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Does QuantumMechanics Apply to One or Many Particles? F. Castaiio, L. ~ a i n M. , N. Sanchez Rayo, and A. Torre1 Dpto. Quimica Fisica (Ciencias), Universidad dei Pais Vasco, Apdo. 644, Biibao. Spain In dealing with quantum mechanics and its applications to soectrosco~v chemistrv textbooks ( 1 ). as well ."some ~hvsical . " as more specialized works (2), pay some attention to concepts such as "transition between states" and "transition prohability." A spectroscopic transition is described as the effect produced by a perturbation-usually the electromagnetic radiation-to a single atom or molecule that was initially in a pure state n before passing to a pure state rn. Time-dependent perturbation theory is the tool used to describe such transitions. When spontaneous and induced Einstein's coefficients A and B are introduced, textbooks speak of transitions of atoms or molecules in a system that has N independent particles-a macroscopic system-in which the interaction between particles is neglected. There are, therefore, two situations, transition in microscopic and macroscopic systems. Both cases are generally included in the same chapter without illuminating the difference (1-2). The purpose of this paper is to analyze the usual formulation of time-dependent perturbation theory to describe transitions and to find the system to which they refer. As it is shown below, some inconsistencies appear when the system is an isolated particle. We will conclude by writing down the consequences derived from the use of time-dependent and time-independent Schrodinger equations in connection with the two mentioned cases. Theory Let H" he the hamiltonian of a time-independent or unperturbed system and V(t) the perturbation induced by the electromagnetic radiation. The time-dependent Schrodinger equation of the system is
*
(H"
a* + V(t))T = ih y "*
*
a linear combination of unperturhed states the meaning of which is puzzling for most students. We shall try to shed some light on it. functions are a set of orthogonal functions Unperturbed and they have the form (2)
*a
*;
= @;e-~E;i/h
L
Integration of eqn. (4) from the time at which the perturbation is applied, t = 0 to a time t gives
Unknown coefficients are on both sides of this equation, therefore, we must make approximations to handle them. It is assumed that before the perturbation starts, t = 0, the system is in a pure state n, and therefore For small perturbations and short times the perturbed and unperturhed systems will not be very different, so the following approximations are made in eqn. (5) (1-2)
The coefficient a,(t) is evaluated as
from which is calculated the "transition prohability" from state n to rn, P,-, as ( I )
Discussion If eqn. (2) refers to a single, isolated particle and this is in a pure state n at t = 0, we can write
At time t, when the transition from state n torn happens, the wavefunction must he (1)
(3)
where (Pk are time-independent functions and EP is the corresponding energy of the unperturhed state k. To obtain the time-dependent coefficients (ak/the expansion (2) is substituted into the Schrodinger equation (1). One gets V,y o k ~ , e - ~ E ; t / h= c hE daL (p'ae -1E;LIh
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and
(1)
where is the time-dependent wavefnnction of the perturbed system. Usually J!' is expanded onto the unperturhed wavein (1-4) functions = ak(t)Y; (2)
{*;/
where
This equation contradicts the assumptions made in eqn. (a), and it is not possible to describe aspectroscopic transition for a microscopic system in a simple way as
a-t
Multiplying this equation by the conjugate wavefunction VK' and integrating over all space gives (2)
q-T;
The original papers that deal with transitions write down the wavefunction in eqn. (2) referring to a macroscopic system. Therefore, each wavefunction is a stationary state of an atom or molecule given by eqn. (3) and lak ( t )1 is the number of particles in this state at time t (5).Because the coefficients ak must be normalized to unity, the wavefunction in eqn. (2) must he applied to an assembly of N similar independent
*a
'
Author to whom correspondence should be addressed. Present address: Department of Chemistry. Room 625, University of Edinburgh, West Mains Road, Edinburgh EH9 3JJ, U.K.
Volume GO
Number 5
May 1983
377
particles. If each of these ah's is multipled by N1I2 it is written (6)
Time-independent systems are usually described by the eigenstates equation (7-8) Horn', = EiQ;
If the wavefunction 'P of eqn. (2) refers to a macroscopic system, and Iak(t)I2 is the fraction of particles in state k , approximations (7) and (8) have a clear physical meaning. Only a small number of particles change in state, most of them remain unchanged so that the population of the initial state n is near unity. This concept of the wavefunction justifies the name "probability" given by eqn. (9) as the ratio of favorable cases (particles that make transitions) and total cases (all particles). We feel the lack in physical chemistry textbooks lies in relating transition probability to the mathematical probability. From a physical point of view the perturbation V ( t )induces transitions in some undetermined warticles thus varying . .the fraction of particles in the corresponding states. In the particular case when the oerturbation is nil-a time independent Hamiltonian-each fraction of particles in the system must he a constant in time. and the wavefunction in relation (2) must now be formulated as ~
~
x =Xca*i
(10)
k
where c k are time-independent constants. We can observe that the wavefunction, X,in eqn. (10) fulfills the time-dependent Schrodinger equation as the pure states ' P a do independently a*; Ha*; = ~ h (11) at
The function x is mathematically a more general solution than the 'Pi's for eqn. (ll), and i t represents a macroscopic system in which all the particles are not necessarily in the same state.
378
Journal of chemical Education
(12)
which .i (leriwd irom rhr time-dependent Srhriidin~ereqn. , I 1) 18).In this , m t , it is not posaihle I,, makc :I linear coml~inatiun or all the func~iuns4'; 11131 i111rill eqn. ,121 !except in the case of (lwmerate eirrniunrtimi~.Functions 9,and eqn. or an assembly of N indepen(12) refer to ;unique dent warticles, all of them in the same state. However, they can neve; represent an assembly of N independent particles in several states. We conclude that the time-dependent Schrodinger equation can describe any system: miroscopic or macroscopic, with nerturhation or without it. with all oarticles in the same state or in different states. The eigenstates equation refers to the narticnlar case of time-indenendent Hamiltonian. and so i t repre-ents mly pure alate. Lrr t his kmd o: sratr and hwnuse uf ncmnali~ationwnditions it is 1131 o ~ ~ s i h1,) l edist:nyui4h between macroscopic or microscopic systems. Obviously, the time-dependent Schrodinger equation is the most powerful tool. L
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Acknowledgment We would like to thank Professor R. J. Donovan and Mr. A. Hopkirk (Edinburgh University) for checking this manuscript. Literature Cited Moore, w.,"Physical Chemistry: 5th Ed., Longman, London, 1972, p. 753. (2) Atkins, P. W., ''Molecular Quantum Mechanics? Clsrendon Press, Oxford, 1970, p. (1)
3,
n
(3) Lel"ne,I. N., "Quantum Chemistry? 2nd Ed., Ally" andBscon, Baston. 1974, p. 141. (4) Fried, V., Blukis. U..and Hameka,H. F., "Physical Chemk1ry.l' Maemilisn, New York 1974. p 399. (5) Dirac. P. A. M.,Pmc.Roy. Soc., (Londan) 112,661 (1926). (6) DBac,P.A.M.,Ploc. Roy. Soc., (London) 114.243 (1927). (7) Atkins.P. W.."PhmiealChemistry?OxfordU.P., Oxford. 1978,p.39(1. (8) Levine, I. N.."Physical Chemistw." McCraw-Hill, New Yark, 1978, p. 518.
