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Modern Physics Letters B

Quantum system: Wave function, entanglement and the uncertainty principle

Alexey V. Melkikh

Published Mar 5, 2021• 8 citations
Publisher

Abstract

A definition of entanglement based on the overlap of wave functions of identical particles is proposed. A definition of a quantum system of particles related to their entanglement with each other and the environment is proposed. It is shown that in the general case, the uncertainty principle cannot be formulated for a system of quantum particles in the form of a single inequality.

DOI
10.1142/s0217984921502225

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A definition of entanglement based on the overlap of wave functions of identical particles is proposed. A definition of a quantum system of particles related to their entanglement with each other and the environment is proposed. It is shown that in the general case, the uncertainty principle cannot be formulated for a system of quantum particles in the form of a single inequality.
Abstract A new uncertainty relation (UR) is obtained for a system of N identical pure entangled particles if we use symmetrized observables when deriving the inequality. This new expression can be written in a form where we identify a term which explicitly shows the quantum correlations among the particles that constitute the system. For the particular cases of two and three particles, making use of the Schwarz inequality, we obtain new lower bounds for the UR that are different from the standard one.
This extended note deals with a pedagogical description of the entangled state of two particles, starting from first principles. After some general remarks about quantum mechanics and physical theories, the single particle case is discussed by defining state, uncertainty relations and wave function in the state space. The system of two particles is then considered, with its possible states, starting from the original papers by Einstein Podolsky Rosen and by Schroedinger. The quantum measurement problem is then introduced, together with its role in the entanglement state. Finally the orthodox solution and the relevant conclusions are drawn.
In quantum mechanics the situation is different because the uncertainty principle does not allow us to know the initial positions and momenta of the particles with desired accuracy. The individual particle is described by a wave packet. Since the wave packet is localized and spreading with time, it is impossible to distinguish two identical particles as their wave functions always overlap at any point in space. Quantum theory describes any object at energies below its excitation threshold with four degrees of freedom. The wave functions of all the states of a given system of identical particles must have the same symmetry; otherwise, the wave function of a state formally a superposition of states of different symmetry would be neither symmetrical nor antisymmetrical. Hence, for a system of identical particles, at high energies and low concentration, probable number of particles per quantum state will be very small. Wave functions of the particles are well separated.
We consider entanglement in a system with a fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle basis chosen. The nonfactorization due to (anti)symmetrization is naturally excluded by using the (anti)symmetrized basis or, equivalently, the particle number representation. The natural degrees of freedom in quantifying the identical-particle entanglement in a chosen single-particle basis are occupation numbers of different single-particle basis states. The entanglement between effectively distinguishable spins is shown to be a special case of the occupation-number entanglement.
The Robertson's formulation of the uncertainty relation is the most widely accepted form of the Heisenberg uncertainty relation (HUR). It gets modified when we consider it for entangled particles. But this formulation does not consider the measurement process itself. There are reformulation of the uncertainty relations called the generalized uncertainty relations by including the measurement process into the uncertainty relation. HUR gets modified for the case of entangled particles. Here it is shown that the limit of the Generalized Uncertainty Relation (GUR) also reduces for entangled particles. So, GUR also shows similar trend as that of the HUR for entangled particles. Also as entanglement increases, the uncertainty reduces and measurement becomes more precise.