The hydrogen atom as an entangled electron–proton system Paolo Tommasini, Eddy Timmermans, and A. F. R. de Toledo Piza
Citation: American Journal of Physics 66, 881 (1998); doi: 10.1119/1.18977 View online: http://dx.doi.org/10.1119/1.18977 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/10?ver=pdfcov Published by the American Association of Physics Teachers
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[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 The hydrogen atom as an entangled electron–proton system Paolo Tommasini and Eddy Timmermans Institute for Theoretical Atomic and Molecular Physics, Harvard–Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138 A. F. R. de Toledo Piza Instituto de Fisica, Universidade de Sa˜o Paulo, C. P. 66318, Sa˜o Paulo, SP 05315-970, Brazil ͑Received 22 October 1997; accepted 6 February 1998͒ We illustrate the description of correlated subsystems by studying the simple two-body hydrogen atom. We study the entanglement of the electron and proton coordinates in the exact analytical solution. This entanglement, which we quantify in the framework of the density matrix formalism, describes correlations in the electron–proton motion. © 1998 American Association of Physics Teachers.
I. INTRODUCTION II. SUBSYSTEMS AND QUANTUM CORRELATIONS
Independent particle models often provide a good starting In very broad terms, giving the state of a physical system point to describe the physics of many-particle systems. In means providing the necessary information to evaluate all its observables quantitatively. The states of a quantum mechani- these models, the individual particles behave as independent cal system are frequently represented by vectors of unit norm particles that move in a potential field that accounts for the in a Hilbert space H, in general of infinite dimension ͑hence interaction with the other particles in an average sense. How- their usual designation as ‘‘state vectors’’͒. If the system is ever, depending on the systems and the properties that are in a state ͉⌿͘, with ͗⌿͉⌿͘ϭ1, the outcome of measurements studied, the results of the independent particle calculations of an observable quantity A has an expectation value ͗A͘ are not always sufficiently accurate. In that case it is neces- equal to sary to correct for the fact that the motion of a single particle depends on the positions of the other particles, rather than on ͗A͘ϭ͗⌿͉Aˆ ͉⌿͘, ͑1͒ some average density. Consequently, in a system of interact- ˆ ing particles, the probability of finding two particles with where we use a caret, ‘‘A,’’ to distinguish the operator rep- given positions or momenta is not simply the product of the resenting the observable from the observable itself. If the vectors w , iϭ1,2,... are a basis of the Hilbert space ,we single-particle probabilities: We say that the particles are ͉ i͘ H ‘‘correlated.’’ can use the completeness relation, 1ϭ͚m͉wm͗͘wm͉, to write the expectation value as In a more general context, the problem can be formulated as the description of interacting subsystems ͑the single par- ticle in the many-particle system being the example of the ͗A͘ϭ ͗⌿͉wm͗͘wm͉A͉wn͗͘wn͉⌿͘. ͑2͒ ͚m,n subsystem͒. A convenient theoretical framework to deal with this problem in quantum mechanics is provided by the den- In matrix notation, the mn-matrix element of the Aˆ operator sity matrix theory, which is almost as old as quantum me- ˆ ˆ in the ͉w͘ basis is equal to Amnϭ͗wm͉A͉wn͘. The remaining chanics itself.1,2 The ever-present interest in density matrix ingredients of Eq. ͑2͒ can be collected as nmϭ͗wn͉⌿͘ theory, in spite of its long history, is justified by the power of 1,4,5 ϫ͗⌿͉wm͘, known as the density matrix, which is recog- its description and the importance of its applications, which nized as the matrix representing the so-called density opera- extends to the study of the very foundations of quantum tor, ϭ͉⌿ ⌿͉, in the chosen basis. The expectation value 1–3 ͗͘ theory. In this article, we briefly review some important of Eq. ͑2͒ can then be written as aspects of density matrix theory and we illustrate its use for describing correlations of interacting subsystems by studying ˆ ͗A͘ϭ͚ Amnnm , ͑3͒ the simple, exactly solvable system of the hydrogen atom. m,n
881 Am. J. Phys. 66 ͑10͒, October 1998 © 1998 American Association of Physics Teachers 881
[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 which is the trace of the product of the A and matrices, An important limiting situation is that in which the overall ͗A͘ϭTr(Aˆ ). In this expression, the state of the system is state vector ͉⌿͘ is itself a product of a u vector and a v represented by the density matrix or, equivalently, by the vector, ͉⌿͘ϭ͉u͉͘v͘. For these product states, the density density operator . The form of this operator, together with matrix ϭ͉⌿͗͘⌿͉ becomes the normalization property of the underlying state vector ͉⌿͘, lead to some important properties, which are of course ϭ͉u͉͘v͗͘v͉͗u͉ϭuv, ͑6͒ shared by the corresponding density matrices. Namely, the density operator is a non-negative ͑meaning that ͉͉͗͘ and the reduced density u is simply given by ͉u͗͘u͉. This is у0 for any vector ͉͒͘ self-adjoint operator (ϭ†), of unit just of the form of a density operator associated with the trace ͓Tr()ϭ1͔, and idempotent ͑meaning that ϭ2, state vector ͉u͘, which can then also be used to describe the which follows from ͉⌿͗͘⌿͉⌿͗͘⌿͉ϭ͉⌿͗͘⌿͉͒. state of the subsystem. In general, the state vector ͉⌿͘ cannot The use of state vectors ͉⌿͘ to describe the state of a be factored in this fashion, however. In this case one says quantum system is, however, not general enough to cover that the two subsystems are in an ‘‘entangled’’ state. The many frequently occurring situations. As we shall discuss density matrix of an entangled state does not factor and ap- below, a suitable generalization is provided by the density pears as a series of u and v products: The subsystems are matrix language, if only we relax the idempotency require- correlated. The number of terms gives some indication of the ment. departure from the uncorrelated density matrix, but as the Often, when probing a system, only part of the total sys- number of terms depends on the choice of basis in the u and tem is subjected to a measurement. For example, in a typical v spaces, this is not a good measure of correlation or en- scattering experiment only the scattered particle is detected. tanglement. A way out of this difficulty is, however, pro- The target system is left behind in a final state that could be vided by a simple and remarkable result due to Schmidt7 ͑see different from its initial state, but this state is not observed also Refs. 2, 8, and 9͒ which essentially identifies a ‘‘natu- directly. It is then natural to divide the total system into the ral’’ basis ͑in the sense of being determined by the structure target system and the single-particle system of the scatterer. of the entangled state itself͒ in which the description of the Also, in many experiments involving mesoscopic, or semi- entanglement is achieved with maximum simplicity. macroscopic, quantum devices, it is very hard, if not impos- The Schmidt basis is a product basis, in the usual sense sible, to ensure efficient isolation from the environment, that it is written as the tensor product of two particular bases, which then plays the role of a second subsystem coupled to one for the H space and another for the H space. In order 6 u v the system of interest. Mathematically, such partitioning of to find these two bases one looks for the eigenvectors of the the system into subsystems means factoring the Hilbert space reduced density matrices. If the state is not entangled, ͉⌿͘ H into the tensor product of the corresponding subspaces, ϭ͉u͉͘v͘, u and v each have one single eigenvector of HϭHu Hv . If the Hu and Hv spaces are spanned, re- nonzero eigenvalue, ͉u͘ for u and ͉v͘ for v, and the nu- spectively, by the bases ͉ui͘, iϭ1,2,... and ͉v j͘, jϭ1,2,..., merical value of the corresponding eigenvalues is equal to 1. then H is spanned by the product states ͉ui͉͘v j͘. Accord- One then completes the basis in each subspace by including ingly, a state vector ͉⌿͘ describing the state of the system enough additional orthonormal vectors which are moreover can be expanded as orthogonal to the single ‘‘relevant’’ vector with nonzero ei- genvalue. Each one of these additional basis vectors is then an eigenvector of the corresponding reduced density with ͉⌿͘ϭ͚ di, j͉ui͉͘v j͘. ͑4͒ i, j eigenvalue zero, and the large degree of arbitrariness in choosing them reflects the large degeneracy of the corre- Suppose now that one wishes to describe the state of one sponding eigenvalue zero. If, on the other hand, the sub- of the subsystems alone, say subsystem u. The observable systems are correlated, then u and v have a spectrum of quantities of this subsystem are represented by operators nonvanishing eigenvalues. Interestingly, as we prove below, which act nontrivially on the vectors of Hu while acting on the eigenvalues of the u and v operators are the same. To the vectors of H simply as the unit operator. The expecta- u v see that, we assume that the matrix in the ͉ui͘ basis is tion value of such operators can be obtained from the re- diagonalized by the unitary transformation U ͚͑ U U* duced density matrix for the u system, which is defined as i il ij ϭ␦ ; ͚ U U*ϭ␦ ͒, the trace of the full density matrix over the v subspace. lj i li ji jl When the entire system is in the state described by ͉⌿͘, this u reads ikUklϭlUil . ͑7͒ ͚k
uϭTrv ͚ ͚ ͉ui͉͘v j͘di, jdk*,l͗uk͉͗vl͉ u u ͩ i, j k,l ͪ The -matrix elements are given by Eq. ͑5͒, ik ϭ͚ijdijd*kj , so that Eq. ͑7͒ can be written as
ϭ ͉ui͘di, jd*k, j͗uk͉. ͑5͒ ͚i,k ͚j dijd*kjUklϭlUil . ͑8͒ ͚k, j We can understand the trace as a sum ͚ j over the probabili- ties for the v subsystem to be in any possible ͉v ͘ state. For j Multiplying Eq. 8 by d* , and summing over i, we obtain example, in the scattering experiment where only the state of ͑ ͒ ilЈ the detected particle is determined, the final state of the tar- get (v) system is not measured, and one has to sum over the d* d d*U ϭ d* U . 9 ͚ ͚ ilЈ ij kj kl l͚ ilЈ il ͑ ͒ probabilities of finding the target in all possible ͉vl͘ states. i, j k i
882 Am. J. Phys., Vol. 66, No. 10, October 1998 Tommasini, Timmermans, and de Toledo Piza 882
[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 Defining V j,lϭ͚kd*kjUkl , we find with Eq. ͑9͒ that the vec- III. SINGLE-PARTICLE SYSTEM tor of components (V1l ,V2l ,...,Vjl ,...) is an eigenvector of v with eigenvalue : For the sake of illustration, we apply the density matrix l description to a single, ‘‘elementary’’ particle system. In spite of the extreme simplicity of this type of system, the v density matrix language enables us to consider states that are ͚ l jV jlϭlVllЈ , ͑10͒ j Ј not accessible within the standard state vector ͑or wave func- tion͒ description—mixed states—which are of interest for where v ϭ͚ d* d . It follows that u and v have the the discussion of correlated many-particle systems. lЈj i ilЈ ij In the coordinate representation, which corresponds to the same eigenvalues . If the nonvanishing eigenvalues are not choice of the eigenfunctions r of the coordinate operator as degenerate, then all these vectors are automatically orthogo- ͉ ͘ basis, the density matrix of a pure single-particle system nal. And if, moreover, the set of eigenvectors with nonvan- represented by the state vector , takes on the form ishing eigenvalue is still not complete, one still may com- ͉⌿͘ plete the bases with additional orthogonal vectors which are again eigenvectors of the respective reduced densities with ͑r,rЈ͒ϭ͗r͉⌿͗͘⌿͉rЈ͘ϭ⌿͑r͒⌿*͑rЈ͒, ͑12͒ eigenvalue zero. Expanding ͉⌿͘ in the product basis con- structed in this way, which thus includes the eigenvectors where r ⌿ ϭ⌿(r) is the wave function. In the general u v ͗ ͉ ͘ ͉u͘ of and ͉v͘ of ,͉⌿͘ϭ͚c͉u͉͘v͘, the reduced case, the single-particle density matrix is an object (r,rЈ) density matrices take on the form with the properties of hermiticity, (r,rЈ)ϭ*(rЈ,r), unit trace, ͐d3r (r,r)ϭ1, and non-negativity, ͐d3r͐d3rЈ u 2 ϫ*(r)(r,rЈ)(rЈ)у0. Note that idempotency is not be- ϭ͚ ͉c͉ ͉u͗͘u͉, ͑11͒ ing required in order to allow for mixed states. We shall be concerned with a particular class of states which is diagonal and shows that the eigenvalues are equal which we shall refer to as homogeneous states. By definition, 2 to ϭ͉c͉ . This also shows that the eigenvectors with zero these states are translationally invariant, meaning that trans- eigenvalue do not participate in the expansion. The eigenval- lation by an arbitrary position vector R leaves the density ues are thus both the probabilities of finding subsystem u matrix, and hence the state itself, unaltered: (rϩR,rЈϩR) ϭ (r,r ). As a consequence, all the density matrix elements in the states ͉u͘ and the probabilities of finding subsystem v Ј 2 depend only on relative positions, specifically, (r,r ) in the states ͉v͘. The set of numbers ͉c͉ can therefore be Ј interpreted as a distribution of occupation probabilities. In ϭ(rϪrЈ). In this case, the Fourier transform ˜(k)of(r fact, the unit trace condition on the complete density opera- ϪrЈ), tor ensures that c 2ϭ1. For uncorrelated subsystems, ͚͉ ͉ 1 there is just one nonvanishing probability and this distribu- 3 ͑rϪrЈ͒ϭ 3 d k ˜͑k͒exp͑ik–͓rϪrЈ͔͒, ͑13͒ tion has a vanishing ‘‘width’’ or standard deviation. It is then ͑2͒ ͵ reasonable to use the standard deviation as a measure of the 10 plays a dual role in the description of the homogeneous state: correlation. Note that when the two subsystems are corre- ͑1͒ it is proportional to the momentum distribution of the lated, i.e., there is more than just one nonvanishing occupa- system, and ͑2͒ it gives the eigenvalues of the density matrix. 2 tion probability ͉c͉ , the reduced density matrix Eq. ͑11͒ is We shall now prove both assertions. no longer idempotent, although all the other properties listed Let us first evaluate the expectation value ͗p͘ of the mo- above for still hold. As a result of this, the state of an mentum. If the system is pure and described by the quantum entangled subsystem cannot be described in terms of a state state ͉⌿͘, then vector in the corresponding Hilbert space, and one is forced to use the density matrix language. In this case one says that ͗p͘ϭ d3r ⌿*͑r͓͒Ϫiបٌ͔⌿͑r͒ the subsystem is in a ‘‘mixed’’ state. ‘‘Pure’’ states, associ- ͵ ated with definite state vectors, are, on the other hand, al- ways associated with idempotent density operators. 3 3 ϭϪiប d rdrЈ␦͑rϪrЈٌ͒r͑r,rЈ͒. ͑14͒ In summary, the foregoing discussion shows that a sub- ͵ system of a larger quantum system is, in general, in a mixed The trace prescription for calculating ͗p͘, obtained above, is state, even if the state of the system as a whole is pure. also valid for mixed states and inserting Eq. ͑13͒, we obtain Furthermore, the Schmidt analysis shows that in this case the state of the complementary subsystem has exactly the same p͘ϭϪiប d3r d3rЈ ␦͑rϪrЈٌ͒ ͑r,rЈ͒͗ degree of impurity, in the sense that both reduced density ͵ ͵ r matrices have the same spectrum. This corresponds to the 1 impurity of both subsystems being due to their mutual en- ϭ d3r d3rЈ d3k͓បk͔˜͑k͒ tanglement in the given overall pure state. To the extent that ͑2͒3 ͵ ͵ ͵ a given quantum system is not really isolated, but interacts ϫexp͑ik ͓rϪr ͔͒␦͑rϪr ͒, with other systems ͑possibly a nondescript ‘‘environment’’͒, – Ј Ј this analysis also shows that assuming the purity of its state ⍀ is inadequately restrictive. In the following section we exam- ϭ d3k͓បk͔˜͑k͒, ͑15͒ ͑2͒3 ͵ ine the spatially homogeneous, but possibly not pure, states of a single quantum particle, independently of any dynamical where we have used a large quantization volume ⍀ to avoid processes that may in fact give real existence to such states. problems related to states of infinite norm. This normaliza- An example of such a dynamical process will be given next. tion scheme, commonly called box normalization, assumes
883 Am. J. Phys., Vol. 66, No. 10, October 1998 Tommasini, Timmermans, and de Toledo Piza 883
[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 that the particle is confined to a large volume ⍀, which is with their inherent observable properties, without regard to taken to be infinite at the end of the calculation, ⍀ ϱ.A any dynamical processes that might have led to their prepa- → ration. In the following section we consider a simple but plane wave state is then normalized as exp(ik–r)/ͱ⍀. Sub- stituting pϭបk in the integration of Eq. ͑15͒, we find that definite composite dynamical system and use the tools devel- oped here in order to study the entanglement effects gener- ated by definite interaction processes. ͗p͘ϭ ͵ d3p pf ͑p͒, ͑16͒ where f (p)ϭ ⍀/(2ប)3 ˜(p/ប). Similarly, the expectation ͓ ͔ IV. HYDROGEN ATOM value of higher-order powers of momentum components are equal to the corresponding higher-order moments of the f (p) We now apply the density matrix formalism to stationary 2 3 2 function, e.g., ͗(pˆ–xˆ) ͘ϭ͐d p(p–xˆ) f (p). Thus, f (p) may states of a system that can display correlation: the two-body be interpreted as a momentum distribution function. hydrogen atom. The hydrogen atom consists of an electron To see that ˜(k) gives the eigenvalues of the density ma- and a proton, interacting by means of the Coulomb potential. trix, we use Eq. ͑13͒ to evaluate The Hamiltonian of the electron–proton system is ˆ 2 ˆ 2 2 exp͑ik rЈ͒ pe pp e 3 – Hˆ ϭ ϩ Ϫ , ͑19͒ d rЈ ͑rϪrЈ͒ 2m 2m r Ϫr ͵ ͱ⍀ e p ͉ e p͉ where pˆ and r denote the momentum and position operators 1 exp ik r 3 ͑ Ј– ͒ 3 in coordinate space, and the subscripts e and p indicate the ϭ 3 d kЈ ˜͑kЈ͒ d rЈ ͑2͒ ͵ ͱ⍀ ͵ electron and the proton. In fact, the hydrogen atom can be treated as two uncorre- exp͑ik–r͒ lated subsystems. Transforming to relative, rϭrpϪre , and ϫexp͑i͓kϪkЈ͔–rЈ͒ϭ˜͑k͒ , ͑17͒ center-of-mass coordinates, Rϭ(m r ϩm r )/M, where M ͱ⍀ e e p p is the total mass, Mϭmeϩm p , the Hamiltonian separates which we recognize as the eigenvalue equation of the density into a center-of-mass term which takes on the form of a matrix Eq. 7 in coordinate representation. From Eq. 17 , ˆ 2 ͓ ͑ ͔͒ ͑ ͒ free-particle Hamiltonian, Hcmϭpˆ R/2M, and an ‘‘internal’’ it follows that the eigenvectors of the density matrix are the Hamiltonian which governs the relative motion of the elec- plane wave states of momentum k and that the corresponding tron and the proton, Hˆ ϭpˆ 2/2m Ϫe2/r, where m is the ˜ int r r r eigenvalues are equal to (k). Ϫ1 Ϫ1 Ϫ1 ˆ The density matrix of a homogeneous single-particle sys- reduced mass, mr ϭme ϩm p . The product of Hcm and ˆ tem does not generally describe a pure system. To see that, Hint eigenstates, we remind the reader that a pure system is characterized by 2 P an idempotent density matrix, ϭ. For the homogeneous ⌿͑R,r͒ϭint͑r͒exp i –R ͱ⍀, ͑20͒ single-particle system, this implies that the eigenvalues of ͩ ប ͪ Ͳ the density matrix satisfy ˜(k)2ϭ˜(k). Consequently, for a is an eigenstate. According to the definitions introduced in pure homogeneous single-particle system, we find that ˜(k) Sec. II, the internal and center-of-mass subsystems for the ϭ0or˜(k)ϭ1. Furthermore, the unit trace condition on the state ͑20͒ are not correlated. If the hydrogen atom is in its density matrix implies atomic ground state, the internal wave function, int ,is equal to 3 1ϭ ͵ d r ͑0͒ 1 1 3/2 int͑r͒ϭ exp͑Ϫr/a0͒ ͱ ͩ a0 ͪ ⍀ 3 ϭ͑0͒⍀ϭ 3 d k ˜͑k͒ ˜͑k͒, ͑18͒ ͑2͒ ͵ →͚k ͑hydrogen in 1s state͒, ͑21͒ 2 2 where in the last step we reconverted the momentum integral where a0 is the Bohr radius, a0ϭប /mre . to the sum of discrete momenta appropriate to the adopted Partitioning the hydrogen atom differently into electron volume quantization. Thus, the homogeneous single-particle and proton subsystems gives an entangled state: system that is pure can only be described by a density matrix P ͓mereϩm prp͔ with a single eigenvalue, say ˜(k ), equal to 1, and all other ⌿͑r ,r ͒ϭ ͑r Ϫr ͒exp i ͱ⍀. 1 e p int e p ប • M eigenvalues ˜(k)ϭ0, kÞk . Such a density matrix repre- ͩ ͪ Ͳ 1 ͑22͒ sents a particle with a definite momentum ͑zero standard deviation of the distribution of density matrix eigenvalues͒, The reduced density matrix for the electron system can be corresponding to a plane wave wave function, ͗r͉⌿͘ obtained by taking the trace over the proton basis set of coordinate eigenfunctions: ϭexp(ik1–r)/ͱ⍀. In contrast, an impure single-particle sys- tem has a spectrum ˜(k) of density matrix eigenvalues. The r ,rЈ ϭ d3r r ,r * rЈ ,r standard deviation of the ˜(k) distribution is a measure of ͑ e e ͒ ͵ p ⌿͑ e p͒⌿ ͑ e p͒ the correlation or entanglement of the single-particle 11 system. meP ϭexp i ͓r ϪrЈ͔ d3r ͑r Ϫr ͒ It should be remarked, finally, that all the above discussion ͩ បM • e e ͪ ͵ p int e p had a purely kinematical character, in the sense that we dealt with possible states of a quantum mechanical system and ϫint* ͑reЈϪrp͒/⍀
884 Am. J. Phys., Vol. 66, No. 10, October 1998 Tommasini, Timmermans, and de Toledo Piza 884
[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 meP tum distribution f int(p) for atoms at rest, by means of a ϭexp i ͓r ϪrЈ͔ ͑r ,rЈ͒, ͑23͒ ͩ បM • e e ͪ int e e simple displacement in momentum space: m where we introduce the electron density matrix for elec- e int f ͑p͒ϭ f int pϪ P , ͑29͒ trons in atoms at rest (Pϭ0). It is interesting to note that the ͩ M ͪ electron subsystem of the hydrogen atom is homogeneous. where To prove that, it is sufficient to show that int depends on re 3 1 64a0 and rЈ as reϪrЈ . Substituting rЈϪrpϭy and writing the e e e f int͑p͒ϭ 3 . ͑30͒ argument of , r Ϫr ,asrϪrЈϩy, we do indeed find ͑2ប͒ a0p int e p e e 1ϩ that ͩ ប ͪ Equation ͑29͒ expresses a Gallilean transformation from the d3r rϪr * rЈϪr ͵ p int͑ p͒int͑ e p͒ center-of-mass reference frame (Pϭ0) to the lab frame of reference in which the atom is moving with velocity P/M.If the electron in the center-of-mass frame has velocity v ϭ d3y r ϪrЈϩy * y , 24 ͵ int͑ e e ͒int͑ ͒ ͑ ͒ ϭp/me , then, in the lab frame, its velocity is observed as vЈϭp/m ϪP/M, corresponding to a momentum equal to which depends solely on r ϪrЈ . Consequently, the reduced e e e m vЈϭpϪ(m /M)P. electron density matrix ͑23͒ only depends on the relative e e position, indicating a homogeneous subsystem. This might Since f and f int are equal up to a translation in momentum appear surprising: We do not usually think of the electron in space, their widths, or standard deviations, are equal. In the hydrogen as a homogeneous system. However, the reduced center-of-mass frame, ͗p͘ϭ0, so that the standard deviation 2 2 density matrix describes the observation of hydrogenic elec- ⌬pϭͱ͗p ͘Ϫ(͗p͘) is equal to trons in the assumption that the proton is ‘‘invisible.’’ Fur- ប thermore Eq. ͑22͒ describes a hydrogen of fixed momentum 3 2 ⌬pϭͱ d ppfint͑p͒ϭ . ͑31͒ ͑e.g., a beam of hydrogen atoms of well-defined velocity͒ so ͵ a0 that the center-of-mass position of the atom is undetermined. Thus, although the electron subsystem is homogeneous, as a It is then equally likely to observe the electron in any posi- consequence of the electron–proton correlation, a measure- tion, and the electron subsystem is homogeneous. ment of the electron momentum can yield a finite range of Consequently, as discussed in the previous section, the values. In accordance with the Heisenberg uncertainty prin- Fourier transform of the density matrix plays a central role. ciple ͑the correlations confine the electron in the center-of- The density-matrix ofa1selectron in a hydrogen atom int mass frame to a region of size ϳa0͒ the size of this region in at rest ͓Pϭ0 in Eq. ͑23͔͒, has a simple analytical Fourier momentum space, or more precisely, the standard deviation transform, ⌬p of the momentum distribution, equals ប/a0 . In the con- 3 text of describing correlating systems, ⌬p is a measure of the 1 64a0 ˜ ͑k͒ϭ . ͑25͒ strength of the correlation: The stronger the electron and pro- int ⍀ 1ϩ͑a k͒2 0 ton are correlated, the higher the value of a0 and the larger The Fourier transform ˜ for an electron in an atom of arbi- the region in momentum space (⌬p) over which the electron momentum is ‘‘spread out.’’ trary momentum P is a translation in k space of ˜int .To show that, we insert the inverse Fourier transform of ˜ in the expression for the density matrix ͑23͒, V. CONCLUSION 1 In this paper, we have reviewed aspects of density matrix 3 ͑re ,reЈ͒ϭ 3 d kЈ ˜int͑kЈ͒ theory which pertain to the description of the entanglement ͑2͒ ͵ of correlated subsystems. The mutual entanglement of sub- systems of a larger ͑pure͒ system is conveniently quantified mep ϫexp i ϩkЈ –͓reϪreЈ͔ . ͑26͒ by the standard deviation of the reduced density matrix ei- ͩ ͫ Mប ͬ ͪ genvalues of the entangled subsystems. For the simple ͑but relevant͒ case of homogeneous single-particle subsystems, The substitution, kϭkЈϩmeP/Mប, then leads to the Fourier transform: the density matrix eigenvalues give the particle’s momentum distribution. Thus the standard deviation of the momentum 1 meP distribution is a measure of the entanglement of the single- ͑r ,rЈ͒ϭ d3k ˜ kϪ particle system, as we have illustrated for the specific ex- e e ͑2͒3 ͵ int Mប ͩ ͪ ample of the electron in the two-body hydrogen atom. ϫexp͑ik–͓reϪreЈ͔͒, ͑27͒ ACKNOWLEDGMENT from which it follows that One of the authors, ET, acknowledges support by the NSF meP through a grant for the Institute for Atomic and Molecular ˜͑k͒ϭ˜ kϪ . ͑28͒ intͩ Mប ͪ Physics at Harvard University and Smithsonian Astrophysi- cal Observatory. Thus, compared to ˜int , ˜ is displaced in k space by meP/Mប. Similarly, the momentum distribution f (p) 1J. von Neumann, Mathematical Foundations of Quantum Mechanics 3 ϭ͓⍀/(2ប) ͔˜int(p/ប), is related to the electron momen- ͑Princeton U.P., Princeton, 1955͒.
885 Am. J. Phys., Vol. 66, No. 10, October 1998 Tommasini, Timmermans, and de Toledo Piza 885
[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 2E. Schro¨dinger, ‘‘Probability Relations Between Separate Systems,’’ Proc. 8203 ͑1992͔͒. This follows a procedure apparently first used by von Neu- Cambridge Philos. Soc., 31 555–563 ͑1935͒; ‘‘Probability Relations Be- mann ͑see Ref. 1͒, which was almost simultaneously adopted by S. tween Separate Systems,’’ 32, 446–452 ͑1936͒. Watanabe, ‘‘Application of the Thermodynamical Notions to Normal State 3 Roland Omne`s, The Interpretation of Quantum Mechanics ͑Princeton of the Nucleus,’’ Z. Phys. 113, 482–513 ͑1939͒ to study the entanglement U.P., Princeton, NJ, 1994͒. of individual nucleons in the actual stationary states of an atomic nucleus. 4 L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Non-relativistic This type of use of an entropy function was later spread by the work of Theory ͑Addision–Wesley, Reading, MA, 1965͒, 2nd ed. The density op- Shannon ͓C. E. Shannon and W. Weaver, The Mathematical Theory of erator was called ‘‘statistical operator’’ by von Neumann in Ref. 1. Communication ͑The University of Illinois Press, Champaign, IL, 1949͔͒. 5U. Fano, ‘‘Description of States in Quantum Mechanics by Density Matrix In more recent times use has also been made of the so-called ‘‘linear and Operator Techniques,’’ Revs. Mod. Phys. 29, 74–93 ͑1957͒. 2 2 6L. Davidovich, M. Brune, J. M. Raymond, and S. Haroche, ‘‘Mesoscopic entropy’’ ϭTr(Ϫ )ϭ1Ϫ͚ . This can be found, e.g., in P. C. Li- Quantum coherences in cavity QED: Preparation and Decoherence Moni- chtner and J. J. Griffin, ‘‘Evolution of a Quantum System Lifetime of a toring Schemes,’’ Phys. Rev A 53, 1295–1309 ͑1996͒; M. Brune, E. Ha- Determinant,’’ Phys. Rev. Lett. 37, 1521–1524 ͑1976͒; W. H. Zurek, S. gley, J. Dreyer, X. Maıˆtre, A. Maali, C. E. Wunderlich, J. M. Raymond, Habib, and J. P. Paz, ‘‘Coherent States via Decoherence,’’ ibid. 70, 1187– and S. Haroche, ‘‘Observing the Progressive Decoherence of the ‘Meter’ 1190 ͑1993͒; J. I. Kim, M. C. Nemes, A. F. R. de Toledo Piza, and H. E. in a Quantum Measurement,’’ Phys. Rev. Lett. 77, 4887–4890 ͑1996͒. Borges, ‘‘Perturbative Expansions for Coherence Loss,’’ ibid. 77, 207– 7 E. Schmidt, ‘‘Zur Theorie Derlinearen und Nichtlinearen Integralgleichun- 210 ͑1996͒. The standard deviation of the eigenvalues of can be written gen,’’ Math. Annalen 63, 433–476 ͑1906͒. as ͚ 2Ϫ1/N2, where N is the number of nonvanishing eigenvalues , 8 M. C. Nemes and A. F. R. de Toledo Piza, ‘‘Effective Dynamics of Quan- and is thus clearly related to the linear entropy. The essential ingredient in tum Subsystems,’’ Physica A 137, 367–388 1986 . ͑ ͒ all these measures is a nonlinear dependence on that makes it sensitive to 9A. Ekert and P. L. Knight, ‘‘Entangled Quantum Systems and the Schmidt the lack of idempotency of the density operator. Decomposition,’’ Am. J. Phys. 63, 415–423 ͑1995͒. 11In the remainder of the paper, we choose to determine the standard devia- 10The degree of impurity of a state described by a density operator is ˜ usually measured in terms of an entropy function, SϭϪTr( log ) tion of the density matrix eigenvalues (k) in a manner that is different 2 2 from the standard deviation, ͚ Ϫ1/N , in Ref. 10. In view of the close ϭϪ͚( log ), ͕͖ being the eigenvalues of ͓see, e.g., V. Buzˇek, H. Moya Cessa, P. L. Knight, and S. J. D. Phoenix, ‘‘Schro¨dinger-Cat States relationship between ˜(k) and the momentum distribution f (p), we in the Resonant Jaymes–Cummings Model: Collapse and Revival of Os- choose to calculate the standard deviation of f (p) as the width of the cillations of the Photon-Number Distribution,’’ Phys. Rev. A 45, 8190– momentum distribution in p space.
886 Am. J. Phys., Vol. 66, No. 10, October 1998 Tommasini, Timmermans, and de Toledo Piza 886
[This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46