Entangled quantum particles in an infinite square well: knowledge of the whole versus knowledge of the parts
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2010 Eur. J. Phys. 31 193
(http://iopscience.iop.org/0143-0807/31/1/017)
Download details:
IP Address: 150.214.103.186
The article was downloaded on 10/12/2009 at 14:41
Please note that terms and conditions apply.
The Table of Contents and more related content is available
Eur. J. Phys. 31 (2010) 193–203 doi:10.1088/0143-0807/31/1/017
Entangled quantum particles in an infinite square well: knowledge of the whole versus knowledge of the parts
R Q Odendaal
1and A R Plastino
2,3,41Physics Department, University of Pretoria, Pretoria 0002, South Africa
2National University La Plata, UNLP-CREG-Conicet, CC 727, La Plata 1900, Argentina
3Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, 18071-Granada, Spain
E-mail:arplastino@maple.up.ac.za
Received 8 September 2009, in final form 22 October 2009 Published 25 November 2009
Online atstacks.iop.org/EJP/31/193 Abstract
Entangled states of composite quantum systems exhibit one of the most distinct and non-classical features of the quantum mechanical description of Nature, first pointed out by Schroedinger: ‘Maximal knowledge of a total system does not necessarily imply maximal knowledge of all its parts’. We provide an elementary illustration of this fundamental aspect of quantum physics by considering a system of two particles in an infinite, one-dimensional square potential well. In contrast to standard introductory presentations of quantum entanglement, our present considerations do not require density matrix formalism, nor explicit use of the tensor product structure for the description of composite quantum systems.
1. Introduction
Nowadays there is wide consensus that the phenomenon of quantum entanglement [1–11]
is one of (if not the) most fundamental features of quantum physics [12,13]. The study of quantum entanglement has led to the discovery of novel quantum information processes such as quantum teleportation and superdense coding that may have important practical applications [13]. The technological impact of quantum entanglement, however, is not limited to the information technologies, but is also at the basis of other important developments, such as quantum metrology [14]. Besides its revolutionary technological implications, quantum entanglement constitutes an essential ingredient in our present understanding of various basic aspects of quantum physics, such as, the foundations of quantum statistical mechanics [15] or the phenomenon of decoherence [16] and its role in the emergence of a classical world picture
4 Author to whom any correspondence should be addressed.
0143-0807/10/010193+11$30.00 2010 IOP Publishing Ltd Printed in the UKc 193
from a quantum mechanical background [17]. The relationship between entanglement and the time evolution of composite quantum systems is another interesting example [3,18].
From both fundamental and practical points of view, it is important to incorporate the concept of entanglement into the teaching of quantum mechanics [19,20]. It must be realized that quantum entanglement is not just a fashionable research subject. Entanglement is a fundamental concept of quantum physics that plays a deep role within all applications of quantum mechanics involving composite systems (e.g. problems involving more than one particle).
The maximum knowledge that we can in principle have about the state of a quantum system is represented by a wavefunction (or, in more abstract terms by a vector in an appropriate Hilbert space). A wavefunction codifies all that we can in principle know about the physical state of the system. Given a wavefunction, the strictures of quantum theory provide a concrete prescription for computing the probabilities of getting different outcomes when measuring any physical observable. One of the most basic and counter-intuitive features of quantum physics is that the most complete possible description of the physical state of a composite quantum system (given, as said, by a wavefunction) is not always accompanied by a description as complete as possible of the states of each of its constituent parts. In Schroedinger’s words [21] ‘Maximal knowledge of a total system does not necessarily imply maximal knowledge of all its parts’. Quantum states of composite systems behaving in this non-classical way are called entangled states.
The aim of the present contribution is to illustrate the aforementioned basic aspect of quantum physics, associated with entangled states, by recourse to one of the most elementary examples of a composite quantum system: two non-interacting quantum particles of zero spin in a one-dimensional quantum well. We shall illustrate, from the physical rather than the mathematical point of view, the fact that having a maximally specified global state of the two particles does not imply that such a description is possible for the individual particles. We are going to prove in a particular example that, in spite of having a global wavefunction for the two particles, no single-particle wavefunction can properly account for the statistics of measurements of both the position and the energy of an individual particle. Our discussion will involve a minimum amount of formalism, at the level of introductory books on quantum mechanics such as the texts by Gillespie [22] or by French [23], or even introductions to modern physics such as [24]. In particular, we shall not employ the density matrix formalism, nor explicitly use the tensor-product structure associated with composite quantum systems.
Our present considerations allow us to introduce some basic ideas on quantum entanglement (and to relate them to more standard aspects of quantum physics) within the context of a problem that can be treated exactly using tools that are normally available to students taking introductory university courses in quantum physics. Our treatment of entangled states of two particles in an infinite potential well provides a strategy for the teaching of a fundamental feature of quantum mechanics that is not usually considered at the elementary level, without the need to introduce mathematical tools or formal developments that are not already covered in the aforementioned basic courses. The present discussion may also be useful, at a more advanced stage, as a starting motivation for introducing the concept of mixed quantum states and the density matrix formalism.
2. Two particles in a one-dimensional infinite square well
The infinite square well potential provides one of the most useful systems to illustrate in an elementary way many important aspects of quantum physics [25]. Here we shall consider a system of two particles of zero spin in an infinite square well as a tool for introducing
Figure 1.Two particles of mass m1and m2in a one-dimensional infinite square well.
some fundamental notions concerning the phenomenon of quantum entanglement. We shall assume that the student is familiar with the basics of quantum wave mechanics necessary for discussing a system consisting of two distinguishable, non-interacting particles of zero spin, with masses m1and m2and coordinates x1and x2, moving in a one-dimensional infinite square well of length L with walls located at x= ±L/2, as illustrated in figure1. These preliminaries are briefly summarized here.
If we only have one particle of mass m in the infinite square well, the maximal possible knowledge about its state is provided by a normalized wavefunction (x) satisfying appropriate boundary conditions,
L/2
−L/2|(x)|2dx= 1,
(−L/2) = (L/2) = 0.
(1)
The probability density of finding the particle within the small interval [x, x + dx] is given by
P (x) dx= |(x)|2dx. (2)
The linear momentum p of the particle and its kinetic energy K = p2/2m correspond, respectively, to the operators
p−→ ¯h i
d dx K−→ −¯h2
2m d2 dx2.
(3)
Any wavefunction complying with (1) can be expanded as a linear combination of the eigenfunctions of the kinetic energy operator, which is actually the Hamiltonian of one-particle in the infinite square well,
(x)=
∞ n=1
cnn(x), (4)
where the coefficients ciare normalized,
∞ n=1
|cn|2= 1. (5)
The eigenfunctions n(x) are orthonormal,
L/2
−L/2
n(x) ∗n(x) dx= δnn, (6)
Φn(x)
x Φ1(x)
Φ2(x) Φ3(x)
Figure 2. The first three eigenfunctions of a quantum particle in a one-dimensional and infinite square well potential.
and comply with the eigenvalue equation
−¯h2 2m
d2
dx2 n(x)= nn(x), n= 1, 2, . . . , (7) where
n(x)= sin
nπ L
x +L
2
, n= ¯h2π2n2
2mL2 , n= 1, 2, . . . . (8) The eigenfunctions n(x) are even functions of x for even values of the quantum number n, and odd functions for odd values of n (see figure2).
According to the standard postulates of quantum mechanics, the possible results obtained when measuring a physical observable are given by the eigenvalues of the associated operator.
The probabilities of getting each of these results are given by the square modulus of the corresponding coefficients of the expansion of the wavefunction as a linear combination of the observable’s eigenfunctions. In particular, the probability of obtaining the value iwhen measuring the kinetic energy of a particle described by the wavefunction (4) is equal to|ci|2. If we now have a system consisting of two particles in the one-dimensional infinite square well, the maximal possible knowledge of the two-particle system is given by a normalized wavefunction (x1, x2) verifying appropriate boundary conditions,
L/2
−L/2
L/2
−L/2|(x1, x2)|2dx1dx2= 1,
(x1,±L/2) = (±L/2, x2)= 0.
(9)
The joint probability of finding the two particles within the coordinate intervals x1, x1+ dx1
and x2, x2+ dx2is
P (x1, x2) dx1dx2= |(x1, x2|2dx1dx2. (10)
The linear momentum pjand kinetic energy Kjof the j -particle (j = 1, 2) correspond to the operators,
pj −→ ¯h i
∂
∂xj
, j = 1, 2,
Kj −→ − ¯h2 2mj
∂2
∂xj2, j = 1, 2.
(11)
The total energy of the two particles corresponds to the Hamiltonian operator
H= K1+ K2, (12)
whose eigenfunctions (satisfying the boundary conditions (1)) and eigenvalues are, respectively,
n1n2(x1, x2)= n1(x1)n2(x2) (13) and
n1n2= ¯h2π2 2L2
n21 m1 + n22
m2
, (14)
where the functions n(x) are the same as in (8). Since the particles do not interact, each particle has its ‘own’ energy. The energy of the i-particle is represented by the Hamiltonian Ki, and the total Hamiltonian H of the two-particle system is just the sum (12) of the individual Hamiltonians of particles 1 and 2.
3. An entangled state of two particles in a one-dimensional square well
Let us consider a quantum state of the two particles (of zero spin) described by the wavefunction
(x1, x2)= c11(x1)1(x2) + c2
√2[2(x1) + i3(x1)]2(x2), (15) where
|c1|2+|c2|2= 1, c1, c2= 0. (16)
Our first task will be to prove that when the two-particle system’s state is given by the wavefunction (15) there is no single-particle wavefunction (x) providing a complete description of the ‘individual’ state of particle 1. It is possible to prove that the two-particle wavefunction (15) is not factorizable. That is, there are no single-particle wavefunctions
1(x1) and 2(x2) such that (x1, x2)= 1(x1)2(x2). In spite of this mathematical fact, students may still wonder whether there exists a single-particle wavefunction accounting for the physical properties of particle 1. After all, in classical probability theory, even when a joint probability distribution describing a composite system is not factorizable, there still are marginal probability distributions describing each subsystem ‘individually’. It is instructive for students to realize that in this respect quantum wavefunctions differ drastically from classical probability distributions. Here we shall prove that, indeed, no single-particle wavefunction satisfactorily accounts for the state of particle 1. Our approach is going to be physical rather than mathematical, in the sense that our arguments are not going to involve explicitly the non-factorizability of (15). Rather, our arguments will be based directly on the impossibility of accounting for the possible results of physical measurements performed upon particle 1 by recourse to a single-particle wavefunction.
The probability P (x1, x2) dx1dx2of finding the particle in the small region corresponding to the coordinate ranges [x1, x1+ dx1] and [x2, x2+ dx2] is given by the probability density
x1
x2
|Ψ(x1,x2)|2
Figure 3.The probability spatial P (x1, x2) corresponding to the two-particle wavefunction (15).
P (x1, x2)= |(x1, x2)|2(see figure3). On the other hand, the probability P (x) dx of finding particle 1 within the small interval [x, x + dx] (regardless of the location of particle 2) is given by the marginal probability density
P (x1)=
L/2
−L/2
P (x1, x2) dx2
= |c1|2|1(x1)|2+ |c2|2
2 (|2(x1)|2+|3(x1)|2), (17) obtaining by integrating the joint probability density P (x1, x2) = |(x1, x2)|2 over the complete range of possible x2-values (see figure4).
Let us now consider the possible results of measuring the energy of particle 1. The wavefunction (15) can be regarded as a linear combination of the wavefunctions 1(x1)1(x2),
2(x1)2(x2) and 3(x1)2(x2), with coefficients c1, c2/√
2 and ic2/√
2, respectively. Now, these three wavefunctions are clearly eigenfunctions of particle 1’s Hamiltonian, represented by the operator−2m¯h2 ∂x∂22
1. The concomitant eigenvalues are, respectively,2m¯h2π2
1L2,2m4¯h2π2
1L2 and 2m9¯h2π2
1L2. Consequently, these are the only possible results when measuring the energy of particle 1, which occur with probabilities respectively equal to|c1|2,12|c2|2and12|c2|2.
Suppose now that the position probability density P (x) (from now on we are going to drop the subindex ‘1’ when referring to the coordinate of particle 1) arises from a wavefunction (x)
P(x1)
x1
Figure 4.Probability density P (x1) associated with particle 1.
fully accounting for the ‘individual’ physical state of particle 1. In that case the wavefunction
(x) must satisfy,
P (x)= |(x)|2, (18)
and, consequently, (x) has to be of the form
(x)=
P (x) exp(iS(x)), (19)
with S(x) being a (possible x-dependent) appropriate phase. A wavefunction such as (19) accounts correctly for the features of the state of particle 1 related to the measurement of position. However, it does not necessarily lead to the correct probabilities for the possible outputs associated with measurements of other observables. Indeed, as we are going to prove shortly, there is no wavefunction (x) providing correct descriptions of both the probabilities associated with the measurement of position and the probabilities associated with the measurement of the energy.
We know that when measuring the energy of particle 1 there are only three possible results.
Consequently, if there exists a wavefunction (x) describing all the physical properties of particle 1, this wavefunction has to be a linear combination of the three eigenfunctions of particle 1’s Hamiltonian K1corresponding to the alluded three possible measurement values for particle 1’s energy. Therefore,
(x)= d11(x) + d22(x) + d33(x). (20) Furthermore, taking into account the probabilities of getting the three values for the energy of particle 1, we also have
|d1|2= |c1|2
|d1|2= |d2|2=12|c2|2. (21)
Now, the (coordinate) probability density associated with the wavefunction (20) is
|(x)|2 = |d1|2|1(x)|2+|d2|2|2(x)|2+|d3|2|3(x)|2+ (d1d2∗+ d1∗d2)1(x)2(x) + (d1d3∗+ d1∗d3)1(x)3(x) + (d2d3∗+ d2∗d3)2(x)3(x). (22)
If the wavefunction (20) provides a correct account of the physical state of particle 1, the probability densities (17) and (22) must be identical. Therefore, comparing the expressions for these two probability densities it follows that
f (x)= (d1d2∗+ d1∗d2)1(x)2(x) + (d1d3∗+ d1∗d3)1(x)3(x)
+ (d2d3∗+ d2∗d3)2(x)3(x)= 0. (23)
Now, the function f (x) can be written as the sum of a symmetric and an antisymmetric part,
f (x)= fs(x) + fa(x), (24)
where
fs(x)=12(f (x) + f (−x))
fa(x)=12(f (x)− f (−x)). (25)
It is clear from (25) that if a function f (x) vanishes for all values of x then both the symmetric and antisymmetric parts of f (x) must vanish separately. Consequently, for the antisymmetric component of (23) we have
fa(x)= (d1d2∗+ d1∗d2)1(x)2(x) + (d2d3∗+ d2∗d3)2(x)3(x)= 0, (26) leading to
(d1d2∗+ d1∗d2)1(x) + (d2d3∗+ d2∗d3)3(x)= 0. (27) Evaluating the left-hand side of (27) at the nodes of 3(x) one obtains
d1d2∗+ d1∗d2= 0, (28)
and, consequently, one also has
d2d3∗+ d2∗d3= 0. (29)
It follows from equations (28) and (29) that the first and third terms of (23) vanish. This, in turn, implies that the second term also vanishes, leading to
d1d3∗+ d1∗d3= 0. (30)
From equations (28) and (29) it follows that d1d2∗= i r12,
d1d3∗= i r13, d2d3∗= i r23,
(31)
where r12, r13and r23are real numbers.
Because of our assumption that the two coefficients c1,2are both different from zero, the coefficients d1,2,3must also be all different from zero. Now, from equation (31) one gets
|d1|2d2d3∗= r12r13
|d2|2d3d1∗= −r12r23
|d3|2d1d2∗= r13r23.
(32)
But equations (32) imply that the three (non-vanishing) quantities d1d2∗, d2d3∗and d3d1∗are all real numbers, which contradicts the previously obtained relations (31) which say that these three quantities are purely imaginary. This contradiction implies that there is no wavefunction
(x) that accounts fully and correctly for the physical state of particle 1. In particular, we proved that no such wavefunction can reproduce correctly both the probabilities associated with the measurement of the position of particle 1 and the probabilities related to the measurement of the energy of particle 1. Even though we have maximal knowledge about the joint state of
the two-particle system, represented by the two-particle wavefunction (x1, x2), we do not have maximal knowledge about the physical state of particle 1. Such maximal knowledge would be represented by a wavefunction (x) for particle 1 accounting for all the features of its physical state. We proved that such a wavefunction does not exist.
Even if the present discussion has been focused on the particular family of wavefunctions (15), the main conclusions arrived at still hold for any joint wavefunction of the two particles that cannot be written in the factorized form (x1, x2) = 1(x1)2(x2). These non- factorizable wavefunctions are said to describe entangled quantum estates of the two particles.
The systematic analysis of general entangled states, however, requires the use of more advanced tools than those used in this paper.
4. Partial knowledge about the physical state of a quantum system
As we have seen in the previous section, even having the maximal knowledge about the state of the composite two-particle system allowed by the laws of quantum mechanics (represented by the wavefunction (15)) the physical state of particle 1 does not permit that kind of description.
We cannot associate a wavefunction with particle 1. In other words, the amount of knowledge that we have about the state of particle 1 is not as complete as permitted by quantum mechanics.
In order to clarify this point, let us consider an operator corresponding to a physical observable A associated with particle 1. Such an operator can be written as
A(x1, p1)= A
x1,¯h
i
∂
∂x1
. (33)
Let us consider the expectation value of the observable A evaluated on the wavefunction (15):
A =
∗(x1, x2)A
x1,¯h
i
∂
∂x1
(x1, x2). (34)
Taking into account that the operator A acts only on the coordinate x1of particle 1, and the fact that the wavefunctions iare orthonormal, it can be verified that
A = |c1|2
∗1(x)A1(x) dx +1 2|c2|2
∗2(x)A2(x) dx +1 2|c2|2
∗3(x)A3(x) dx.
(35) In order to understand the meaning of (35), first note that the integral
∗i(x)Ai(x) dx is equal to the expectation value of A corresponding to a particle whose state is given by the wavefunction i(x). Now, suppose that one has a quantum particle to whom one cannot assign a precise state because the particle was not prepared in a state described by a specific wavefunction. Instead, let us assume that the particle was prepared in one of the three states represented by the wavefunctions
1(x), 2(x), 3(x), (36)
with the probabilities of having been prepared in each of the three possible states respectively equal to
|c1|2,1
2|c2|2,1
2|c2|2. (37)
Under these circumstances, the expected mean value of the measurement of the observable A is given precisely by equation (35). This interpretation is valid for any physical observable A measured on particle 1. Therefore, when the two particles (1 and 2) are jointly described by the wavefunction (15), particle 1 behaves as if it were prepared in one of the three states (36) with probabilities (37). In other words, the ‘individual’ state of particle 1 is not fully determined,
in the sense that we cannot assign particle 1 a specific wavefunction ‘of its own’. Instead, the physical state of particle 1 can be characterized by the set of possible wavefunctions (36) with the associated probabilities (37). This state of affairs corresponds to what is known as a mixed quantum state, as opposed to a pure quantum state, which is given by a single, well-defined wavefunction . The mixed state characterizing particle 1 is said to be given by a statistical mixture of the states (1, 2, 3)with weights (probabilities) Pirespectively equal to
P1= |c1|2, P2= 12|c2|2, P3=12|c2|2 .
Interestingly, the above state of affairs is similar to that occurring in Statistical Mechanics, when one does not know precisely the system’s state, but one is able to assign probabilities Pi
to the possible states of the system. Indeed, important recent developments on the foundations of Statistical Mechanics [15] emphasize the key role of quantum entanglement in providing a physical justification for the thermo-statistical description of systems in equilibrium with a heat bath. If we have a system S interacting with a large heat bath B, even if the composite S+B is in a pure quantum state, for the overwhelming majority of these states the entanglement between the system and the bath will render the state of S non-accountable by a pure quantum state and, consequently, a description in terms of a statistical mixture is necessary. We can establish an analogy between our present illustrative example of an entangled state and the Statistical Mechanical scenario: one of the two particles plays the role of the ‘system’ and the other one plays the role of the ‘heat bath’.
Summing up, even if the two-particles 1 and 2 are jointly described by a maximally determined quantum state, corresponding to the wavefunction (x1, x2), the state of particle 1 is described by a statistical mixture of three wavefunctions. To appreciate the highly non- classical and counter-intuitive nature of this aspect of quantum mechanics, it is instructive to compare the behaviour of the two-particle quantum system with the behaviour of a classical two-particle system. Let us consider a classical system consisting of two particles moving in the one-dimensional x-axis. The maximal knowledge that we can have about the state of these particles corresponds to the knowledge of the coordinates and momenta of the two particles:
(x1, x2, p1, p2). This can be represented as a point in a four-dimensional classical phase space.
On the other hand, a situation of partial or incomplete knowledge of the state of the system is represented by a normalized phase-space probability density F (x1, x2, p1, p2),
F (x1, x2, p1, p2) dx1dx2dp1dp2, (38) being the probability of finding the system with its phase-space variables within the ranges [xi, xi+ dxi] and [pi, pi+ dpi], (i = 1, 2).
Now, it is clear that if we have complete knowledge of the joint state of the two-particle system (which is represented by a point (x1, x2, p1, p2) in the four-dimensional phase space), we also have complete knowledge of the state of each particle. It is evident that when the state of the two particles corresponds to the phase-space point (x1, x2, p1, p2), particle 1, say, also has a well-determined dynamical state given by the concomitant coordinate and momentum (x1, p1). We have seen, however, that the composite quantum system does not behave like this.
5. Conclusions
A remarkable feature of the quantum mechanical description of Nature is that maximal knowledge about the state of a composite system does not necessarily imply maximal knowledge of the states of its subsystems. In quantum mechanics, the maximum possible knowledge about the state of a system is provided by a wavefunction . According to the aforementioned, counter-intuitive feature of quantum physics, even when the state of a
composite system is characterized by a joint wavefunction it may happen that it is not possible to assign an individual wavefunction to each of its subsystems. Wavefunctions of a composite system behaving in this way are called entangled. In this paper, we have discussed in detail an elementary illustration of this basic aspect of quantum physics, in connection with a system of two quantum particles in an infinite square well.
Acknowledgment
This work was partially supported by the Project FQM-2445 of the Junta de Andalucia (Spain).
References
[1] Walck S N and Hansell N C 2001 Characterization and visualization of the state and entanglement of two spins Eur. J. Phys. 22 343–50
[2] Roston G B, Casas M, Plastino A and Plastino A R 2005 Quantum entanglement, spin 1/2 and the Stern–Gerlach experiment Eur. J. Phys.26657–72
[3] Curilef S, Zander C and Plastino A R 2006 Two particles in a double well: illustrating the connection between entanglement and the speed of quantum evolution Eur. J. Phys.271193
[4] Tommasini P, Timmermans E and Toledo Piza A F R 1998 The hydrogen atom as an entangled electron–proton system Am. J. Phys.66881–6
[5] de la Torre A C 1994 Contextuality in quantum systems Am. J. Phys.62808–12
[6] Ekert A and Knight P L 1995 Entangled quantum systems and the Schmidt decomposition Am. J.
Phys.63415–23
[7] Batle J, Casas M, Plastino A R and Plastino A 2002 Entanglement, mixedness and q-entropies Phys. Lett. A 296251–8
[8] Penrose R 2004 The Road to Reality (London: Jonathan Cape)
[9] Taylor J C 2002 Hidden Unity in Nature’s Laws (Cambridge: Cambridge University Press) [10] Mermin D 1990 Boojurns All the Way Through (Cambridge: Cambridge University Press)
[11] Paty M 1999 Are quantum systems physical objects with physical properties? Eur. J. Phys.20373–88 [12] Bengtsson I and Zyczkowski K 2006 Geometry of Quantum States: An Introduction to Quantum Entanglement
(Cambridge: Cambridge University Press)
[13] Nielsen N and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[14] Giovannetti V, Lloyd S and Maccone L 2006 Quantum metrology Phys. Rev. Lett.96010401 [15] Gemmer J, Michel M and Mahler G 2004 Quantum Thermodynamics (Berlin: Springer)
[16] Lovett B W and Nazir A 2009 Aspects of quantum coherence in nanosystems Eur. J. Phys.30S89–S100 [17] Schlosshauer M 2007 Decoherence and the Quantum-to-Classical Transition (New York: Springer)
[18] Borras A, Zander C, Plastino A R, Casas M and Plastino A 2008 Entanglement and the quantum brachistochrone problem EPL8130007
[19] Isham C J 2001 Lectures on Quantum Theory: Mathematical and Structural Foundations (London: Imperial College Press)
[20] Peres A 1993 Quantum Theory: Concepts and Methods (Dordrecht: Kluwer)
[21] Moore W 1990 Schrodinger: Life and Thought (Cambridge: Cambridge University Press) [22] Gillespie D T 1970 A Quantum Mechanics Primer (Scranton, PA: International Textbook Company) [23] French A P and Taylor A F 1978 Introduction to Quantum Physics 2nd edn (New York: W W Norton and Co.) [24] Tipler P A and Llewellyn R A 2008 Modern Physics 5th edn (New York: Freeman)
[25] Naqvi K R, Waldenstrom S and Hassan T H 2001 Fractional revival of wave packets in an infinite square well:
a Fourier perspective Eur. J. Phys.22395–402
