQ2(0)sin2 + 2[exp wt
hw (hwlkT)
- 11
(44)
As it appears, the above average energies are the sum of three terms. The first one is half the zero-point energy, the second one behaves, respectively, in eqs 43 and 44, as the time-dependent potential and kinetic energies of a classical harmonic oscillator, whereas the last one may be recognized according to eqs 38 and 39 to be the thermal average energy of an ensemble of quantum oscillators. Classical Behavior at Zero Absolute Temperature
Now, suppose that the parameter ao appearing in eq 42 is very great and that the temperature is zero. Then, the aver-
By maximization of the entropy of a population of quantum harmonic oscillators, subject to constraints concerning the average position, average momentum, and average energy at a given time, we have obtained a mixed state for which the Heisenberg uncertainty is minimum. Then, we have found that the time evolution of the average values describing the position, momentum, and kinetic and potential energies of this state, reduce for zero temperature to those of a classical oscillator. It would appear that this is beautiful and easy to obtain classical behavior in the framework of quantum mechanics with the aid of a concept of statistical mechanics that originates from thermodynamics. Acknowledgment
We thank P. W. Atkins for discussion on the quasi-classical properties of coherent states. Literature Cited 1. B1aise.P.; Hem-Rouaeau. 0. J. Ckm. Ed-. L986,63,31. 1. Pauiing. L.; Wilson, W. Introduction to Quantum Mechaniea: MeCraa-Hill: N s a York, 1935: p62. ofRodiotiom Wiley: NearYorh, 1-3; 3. h i s e l l , W. H. BuontvmStotisticalPlop~rfie~
" ,M
4. Pong,F.K.T h y ofMokculor Relordon; Wiley: NsaYork, 1975:pp2732
5. Atkins. P.W.Physieo1 Chemistry. Oxford Univ. Oxford, 1978: pM3. 6. Ref 4, pp 32-S8 7. Dwydw. A. S.Quonlum Mechanics; Pezgmm:Oxford, 1972:~122. 8. Ref7. p 55. 9. Bamoa. G.M.Phy.iml Chomi~fly;MeCmw-Hill: Kogekusha. 1973: p 107. lo. Carruthers, P.;Nioto. M. M. Am. J.Phw. 1965,33,537.
Volume 66 Number 6 June 1989
469
Appendix A Determination of c First, let us look at the partition function 1h appearing in expression 22, which ensures the normalization condition 7. We may write:
Appendix B In the basis {lm)),where c+c is diagonal, we have the following properties: (mln) = 6.,
(B1)
=+elm) = mlm) r tr exp [-Xc+c] = 1= e
(ml exp [-Xc+c]lrn)
(Al)
clm) = f i l m
"2
where we have performed the trace on the basis (1rn)Jwhere cfc is diagonal. Next, owing to the result BE, eq A1 leads to the usual expression far the partition function of a quantum harmonic oscillator:
(B2)
- 1)
c+lm) = ,&Film
(B3)
+ 1)
(84)
As a consequence, we have using the above relations:
(mlc+clm) = m6,
(B5)
(ml(e+e)'lm) = m'6,
(B6)
(mlc21m) = (mlcVm) = (mlclm) = (mlc+lm) = 0 (81) Next, owing to the expansion property of the exponential and to eq B6, we have Determination of 6(t) a n d p(t) In terms of the time-dependent operators c(t) and c+(t) given by eqs 9 and 30, and owing to the definitions 2 and 3 in which we must take a sad a+ as time dependent, we have:
Moreover, according to the definition 23, we have:
Moreover, b y aid of t h e completeness relation, using the basis {lm)),we have:
(B9) Next, using the distribution properties and the results BE and B12, we find eq 31. By asimilar inference, it is possible u>obtain the result 32.
a result that, according to eqs B6 and B8,gives:
.
(ml(exp[-Xcfc]J ctclm) = m exp [-Am]
(B10)
O n t h e other hand, using in a similar way the completeness relation and the result B8, we have: Determination of v(t)andT(t) We may express the mean value of V(t) in terms of c(t) and c+(t) owing to the definition of p(0) and V and to defmition 2 at a given time: = exp [-Arn]6,(nk+lm)
which will appear t o b e zero because of.,6
and t h e relation
B7. Of course, t h e result is t h e same if we take c+, c, c+c+, or cc. As a consequence, we may write: Next by aid oleqs BE, BIO, BIZ, and B13, we get the result 35. Ry a similar inference, it is possible ta obtain the result 39.
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Journal of Chemical Education
(mllexp [-Ac'c]).
cflm) = (mllexp [-Actel). elm) = 0 (B11)
(mllexp [-Ac+c]l- c+c+lm) = (mllexp [-Ac+cll. cclm) = 0 (Bl2)
