Objective Collapse Induced by a Macroscopic Object

Abstract

The collapse of the wavefunction is arguably the least understood process in quantum mechanics. A plethora of ideas—macro-micro divide, many worlds and even consciousness—have been put forth to resolve the issue. Contrary to the standard Copenhagen interpretation, objective collapse models modify the Schrödinger equation with nonlinear and stochastic terms in order to explain the collapse of the wavefunction. In this paper we propose a collapse model in which a particle’s wavefunction has a possibility of collapsing when it interacts with macroscopic objects, without the intervention of a conscious observer. We propose four possible conditions of collapse of the wavefunction and make testable predictions which differ from standard quantum mechanics.

Appendix A Numerical Methods

We numerically solve the time-dependent Schrödinger equation for this system using the finite difference method by discretizing the range $[-30,30]$ of the position basis into 1,500 segments. We make this choice after checking that the wavefunction does not have appreciable magnitude beyond $x=\pm 30$. We construct the Hamiltonian matrix using the 8-th order central difference formula for the kinetic energy operator. The eigenvalues and eigenvectors of this matrix are computed using standard routines. The initial wavefunction is then decomposed into its components along the eigenvectors. The time-evolution is then trivial to compute

$$\begin{aligned} \vert {\psi (t)} \rangle = \sum _{n=0}^{N_{\textrm{cutoff}}} e^{-iHt/\hbar } \vert {\phi _n} \rangle \langle {\phi _n}\vert {\psi (0)}\rangle \end{aligned}$$

(A1)

For practical purposes, the infinite series is truncated at a cut-off energy such that the error is below machine precision. The first 150 eigenfunctions was found to be sufficient. We start from an initial wavefunction, which is a Gaussian wave-packet centered at $x=-5.0$, and standard deviation 1.0. This initial state corresponds, in the classical picture, to releasing the mass from the point $x=-5.0$, which would subsequently graze the wall located at $x=5.0$. The other parameters are taken as $m=1$, $k_1=1$, $k_2=10$. All quantities in this work are in units where $\hbar =1$.

For the first and third collapse postulates, the value of r is taken as 0.5. The post-collapse wavefunction is supposed to be an eigenfunction of the position operator, i.e., a delta function. However, the numerical routine would not work with such discontinuous functions. So we consider the post-collapse wavefunction to be a narrow Gaussian function of standard deviation 0.25 (Fig. 4). At the instant of collapse, the wavefunction is replaced with the collapsed wavefunction at the position dictated by the respective postulate. This is then decomposed into the energy eigenfunctions and its evolution is calculated using equation A1. The parameter values are taken as: mass of the quantum particle $m = 1$, spring constant of the spring attached to the mass $k_{1} = 1$, the spring constant corresponding to the soft wall $k_{2}=10$, time step $\delta t=0.1$. We calculate for a total number of 10,000 timesteps. In the case of the classical ensemble, 10,000 particles are initialized with zero velocity and positions drawn from a normal distribution with mean $-5$ and standard deviation 1. The dynamics is simulated using the classical equations of motion and integrated using the Runge–Kutta-Fehlberg method.