Infinite quantum well: a coherent state approach
A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.
keywords:, Vector coherent states, quantization, infinite square well
Even though the quantum dynamics in an infinite square well potential represents a rather unphysical limit situation, it is a familiar textbook problem and a simple tractable model for the confinement of a quantum particle. On the other hand this model has a serious drawback when it is analyzed in more detail. Namely, when one proceeds to a canonical standard quantization, the definition of a momentum operator with the usual form has a doubtful meaning. This subject has been discussed in many places (see  for instance), and the attempts of circumventing this anomaly range from self-adjoint extensions  to symmetry approaches .
First of all, the canonical quantization assumes the existence of a momentum operator (essentially) self-adjoint in that respects some boundary conditions on the boundaries of the well. As has been shown, these conditions cannot be fullfilled by the usual derivative form of the momentum without the consequence of losing self-adjointness. Moreover there exists an uncountable set of self-adjoint extensions of such a derivative operator which makes truly delicate the question of a precise choice based on physical requirements [1, 3].
When the classical particle is trapped in an infinite well of real interval , the Hilbert space of quantum states is and the quantization problem becomes similar, to a certain extent, to the quantization of the motion on the circle . Notwithstanding the fact that boundary conditions are not periodic but impose instead that the wave functions in position representation vanish at the boundary, the momentum operator for the motion in the infinite well should be the counterpart of the angular momentum operator for the motion on the circle. Since the energy spectrum for the infinite square well is , we should expect that the spectrum of should be , like the one for without the null eigenvalue. This similarity between the two problems will be exploited in the present paper. We will adapt the coherent states (CS’s) on the circle [4, 5, 6] to the present situation by constructing two-component vector CS’s, in the spirit of , as infinite superpositions of spinors eigenvectors of .
In the present note, we first describe the CS quantization procedure. We recall the construction of the CS’s for the motion on the circle and the resulting quantization. We then revisit the infinite square well problem and propose a family of vector CS’s suitable for the quantization of the related classical phase space. Note that various constructions of CS’s for the infinite square well have been carried out, like the one in  or yet the one resting upon the dynamical symmetry . Finally, we present the consequences of our choice after examining basic quantum observables, derived from this quantization scheme, like position, energy, and a quantum version of the problematic momentum. In particular we focus on their mean values in CS’s (“lower symbols”) and quantum dispersions. As will be shown, the classical limit is recovered after choosing appropriate limit values of some parameters present in the expression of our CS’s.
2 The approach via coherent state quantization
Coherent state quantization [11, 12, 13, 14, 15, 16, 17] is an alternative way of representing classical observables into a quantum system. The states used in it include Glauber and Perelomov CS’s but lie in a wider definition that admits a large range of state families resolving the identity. Identity resolution is here the crucial condition.
In fact, these states form a frame of reference well suited to represent classical quantities and, in that sense, work as a natural quantization procedure which is in one-to-one correspondence with the choice of the frame. The validity of a precise frame choice is asserted by comparing spectral characteristics of quantum observables with data from the observational space. Unlike canonical quantization where the whole model rests upon a pair of conjugated variables within the Hamilton formalism , here we need the following elements.
First of all let be a set equipped with a measure , and let be the Hilbert space of square integrable functions on :
The set can be taken as the phase space of a particular problem as will be the case in this paper. Next we need a finite or infinite orthonormal set , selected among the elements of . This set spans, by definition, the separable Hilbert subspace and must obey the following condition:
Now let us define the family of coherent states in through the following linear superposition:
where the states are in one to one correspondence with the functions in the set . This is an injective map (which should be continuous with respect to some minimal topology affected to for which the latter is locally compact): These coherent states have two main features: they are normalized, and crucially, they resolve the identity in
The CS quantization of a classical observable on , consists then in associating to the operator
This “diagonal” decomposition (in a topological weak sense) may reveal to be valid for a wide class of operators. The function is called upper (or contravariant) symbol of the operator and is non-unique in general. On the other hand, the mean value is called lower (or covariant) symbol of .
3 Quantization of the particle motion on the circle
The motion in the infinite square well potential can be seen as a particular case of the motion on the circle , once we have identified the boundaries of the well with each other and imposed Dirichlet conditions on them. Functions on this domain will behave as pinched waves on a circle so it is useful to expose first the more general case.
Applying our scheme of quantization we can define the CS’s on the circle. The measure space is the cylinder , i.e. the phase space of a particle moving on the circle, where and are canonically conjugate variables. We consistently choose the measure on as the usual one, invariant (up to a factor) with respect to canonical transformations: . The functions forming the orthonormal system needed to construct CS’s are suitably weighted Fourier exponentials:
where can be arbitrarily small. This parameter includes the Planck constant together with the physical quantities characterizing the classical motion (frequency, mass, etc.). Actually, it represents a regularization. Notice that the continuous distribution is the normal law centered at (for the angular momentum variable ). We establish a one-to-one correspondence between the functions and the states which form an orthonormal basis of some generic separable Hilbert space that can be viewed or not as a subspace of . CS’s, as vectors in , read then as
where the normalization factor
is a periodic train of normalized Gaussian functions and is proportional to an elliptic Theta function. Applying the Poisson summation yields the alternative form:
From this formula it is easy to prove that .
The CS’s (7) have been previously proposed, however through quite different approaches, by De Bièvre-González (1992-93) , Kowalski-Rembieliński-Papaloucas (1996) , and González-Del Olmo (1998) .
3.1 Quantization of classical observables
The quantum operator acting on , associated to the classical observable , is obtained as in (5). For the most basic one, i.e. the classical observable itself, the procedure yields
and this is nothing but the angular momentum operator, which reads in angular position representation (Fourier series): .
For an arbitrary function , we have
where is the -th Fourier coefficient of . In particular, we have for the angular position operator :
The shift operator is the quantized counterpart of the “Fourier fundamental harmonic”:
and is canonical in the sense that it is in exact correspondence with the classical Poisson bracket
Some interesting aspects of other such correspondences are found in . For arbitrary functions of the commutator
can arise interpretational difficulties. In particular, when , i.e. for the angle operator
the comparison with the classical bracket is not direct. Actually, these difficulties are only apparent if we consider instead the -periodic extension to of . The position observable , originally defined in the interval , acquires then a sawtooth shape and its periodic discontinuities are accountable for the discrepancy. In fact the obstacle is circumvented if we examine, for instance, the behaviour of the corresponding lower symbols at the limit . For the angle operator we have
where we recognize at the limit the Fourier series of . For the commutator, we recover the canonical commutation rule modulo Dirac singularities on the lattice .
4 Quantization of the motion in an infinite well potential
4.1 The standard quantum context
Any quantum system trapped inside the infinite square well must have its wave function equal to zero beyond the boundaries. It is thus natural to impose on the wave functions the conditions
Since the motion takes place only inside the interval , we may as well ignore the rest of the line and replace the constraints (20) by the following ones:
Moreover, one may consider the periodized well and instead impose the cyclic boundary conditions .
In either case, stationary states of the trapped particle of mass are easily found from the eigenvalue problem for the Schrödinger operator with Hamiltonian:
where obeys the eigenvalue equation
together with the boundary conditions (21). Normalized eigenstates and corresponding eigenvalues are then given by
where is the “revival” time to be compared with the purely classical round trip time.
4.2 The quantum phase space context
The classical phase space of the motion of the particle is the infinite strip equipped with the measure: . A phase trajectory for a given non-zero classical energy is represented in the figure 1.
Typically, we have two phases in the periodic particle motion with a given energy: one corresponds to positive values of the momentum, while the other one is for negative values, . This observation naturally leads us to introduce the Hilbert space of two-component complex-valued functions (or spinors) square-integrable with respect to :
We now choose our orthonormal system as formed of the following vector-valued functions , ,
and the half-width is a parameter which has the dimension of a momentum, say with a dimensionless parameter. This parameter can be arbitrarily small (like for the classical limit) and, of course, arbitrarily large (for a very narrow well, for instance).
The functions are continuous, vanish at the boundaries and of the phase space, and obey the essential finiteness condition (2):
The expression of simplifies to :
It then becomes apparent that and can be expressed in terms of elliptic theta functions. Function has no physical dimension whereas has the same dimension as , that is the inverse of an action.
We are now in measure of defining our vector CS’s . We set up a one-to-one correspondence between the functions ’s and two-component states
forming an orthonormal basis of some separable Hilbert space of the form . The latter can be viewed also as the subspace of equal to the closure of the linear span of the set of ’s. We choose the following set of diagonal real matrices for our construction of vectorial CS’s:
Note that . Vector CS’s, , are now defined for each and by the relation
In particular, we single out the two orthogonal CS’s
By construction, these states also satisfy the infinite square well boundary conditions, namely . Furthermore they fulfill the normalizations
and the resolution of the identity in :
where denotes the identity matrix consistently with the Pauli matrix notation to be used in the following.
4.3 Quantization of classical observables
The quantization of a generic function on the phase space is given by the expression (5), that is for our particular CS choice:
For the particular case in which is function of only, , the operator is given by
Note that this operator is diagonal on the basis.
4.3.1 Momentum and Energy
In particular, using , one gets the operator
where is a Pauli matrix.
For , which is proportional to the Hamiltonian, the quantum counterpart reads as
Note that this implies that the operator for the square of momentum does not coincide with the square of the momentum operator. Actually they coincide up to O.
For a general function of the position our quantization procedure yields the following operator:
In particular, for we get the “position” operator
with . Note the appearance of the classical mean value for the position on the diagonal.
4.3.3 Commutation rules
Now, in order to see to what extent these momentum and position operators differ from their classical (canonical) counterparts, let us consider their commutator:
This is an infinite antisymmetric real matrix. The respective spectra of finite matrix approximations of this operator and of position and momentum operators are compared in figures 2 and 3 for various values of the regulator in units , . When takes large values, one can see that the eigenvalues of accumulate around , i.e. they become almost canonical. Conversely, when all eigenvalues become null, which corresponds to the classical limit.
4.3.4 Evolution operator
The Hamiltonian of a spinless particle trapped inside the well is simply . Its quantum counterpart therefore is . The unitary evolution operator, as usual, is given by
Note the appearance of the global time-dependent phase factor with frequency which can be compared with the revival frequency
5 Quantum behaviour through lower symbols
Lower symbols are computed with normalized CS’s. The latter are denoted as follows
Hence, the lower symbol of a quantum observable should be computed as
This gives the following results for the observables previously considered :
In the same way, the mean value of the position operator in a vector CS is given by:
where we can distinguish the classical mean value for the position corrected by the function
This function depends on the parameter as we show in figure 4 with a numerical approximation using finite matrices. As for , we calculate the dispersion defined as
Its behaviour for different values of is shown in figure 6.
5.0.2 Time evolution of position
The change through time of the position operator is given by the transformation , and differs from by the insertion of an oscillating term in the series. Its lower symbol is given by
where this time the series have the form
Note that the time dependence manifests itself in the form of a Fourier series of with frequencies . This corresponds to the circulation of the wave packet inside the well.
The mean value of the momentum operator in a vector CS is given by the affine combination:
This function reproduce the profile of the function , as can be seen in the figure 5. We calculate then the dispersion , defined as
using the mean values in a CS . Its behaviour as a function of is shown in figure 7.
5.0.4 Position-momentum commutator
The mean value of the commutator in a normalized state is the pure imaginary expression:
Given the symmetry and the real-valuedness of states (36), the mean value of the commutator when is one of our CS’s vanish, even if the operator does not. This result is due to the symmetric spectrum of the commutator around . As is shown in Part c) of figures 2, the eigenvalues of the commutator tend to as , i.e. , increases. Still, there are some points with modulus less than . This leads to dispersions in CS’s that are no longer bounded from below by . Actually, the lower bound of this product, for a region in the phase space as large as we wish, decreases as diminish. A numerical approximation is shown in figure 8.
From the mean values of the operators obtained here, we verify that our CS quantization gives well-behaved momentum and position operators. The classical limit is reached once the appropriate limit for the parameter is found. If we consider the behaviour of the observables as a function of the dimensionless quantity , at the limit and when the Gaussian functions for the momentum become very narrow, the lower symbol of the position operator is . This corresponds to the classical average value position in the well. On the other hand, at the limit , for which the involved Gaussians spread to constant functions, the mean value converges numerically to the function . In other words, our position operator yields a fair quantitative description for the quantum localization within the well. The lower symbol behaves as a stair-step function for close to and progressively fits the function when increases. These behaviours are well illustrated in the figures 4 and 5. The effect of the parameter is also noticeable in the dispersions of and . Here, the variations of the full width at half maximum of the Gaussian function reveal different dispersions for the operators. Clearly, if a classical behaviour is sought, the values of have to be chosen near . This gives localized values for the observables. The numerical explorations shown in figures 6 and 7 give a good account of this modulation. Consistently with the previous results, the behaviour of the product at low values of shows uncorrelated observables at any point in the phase space, whereas at large values of this parameter the product is constant and almost equal to the canonical quantum lowest limit . This is shown in figure 8.
It is interesting to note that if we replace the Gaussian distribution, used here for the variable in the construction of the CS’s, by any positive even probability distribution such that the results are not so different! The momentum spectrum is still and the energy spectrum has the form . In this regard, an interesting approach combining mathematical statistics concepts and group theoretical constructions of CS’s has been recently developed by Heller and Wang [20, 21].
The work presented here has possible applications to those particular physical problems where the square well is used as a model for impenetrable barriers , in the spirit of what has been done in .
The generalization to higher-dimensional infinite potential wells is more or less tractable, depending on the geometry of the barriers. This includes quantum dots and other quantum traps. Nevertheless, we believe that the simplicity and the universality of the method proposed in the present work should reveal itself useful for this purpose.
Author García de León wishes to acknowledge the Consejo Nacional de Ciencia y Tecnología (CONACyT) for its support.
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