Quantum mechanical Gaussian wavepackets of single relativistic particles

In relativistic quantum mechanics, relativistic particles are described by the Klein-Gordon(KG) equation, the Dirac equation, etc., depending on their spins [1]. When the energy density of a particle or experienced by the particle is not too high, while the KG or the Dirac equation may not be reduced to the Schrödinger equation in some situations in atomic and nuclear physics, relativistic quantum mechanics can work nicely with the single-particle interpretation [1]. At a sufficiently high energy density, however, the KG wavefunctions and the Dirac spinors exhibit some weird behaviors in the viewpoint of non-relativistic quantum mechanics, such as the non-positive-definite charge density and the Klein paradox [1, 2]. Those behaviors indicate particle-antiparticle production [1, 3, 4] and thus the breakdown of the single-particle interpretation in quantum mechanics. In those cases, one should consider relativistic quantum field theory [2, 5] to consistently deal with the many-body nature of the system.

Recently, a linearized effective theory [6, 7] for single electrons moving in quantum electromagnetic(EM) fields has been constructed in order to address the issue of the Unruh effect [8, 9] on the motion of relativistic charged particles [10]. That effective theory starts with the total action consisting of (25) and the one for free EM fields. Then the dynamical variables are quantized canonically by introducing the equal-time commutation relations

where $\hat{z}^{i}$ and $\hat{A}^{\mu}_{\bf x}$ are the operators for the deviations of the electron position and EM fields from their classical solutions, respectively, and $\hat{p}^{i}$ and $\hat{\pi}^{\mu}_{\bf x}$ represent their conjugate momenta. Such a Dirac quantization implicitly assumes that the single-particle interpretation and the associated Born’s rule apply to the (reduced) quantum state of the electrons in the Schrödinger representation of this effective theory.

Further, the authors of [6, 7] considered Gaussian states of single electrons and EM fields. The reasons are obvious: Gaussian states of quantum systems are mathematically simple yet capable of describing a broad range of physics, including the ground state and squeezed states of harmonic oscillators(HOs), and the vacuum and thermal states of quantum fields. For Gaussian states of two HOs, the degree of quantum entanglement between the HOs is well defined even when the two-HO system is in a mixed state [11, 12]. Taking these advantages, a particle-field interacting system in a Gaussian state can be analyzed in details and even non-perturbatively in a long period of evolution if the combined system is linear (e.g. [12, 13, 14]).

Nevertheless, it is known that in relativistic quantum mechanics a wavefunction that initially looks like a Gaussian function can evolve very differently from the Gaussian wavepackets of the Schrödinger equation [15, 16, 17]. A closed-form example has been provided by Rosenstein and Usher in Ref. [15], where they explicitly demonstrated that the probability density of the Salpeter wavefunction for a particle at rest in (1+1) dimensional Minkowski space with initial width smaller than its Compton wavelength $\lambda_{C}$ will not concentrate around the classical trajectory after the initial moment, but rather splits into peaks around the lightcone started with the initial position. Then the shape of the probability density of the wavepacket becomes highly non-Gaussian (similar to the behavior of the KG charge density in Figure 1.)

This generated the question: In what conditions is a Gaussian probability density of the wavefunctions (in the Schrödinger representation of a quantized effective theory such as Refs. [6, 7]) justified to mimic a nearly Gaussian charge density of the KG wavefunction or Dirac spinor? In other words, what is the range of validity or the cutoff for an effective theory like Refs. [6, 7] for single electrons moving in EM fields?

To answer the above question about moving single electrons, merely the wavepackets in Refs. [15, 18] for particles at rest are not sufficient. Thus in this paper, we first generalize the Salpeter wavepacket solution in Ref. [15] to the KG wavepacket for relativistic particles in uniform motion, and then construct KG wavepackets initially Gaussian for free particles at constant speed, and for accelerated particles in a uniform electric field. We calculate the KG charge densities of our moving KG wavepacket solutions and see whether their shapes look like Gaussian functions in a long period of time. If they do, the approximated description using Gaussian wavepackets in the Schrödinger representation of an effective theory like [6, 7], associated with Gaussian probability densities in the single-particle interpretation, will hopefully be justified. Then we can identify the range of validity the effective theory for moving single electrons after exploring the parameter space.

This paper is organized as follows. In Section 2.1, we give our wavepacket solutions for free relativistic particles in uniform motion. Our solutions are generalized from the solutions to the Salpeter equation given in [15]. Since these solutions also satisfy the KG equation for free particles, we can simply treat them as the KG wavefunctions and calculate the KG charge densities of them. After we study the properties of these KG charge densities, in Section 2.2 we consider an alternative class of the KG wavefunctions for free particles in uniform motion, which are exactly Gaussian at the initial moment. Similar wavepacket solutions to the KG equation for relativistic charged particles accelerated in a uniform electric field are given and studied in Section 3. Combining our results, we find that the minimal initial width for the KG charge density of a Gaussian KG wavefunction evolving like a Gaussian wavepacket centered at its classical trajectory is about $\lambda_{C}/\gamma_{0}$, where $\gamma_{0}=1/\sqrt{1-(v_{0}/c)^{2}}$ is the Lorentz factor of the particle initially at speed $v_{0}$. Our findings are summarized in Section 4. Finally in A, we remark on how the phase of a KG wavepacket solution, which are evaluated around the particle’s classical trajectory, evolves in a manner akin to the corresponding classical action.

The KG equation for relativistic particles of mass $m$ and charge $q$ moving in EM fields $A^{\mu}$ reads [1]

with the signature $(-,+,+,+)$ and $\hat{p}_{\mu}=-i\hbar\partial_{\mu}$. The KG charge density is defined as

which indicates the dominance of anti-particle in the regions where $\rho$ is negative [1]. If $\rho$ is approximately positive definite and normalizable, then $\rho(t,{\bf x})$ could approximately represent the probability density of finding the particle at $(t,{\bf x})$

For free relativistic particles, (2) reduces to

The solutions of positive energy to (4) also satisfies the Salpeter equation or the square-root KG equation [19],

whose form is similar to the Schödinger equation.

Although the probability density in the context of the Salpeter equation is given as $|\Psi(t,x)|^{2}$, which is positive definite, we will not proceed along this line in this paper. For free particles, while a wavepacket solution $\Psi$ to the Salpeter equation (5) must be a solution to the KG equation (4), we will only focus on the well accepted KG charge density $\rho$ of the wavepacket $\Psi$ in (3). In the presence of background EM fields, the KG equation and the Salpeter equation are not equivalent. In Section 3, we will only consider the solutions to the KG equation and the corresponding KG charge density for particles accelerated in a uniform electric field. These KG wavefunctions will not satisfy the Salpeter equation with the same background fields.

In shaping the initial wavefunction at $t=0$ for a charged particle in a uniform electric field, it is convenient to choose the EM four-potential $A^{\mu}=(0,-{\cal E}t,0,0)$, which gives the electric field ${\bf E}=-cF_{0j}\hat{x}^{j}={\cal E}\hat{x}^{1}$. The corresponding KG equation reads ($c=\hbar=1$, ${\bf x}\equiv(x,y,z)$)

Inserting the ansatz $\Psi=e^{i{\bf p}\cdot{\bf x}}\psi_{\bf p}(t)$ into the above equation, one has

with $M^{2}=m^{2}+p_{y}^{2}+p_{z}^{2}$. Below we take $p_{y}=p_{z}=0$ for simplicity. The general solution to the above equation is [4]

where $D_{\nu}(z)$ are the parabolic cylinder functions, $F\equiv q{\cal E}$, and $c^{\pm}_{\bf p}$ are constants of time. Suppose the initial wavefunction at $t=0$ is Eq. (17) again. Then the wavefunction for $t\geq 0$ will be

with the constants of time

where ${\cal N}$ is the normalization constant.

Since $A^{0}=0$ in this gauge, the charge density (3) is simply $\rho=\frac{q\hbar}{mc^{2}}{\rm Re}\,\,i\Psi^{*}\partial_{t}\Psi$ here. In Figure 6, we show three examples of the charge density $\rho$ of wavefunction (23) with (24), $F=0.1$, and other parameter values similar to those in Figure 4. These examples correspond to the classical trajectories $\bar{x}(t)=c\sqrt{\alpha^{-2}+t^{2}}$ (the upper plots) and $\bar{x}(t)=c\sqrt{\alpha^{-2}+(t+t_{0})^{2}}-c\sqrt{\alpha^{-2}+t_{0}^{2}}+c\alpha^{-1}$ (the lower-right plot) with $\alpha\equiv F/(mc)$ and $t_{0}\equiv p_{0}/F$ for uniformly accelerated charges, and so $\bar{x}(0)=c\alpha^{-1}=10$ in all the three examples in Figure 6.

For a charged particle initially at rest, we find that $\rho$ can be close to a Gaussian function with the value of $\sigma_{0}$ as small as 3 [Figure 6 (upper-left)], i.e. with the initial width about $2\sigma_{0}=6$ in the unit of the reduced Compton wavelength of the particle. This result is the same as we observed in Figure 4 (upper-left) for free particles.

For $\sigma_{0}<3$, as $\sigma_{0}$ decreases, deflection of the charge density $\rho$ from a Gaussian function becomes more and more significant at $t=0$ (though $|\Psi(0,x)|^{2}$ is still Gaussian), and the non-Gaussianity of $\rho$ grows more and more quickly as $t$ increases [Figure 7 (left).] In particular, as shown in Figure 6 (upper-right), the KG charge density of a wavepacket with initial momentum $p_{0}=0$ and initial width $\sigma_{0}$ well below the Compton wavelength will quickly evolve to a double-peak structure, though the expectation value of position still moves around $\bar{x}(t)$ like a classical accelerated charge. This behavior is similar to those of free wavepackets with small initial widths in Figures 1 and 4. The significant tail of the corresponding momentum spectrum $\tilde{\rho}$ (blue curves) in the region of $p<-mc=-1$ in Figure 6 (lower-left) is also similar to the momentum spectra of those wavepackets of free particles with initial widths well below the Compton wavelength in Figure 3.

When both the initial speed and the strength of the uniform electric field are small, the charge density $\rho$ of wavefunction (23) with initial width $2\sigma_{0}$ above $O(\lambda_{C})$ will spread as $t$ increases, while the spreading rate decreases as the particle’s classical speed $\dot{\bar{x}}(t)$ goes to the speed of light [Figure 6 (upper-left)]. In Figure 7 (right) we can see that the half-width $\sigma$ of the charge density $\rho$ in this case (black curve) evolves like $t/\bar{\gamma}(t)$ (red-dotted), rather than the proper time $\tau=\alpha^{-1}\sinh^{-1}\alpha t$, for sufficiently large $t$ ¹¹1The worldline of our uniformly accelerated charge at proper acceleration $a$ in Figure 6 (upper-left) and (upper-right) is $\bar{z}^{\mu}(\tau)=(c\alpha^{-1}\sinh\alpha\tau,c\alpha^{-1}\cosh\alpha\tau,0,0)$ parametrized by its proper time $\tau$, or $\bar{z}^{\mu}(t)=(ct,c\sqrt{\alpha^{-2}+t^{2}},0,0)$ parametrized by the Minkowski time $t$, with $\alpha\equiv a/c$ and $\bar{z}^{\mu}(0)=(0,c\alpha^{-1},0,0)$. It is straightforward to obtain the three velocity $\bar{v}^{i}\equiv d\bar{z}^{i}(t)/dt=(ct/\sqrt{\alpha^{-2}+t^{2}},0,0)$, and the Lorentz factor $\bar{\gamma}(t)=\big{(}1-\frac{v_{i}}{c}\frac{v^{i}}{c}\big{)}^{-1/2}=\sqrt{1+(\alpha t)^{2}}$.. This may be considered as a manifestation of length contraction, rather than time dilation, from the case of free particles at zero speed, whose half-width evolves like $t$ for sufficiently large $t$ (cf. Figure 5).

If the initial speed $v_{0}$ of the charged particle at $t=0$ is close to the speed of light, then again, the minimal initial width for a long-lasting Gaussian charge density goes down to $O(1/\gamma_{0})$ of the Compton wavelength as shown in Figure 6 (lower-right), where $\sigma_{0}=3/\gamma_{0}=0.3$. This may also be considered as a manifestation of length contraction.

We have demonstrated with selected examples that the charge density $\rho$ of the Klein-Gordon wavepacket of a relativistic particle in uniform motion could hopefully be approximated by the probability density of a Gaussian wavefunction in the Schrödinger representation of an effective theory with the single-particle interpretation, if the initial width of $\rho$ is above $O(\lambda_{C})/\gamma_{0}$, where $\lambda_{C}$ is the particle’s Compton wavelength and $\gamma_{0}$ is its Lorentz factor at the initial moment. For a particle with non-negative initial momentum $p_{0}$, an initial width well above $\lambda_{C}/\gamma_{0}$ in position space corresponds to a momentum spectrum with a tail negligibly small in the region of $p<-mc$ in momentum space. For the wavepackets of free particles in uniform motion with all parameter values the same except the initial speed, Gaussian approximation could be good for a longer time for a particle at a higher initial speed in the laboratory (rest) frame.

The wavepackets of relativistic particles linearly accelerated in a uniform electric field show similar behaviors both in position space and momentum space. With an initial width of $\rho$ above $O(\lambda_{C})/\gamma_{0}$, the spreading of the charge density of a uniformly accelerated Gaussian wavepacket gets frozen in the laboratory frame as the group velocity of the wavepacket approaches the speed of light. We find that this is a manifestation of length contraction, rather than time dilation.

Since our initial condition always produces wavepacket solutions spreading as time increases, the minimal initial width of $\rho$ we found in this paper actually implies that, to keep the single-particle interpretation and the Gaussianity, the charge density of a nearly Gaussian KG wavepacket at each moment $t$ can at most be squeezed to a width about $\lambda_{C}/\bar{\gamma}(t)$. This suggests that the UV cutoff of the electron-photon interaction could be about $\lambda_{C}/\bar{\gamma}(t)$ in an effective theory quantized in Minkowski coordinates for single electrons interacting with EM fields [6, 7].