We explore the properties of quantum states and operators that are conjugate to the Hamiltonian eigenstates and operator when the Hamiltonian spectrum is continuous, i.e., we find time-like operators T^ such that [T^,H^]=iℏ. This is a property expected for a time operator. We explicitly unfold the momentum sign degeneracy of energy states. We consider the free-particle case, and we find, among other things, that the time states are also the solution of the quantized version of the classical motion of the particle.
- time operator
- time eigenstates
- conjugate states
- free-particle time eigenstates
The problem of the time operator in quantum mechanics has been studied by numerous researchers for many years and remains a subject of current research. There are many instances in which a time variable is useful. An example of such a situation is calculating the tunneling time of a particle passing through a barrier. This time was recently measured, and it was shown to vanish [1, 2].
There are several approaches in this area that were developed by Kijowski , Hegerfeldt et al. , Weyl , Galapon , Arai and Yasumichi [7, 8], Strauss et al. [9, 10], and Hall , among others. The work by these authors may appear to be in four differing approaches; however, we shall show that they are simply different approaches to the same theme, approximated ones.
Some of these approaches are similar to the work of Weyl on periodic functions . Weyl defined the Hermitian form
We introduce time-like kets and operators following a different route. We search for the states that are conjugate to the energy eigenstates, which is a natural approach to this subject. We find time kets and operators that are valid over the entire Hilbert space. We also find that we can make contact with the operators defined by other authors. These operators lack the oscillatory function found in this work.
Time is typically viewed as a parameter and not as a dynamical variable in classical and quantum mechanics. However, the characteristics of the time variable depend on the representation being considered. In classical mechanics, we have shown that we can talk of translations along the energy direction; in that case, the energy variable becomes a parameter, and time becomes a dynamical variable, a function of the phase-space variables .
For comparison, let us consider the coordinate representation of quantum mechanics. If a variable,
The basis for this work is that time is another coordinate that has to be determined. The conjugate pair coordinate-momentum is a pair of conjugate coordinates that are used to define representations of wave functions and operators. Similarly, energy and time can be used as an alternative coordinate set, but the time coordinate has to be defined. As coordinate and momentum eigenstates, the time eigenstates will also be nonnormalizable, and their peculiarities originate from the type of coordinate that energy is a semibounded quantity.
In Section 2, we use the rewriting of the identity operator in terms of energy eigenstates to define the states that are conjugate to the energy eigenstates and subsequently determine some of their properties and several time-like operators. We define time states for negative and positive momentum values.
Section 3 is devoted to time-like operators and their properties. Time operators are written in three different forms. We verify that the time kets are eigenkets of the time operators. We find “evolution equations” for time kets and note that the time operators are the generators for translations along the energy direction. We also discuss how a wave packet is shifted along the energy direction.
In numerical calculations, we have to address finite regions of variables and not infinite intervals. Therefore, we focus our attention on approximate expressions for time operators in Section 4. We find approximate expressions of time operators that can be used in numerical calculations and are of help in the understanding of the expressions found by other authors.
The free-particle problem is analyzed in Section 5. We find expressions for the time kets for the free particle. The coordinate matrix elements of the time operators are also found, and we learn that the time states are also a solution to the quantum analog of the classical motion. The support of the time states embodies the classical trajectories, and as , we recover the classical motion.
We conclude the chapter with some concluding remarks.
2. Time eigenstates
In this section, we define the states that are conjugate to the energy eigenstates and the corresponding conjugate operator to a given quantum Hamiltonian . We also derive some of their properties. The definition of conjugacy between the operators and that we will use here is the usual one, i.e., that these operators should comply with the constant commutator relationship . We will consider the case of a purely continuous energy spectrum with a Hamiltonian operator of the form , where is the momentum operator, is the coordinate operator, and is the potential energy operator. We will also consider that the sign of the momentum operator commutes with the Hamiltonian. The continuous eigenvalues of the Hamiltonian are denoted by
We will base our definition of time states on rewriting the identity operator in terms of energy eigenstates and using the integral representation of the Dirac delta function. We assume that the Hamiltonian is self-adjoint. Thus, we will work on the span of the Hamiltonian eigenstates, denoted by
We assume that the closure relationship for the energy eigenstates holds, . The
We start with the rewriting of the identity operator in terms of the energy eigenkets,
where we have made use of the properties of the Dirac delta function. We can separate the negative and positive momentum parts of the above expression by means of the closure relationship for the momentum states, obtaining
Thus, we define time-like kets as
With these kets, the identity operator is written as
Then, the identity operator is written in terms of the time evolution of some bras and kets, which are composed of all the energy eigenstates.
Now, we define time-like operators and by introducing a factor
The function exists only for , so that, for the sake of simplicity of notation, we, sometimes, will include explicitly the function , where is the step function, when necessary, otherwise we will omit this factor.
The commutator between these operators and the Hamiltonian operator is
where we have made use of the integration by parts. This is one of the properties that a time operator should comply with—the constant commutator with the Hamiltonian. We also have that
is a time-like operator in the energy representation, which is symmetric in the interval regardless of the boundary conditions at
Thus, we can say that the kets
can be considered as time-like kets. We will study some of their properties in what follows.
The inner product between time states is
Thus, the time states are not orthogonal due to the bounded nature of the Hamiltonian operator.
2.1. Properties of the transformation function between energy and time states
The transformation function between energy and time representations is given by
A property of this transformation function is that it is a sort of eigenfunction of the time-like operator, when the functions exists in the interval
and it is also an eigenfunction of the energy operator, ,
This is similar to the corresponding properties of the transformation function between coordinate and momentum representations. The squared modulus of the transformation function is constant for all values of
Time kets can be used as a coordinate system for quantum systems and are similar to coordinate or momentum eigenkets. The norm of a wave packet in the time representation is (see Eq. (5))
Thus, we will obtain well-defined quantities if the wave packet is normalized in the energy representation, i.e., if . We also note that the transformation from energy to time representations is norm preserving, i.e., it is unitary.
2.2. The time eigenstates are conjugate to the energy eigenstates
Now, the Fourier transform of the time states is
Thus, the kets and are conjugate indeed, i.e., the definition (9) is consistent; and are the Fourier transforms of each other, and then an eigenstate contains all the conjugate eigenstates with the same weight.
3. Time operators
We now focus on the time operators obtained from the time kets of the previous section and on their properties. Time operators for negative, positive, and any value of the momentum are defined as
The last construction was also introduced, from another perspective, by Hegerfeldt et al. . Our construction is different from that of Hegerfeldt et al. because it involves all the energy eigenstates and not only those that are time reflection invariant. Our time operator exhibits the time reversal property already.
Time operators can be written in three equivalent forms in the energy representation. One form is
where we have performed an integration by parts. We also have that
These are the forms in which the time operators act on energy eigenkets, but they take a different form when they act on states or on both, eigenstates and wave packets.
A second energy representation of time operators is
These are the various forms in which the time operators can act on states in the energy representation. The difference with the time operators when acting on energy eigenkets is a minus sign.
Other symmetric expressions for the time operators can also be obtained:
The domain of our time operators is
3.1. Time matrix elements of the Hamiltonian
The matrix elements of the Hamiltonian in the time representation are given by
This is the Schrödinger equation for time kets in the time representation.
3.2. The time ket is the eigenstate of the time operator
We can find the characteristic operator of the commutators and . Because (see Eq. (7a)), the commutator between the operator , , and the Hamiltonian is
Similarly, the commutator between the time operator and the time propagator is
The time ket is the time propagation of a zero time ket ,
Thus, according to Eq. (22), we can say that the time ket is an eigenstate of the time operator
where we have set because is the zero-time state.
An “evolution equation” for the energy eigenstate is (see Eq. (15))
Thus, the time operator is the generator of translations along the energy direction. All quantities are well defined as long as
3.3. Shifting of operators
The shifting of the Hamiltonian along the energy direction is (see Eq. (21))
where . For the translation of the time operator (see Eq. (22)), we have
that is, in the energy-time representations,
Therefore, in the energy-time representations,
Thus, the use of energy and time eigenkets and operators instead of coordinate and momentum eigenkets and operators is similar to going from a parametric representation of curves, with time being the parameter of evolution, to a nonparametric representation in which time is now one of the coordinates.
4. Approximate expressions
In this section, we make contact with other expressions that have been used by other authors. Other works have not made use of the Sa(
4.1. Approximating the integral in an infinite interval
As an approximation, we replace the integral in an infinite interval with the integral in the finite interval , . Then,
where the Sa function of type one is defined as
A plot of this function can be found in Figure 1. This function is zero at
5. The free particle
As an example of the time kets provided by our method, let us apply the derived results to the free-particle system. We find expressions for time eigenkets, including the case when a distinction of the sign of the momentum is needed. In this model, the momentum operator commutes with the Hamiltonian operator , indicating a symmetry, allowing for some simplifications.
A set of energy eigenfunctions, in the coordinate representation, for the free-particle model is
The subscripts in these functions indicate the sign of the momentum of the particle.
Thus, the zero-time eigenstate for the free particle is given as
where we have made the change in variable . The unit of the last ket is time−1. Various other authors have used kets obtained by direct quantization of the classical expression for the time variable and have obtained a time ket with units of time1/2. However, our kets exhibit the properties discussed in this chapter.
Figure 2 shows a three-dimensional plot of the approximation of the squared modulus of the time states and , obtained by not integrating from
For the sake of completeness, we write down the matrix elements of the time operators in the coordinate representation. They are
5.1. Solution to the quantized version of the classical motion of a free particle
The following calculation shows that the time states can also be the solution to the quantized classical expression for the motion of a free particle initially located at
We can think of the last two terms in the above equations as quantum corrections to the classical trajectory of a free particle. These correction terms seem to vanish when .
On the other hand, the straightforward solution to the quantized version of the classical expression for the motion of a free particle gives a quite different function. The solution to the differential equation
is not a localized function either; it actually is proportional to the transformation function between energy and time representations, in momentum representation. Thus, the route of forming conjugate states to the energy eigenstates seems to be a better path for obtaining appropriate time eigenstates.
We have introduced time-like states and time-like operators that are conjugate to the energy eigenstates and Hamiltonian operator, respectively. We have also given an interpretation of the obtained states and operators, and we have found that expressions obtained via other approaches to finding time eigenstates can be related to our expressions. However, the oscillatory Sa factor that we use solves many difficulties found in previous treatments. We have found the form of the time states for the free particle and a time operator that is valid for any
The approximation to time operators that we have introduced in this chapter uses expressions that can be adapted to the case of discrete energy spectra. We will explore this possibility in a later paper. From the literature on time operators, it might be believed that the treatment for a continuous energy spectrum is different from that for discrete energy spectrum systems. But, the results of this study suggest that both types of systems can be addressed in a similar manner.
Finally, we have found that the spectral measure of is a nonorthogonal resolution of the identity defined by
This measure exhibits the covariance property, as was previously stated by Holevo .
Galapon EA: Only above barrier energy components contribute to barrier traversal time. Phys Rev Lett. 2012; 108:170402. DOI: 10.1103/PhysRevLett.108.170402
Eckle P, Pfeiffer AN, Cirelli C, Staudte A, Dörner R, Muller HG, Büttiker M, Keller U: Attosecond ionization and tunnelling delay time measurements in Helium. Science. 2008; 322:1524–1529. DOI: 10.1126/science.1163439
Kijowski J: On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time. Rep Math Phys. 1974; 6:361–386.
Hegerfeldt GC, Muga JG, Muñoz J: Manufacturing time operators: Covariance, selection criteria, and examples. Phys Rev A. 2010; 82:012113. DOI: 10.1103/PhysRevA.82.012113
Weyl H: The Theory of Groups and Quantum Mechanics. 2nd ed. USA: Dover Publications, Inc.; 1950. 447 p.
Galapon EA: Self-adjoint time operator is the rule for discrete semi-bounded Hamiltonians. Proc R Soc Lond A. 2002; 458:2671–2690. DOI: 10.1098/rspa.2002.0992
Arai A, Yasumichi M: Time operators of a Hamiltonian with purely discrete spectrum. Rev Math Phys. 2008; 20:951–978.
Arai A: Necessary and sufficient conditions for a Hamiltonian with discrete eigenvalues to have time operators. Lett Math Phys. 2009; 87:67. DOI: 10.1007/sl 1005-008-0286-z
Strauss Y, Silman J, Machnes S, Horwitz LP: An arrow of time operator for standard Quantum Mechanics. 2008. quant-ph 0802.2448.
Strauss Y: Forward and backward time observables for quantum evolution and quantum stochastic processes – I: The time observables. 2007. math-ph 0706.0268v1.
Hall MJW: Comment on “An arrow of time operator for standard quantum mechanics” (a sign of the time!). 2008. quant-ph 0802.2682.
Torres-Vega G: Conjugate dynamical systems: Classical analogue of the quantum energy translation. J Phys A: Math Theor. 2012; 45:215302. DOI: 10.1088/1751-8113/45/21/215302
Holevo AS: Estimation of shift parameters of a quantum state. Rep Math Phys. 1978; 13:379–399.
Martínez-Pérez A, Torres-Vega G: Translations in quantum mechanics revisited. The point spectrum case. Can J Phys. 2016; 94:1365.