Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Determination of the Quantum State by Measurements

W. A. Gale, Eugene Guth, G. T. Trammell

Published Jan 25, 1968• 64 citations
Publisher

Abstract

A question raised by Pauli as to whether or not the probability densities in space and momentum determine the wave function is answered negatively. The assertion by Kemble that the probability density and its time derivative determine the wave function is shown to be not generally true. It is shown that measurement of the probability density and the probability current determine the wave function of a spinless particle. The measurement of the probability current is discussed. Measurements which determine the spin state and the density matrix of a mixture are also considered.

DOI
10.1103/physrev.165.1434

Related papers

Measuring the quantum mechanical wave function

M. G. Raymer

In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.

Contemporary Physics•Jan 1, 2009•32 citations

Measuring the quantum mechanical wave function

Michael G. Raymer

In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.

Contemporary Physics•Sep 1, 1997•86 citations

How to determine a quantum state by measurements: The Pauli problem for a particle with arbitrary potential

Stefan Weigert

The problem of reconstructing a pure quantum state \ensuremath{\Vert}\ensuremath{\psi}〉 from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval \ensuremath{\Delta}t later, \ensuremath{\Vert}\ensuremath{\psi}(x,t+\ensuremath{\Delta}t)${\mathrm{\ensuremath{\Vert}}}^{2}$, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, ${\mathcal{T}}^{\mathit{M}}$. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Apr 1, 1996•49 citations

Derivation of the Meaning of the Wave Function

Shan Gao

We show that the physical meaning of the wave function can be derived based on the established parts of quantum mechanics. It turns out that the wave function represents the state of random discontinuous motion of particles, and its modulus square determines the probability density of the particles appearing in certain positions in space.

•Nov 27, 2011•5 citations

A New Answer to Pauli's Question: Almost All Quantum States Can Be Uniquely Determined by Measuring Location and Momentum

Donald A. Jackson, Olga M. Kosheleva

In classical mechanics, we can uniquely reconstruct the state of each particle by measuring its spatial location and momentum. In his 1958 paper, W. Pauli, one of the founders of quantum mechanics, conjectured that the same should be true in the quantum case as well: that every quantum state can be uniquely determined by measuring spatial location and momentum. This conjecture was disproven: there are pairs of physically different states that cannot be distinguished if we only measure special location and momentum. A natural question is: how frequent are such pairs? In this paper, we show that almost all quantum states can be uniquely reconstructed by measuring spatial location and momentum. Thus, in practice, Pauli's conjecture is true.

Journal of Uncertain Systems•Jan 1, 2010•1 citation

Measurement of the Wigner Function

Antoine Royer

An operational procedure is given for determining experimentally, in principle at least, the Wigner function for an ensemble of particles. This manner of "measuring" a quantum state, whether pure or mixed, via its Wigner function, seems the simplest possible, and closely parallels the method one might use in classical mechanics to determine a (true) phase-space probability density.

Physical Review Letters•Dec 16, 1985•76 citations
A question raised by Pauli as to whether or not the probability densities in space and momentum determine the wave function is answered negatively. The assertion by Kemble that the probability density and its time derivative determine the wave function is shown to be not generally true. It is shown that measurement of the probability density and the probability current determine the wave function of a spinless particle. The measurement of the probability current is discussed. Measurements which determine the spin state and the density matrix of a mixture are also considered.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
The problem of reconstructing a pure quantum state \ensuremath{\Vert}\ensuremath{\psi}〉 from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval \ensuremath{\Delta}t later, \ensuremath{\Vert}\ensuremath{\psi}(x,t+\ensuremath{\Delta}t)${\mathrm{\ensuremath{\Vert}}}^{2}$, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, TM{\mathcal{T}}^{\mathit{M}}TM. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. \textcopyright{} 1996 The American Physical Society.
In classical mechanics, we can uniquely reconstruct the state of each particle by measuring its spatial location and momentum. In his 1958 paper, W. Pauli, one of the founders of quantum mechanics, conjectured that the same should be true in the quantum case as well: that every quantum state can be uniquely determined by measuring spatial location and momentum. This conjecture was disproven: there are pairs of physically different states that cannot be distinguished if we only measure special location and momentum. A natural question is: how frequent are such pairs? In this paper, we show that almost all quantum states can be uniquely reconstructed by measuring spatial location and momentum. Thus, in practice, Pauli's conjecture is true.