We present a simplified analysis suitable for a beginners' course in quantum mechanics of a recently presented model of positional decoherence in a gas of scatterers. As such, no reference is made to the density matrix formalism, many body theory, or even operator algebra. We only make use of the properties of quantum states and the position and momentum wavefunctions, which students typically encounter in a first quantum mechanics course.

Quantum mechanics describes nature in terms of state vectors $|ϕ⟩$ belonging to a Hilbert space, which in the Schrödinger picture evolve in time. This description has the apparent problem that many quantum mechanical states do not correspond to classical macroscopic objects. For example, a particle in state $|ϕX⟩$ localized at position X has a classical counterpart, while a particle in state $|Φ⟩=(|ϕX⟩+|ϕY⟩)/2$ appears to be located at both X and Y simultaneously and thus has no classical counterpart. Furthermore, macroscopic objects initially in classical-like states can evolve into states with no classical analogs. This is demonstrated in the famous Schrödinger-cat paradox, where a macroscopic cat, initially in a live state, evolves into states where it is both dead and alive.1

This problem can be addressed by noting that observable predictions of the quantum theory are expectation values of Hermitian operators A so that physically measurable consequences of the non-classical nature of states, such as $|Φ⟩$, are contained in the interference term $⟨ϕY|A|ϕX⟩$. The decoherence theory demonstrates that an interaction resulting in an entanglement with a many-particle system can produce rapid decay of such interference terms, leading to classical behavior at the macroscopic level. Here, we show how the action of a gas of scatterers on a macroscopic particle via the Schrödinger equation leads to the approximately exponential decay of $⟨ϕY|A|ϕX⟩$ using a simplified scattering model.2 In physical terms, scattering effectively “measures” the position of the particle, forcing it into a position eigenstate.

Consider a particle with the center of mass at X, represented by state $|ϕX⟩$ with the corresponding position wavefunction $ϕX(y)=⟨y|ϕX⟩$ and surrounded by a gas of scatterers, each in state $|φ⟩$ and with the corresponding position wavefunction $φ(x)=⟨x|φ⟩$. For simplicity, we assume that scattering is one dimensional and the scatterers are much lighter than the particle.2 Scattering is then an elastic reflection from an infinite potential located at x = X, with the momentum of the scatterer changing the sign (see Fig. 1).

Fig. 1.

One-dimensional scattering of an environmental particle with wavefunction $φ(x)$ from a very heavy particle in state $|ϕX⟩$ located at x = X. Due to the mass of the particle, scattering is equivalent to reflection from an infinite potential.

Fig. 1.

One-dimensional scattering of an environmental particle with wavefunction $φ(x)$ from a very heavy particle in state $|ϕX⟩$ located at x = X. Due to the mass of the particle, scattering is equivalent to reflection from an infinite potential.

Close modal

Because $p=m dx/dt$, the (scattering) transformation $p→S−p$ is effected by a change in the sign of the position, accompanied by a possible translation, $x→Sa−x$. This transformation is accomplished by the scattered wavefunction $φ(a−x)$, which reverses the momentum distribution associated with $φ(x)$. Because the potential is infinite, the wavefunction must vanish at the wall;1 thus, the constant a is determined3 by the condition $φ(X)−φ(a−X)=0$ so that $a=2X$. The result is that scattering changes the product of particle and scatterer wavefunctions as

$ϕX(y)φ(x)→S−ϕX(y)φ(2X−x).$
(1)

Consider the wavefunction for a superposition of two orthonormal states corresponding to the particle centered at X and Y

$Φ(y,x)=12[ϕX(y)+ϕY(y)]φ(x).$
(2)

Combining Eqs. (1) and (2), scattering changes the superposition of Eq. (2) as

$[ϕX(y)+ϕY(y)]φ(x)→S−ϕX(y)φ(2X−x)−ϕY(y)φ(2Y−x).$
(3)

Here, $φ(x)$ cannot be factored from the scattered state, which entangles the scatterer and particle states, and this is a key feature of decoherence.

Non-classical behavior is represented by interference of the two states in expressions for expectation values of Hermitian operators A

$⟨Φ|A|Φ⟩=12(⟨ϕX|A|ϕX⟩+⟨ϕY|A|ϕY⟩+r+r*),$
(4)

where we have defined the interference term $r=⟨ϕY|A|ϕX⟩$ and assumed that the expectation value of A does not depend on the state of the scatterer $|φ⟩$. We now show that the contribution of interference terms to the expectation value tends to zero in the presence of scattering when the expectation value does not depend on $|φ⟩$. The interference amplitude of the scattered state is evaluated as follows

$r→Sr∫−∞∞dx φ*(2Y−x)φ(2X−x)$
(5)
$=r∫−∞∞dy φ*(y)φ(2X−2Y+y)$
(6)
$=r∫−∞∞dp |φ̃(p)|2e2ip(X−Y)/ℏ,$
(7)

where we have used the properties of the Fourier transform of a complex convolution in Eq. (7) and $φ̃(p)$ is the momentum wavefunction, that is, the Fourier transform of $φ(x)$.

As an aside, Eq. (7) is a special case of the standard positional decoherence expression for environmental scattering4,5 when the scattering amounts to reflection, that is, the T-matrix satisfies $|T(p,q)|2=Rt δ(p+q)$, where $δ(p)$ is the delta function and R is the scattering rate.

The exponential in Eq. (7) can be expanded in a Taylor series as $e2iαp=1+2iαp−2α2p2+⋯$, giving

$r→Sr[1+2i(X−Y)⟨p⟩/ℏ−2(X−Y)2⟨p2⟩/ℏ2+⋯],$
(8)

where $⟨pn⟩$ denotes the expectation value of pn. Since scattering is assumed to be symmetric with respect to both sides of the particle, we have $⟨p⟩=0$. At a scattering rate R, a total of N = Rt independent scatterers will collide with the particle in time t,2 each with initial wavefunction $φ(x)$, so that r varies as

$r(t)=r(0)[1−2(X−Y)2⟨p2⟩/ℏ2]Rt$
(9)
$≈r(0)e−2Rt(X−Y)2⟨p2⟩/ℏ2,$
(10)

provided that $(X−Y)2⟨p2⟩/ℏ2$ is small. Therefore, if scattering events are independent, which typically occur when the environment consists of a large number of scatterers, interference of particle states decays approximately exponentially, at a rate proportional to the square of the separation for small separations. The effect in Eq. (4) is that after a time inversely proportional to $(X−Y)2$, only the first two terms contribute to the expectation value, and these are the expectation values of A with the particle at X or at Y but not both at the same time. This is what one would expect for a classical mixture of particles in states $|ϕX⟩$ and $|ϕY⟩$, with equal weight 1/2 for both states.

Anticipating a more advanced treatment, we note that the expectation value in Eq. (4) can be written as the trace of a product of matrices

$⟨Φ|A|Φ⟩=tr[(⟨ϕX|A|ϕX⟩⟨ϕX|A|ϕY⟩⟨ϕY|A|ϕX⟩⟨ϕY|A|ϕY⟩)(1/21/21/21/2)],$
(11)

where the second matrix is the density matrix representing the particle state in Eq. (2). It follows from the above paragraph that decoherence diminishes the off-diagonal elements of the density matrix so that with time it tends to a diagonal matrix. Such a density matrix cannot be obtained from a pure quantum state as in Eq. (2) but rather resembles a classical mixture of two quantum states of the particle.

In summary, we have shown that as a consequence of decoherence, the interference of a particle with itself can generally be neglected for macroscopic particles. Ultimately, this leads to a localization of the particle and the fact that classical particles have trajectories, that is, they appear to transition through a set of position eigenstates.5

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The wavefunction $φ(x+vt)−φ(2X−x+vt)=ψ(x,t)$ represents a non-dispersive wave-packet $φ(x)$ and its mirror image about x = X traveling in opposite directions. This solution satisfies both the initial condition $ψ(x,t)=φ(x+vt)$ for large negative times and x > X, and the boundary condition $ψ(X,t)=0$. The wave-packet $φ(x)$ is assumed to vanish at infinity.

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