Conditional probabilities in quantum theory, and the tunneling time controversy

Abstract

It is argued that there is a sensible way to define conditional probabilities in quantum mechanics, assuming only Bayes's theorem and standard quantum theory. These probabilities are equivalent to the ``weak measurement'' predictions due to Aharonov {\it et al.}, and hence describe the outcomes of real measurements made on subensembles. In particular, this approach is used to address the question of the history of a particle which has tunnelled across a barrier. A {\it gedankenexperiment} is presented to demonstrate the physically testable implications of the results of these calculations, along with graphs of the time-evolution of the conditional probability distribution for a tunneling particle and for one undergoing allowed transmission. Numerical results are also presented for the effects of loss in a bandgap medium on transmission and on reflection, as a function of the position of the lossy region; such loss should provide a feasible, though indirect, test of the present conclusions. It is argued that the effects of loss on the pulse {\it delay time} are related to the imaginary value of the momentum of a tunneling particle, and it is suggested that this might help explain a small discrepancy in an earlier experiment.

11 pages, latex, 4 postscript figures separate (one w/ 3 parts)

References in corpus (6)

• How much time does a tunneling particle spend in the barrier region?
• Measurement of the Schrodinger wave of a single particle
• Sub-femtosecond determination of transmission delay times for a dielectric mirror (photonic bandgap) as a function of angle of incidence
• The Reality and Measurement of the Wavefunction
• Measurements, errors, and negative kinetic energy
• Complex--Barrier Tunnelling Times

Cited by in corpus (58)

• Complex weak values in quantum measurement
• Direct measurement of general quantum states using weak measurement
• How much time does a tunneling particle spend in the barrier region?
• The significance of the imaginary part of the weak value
• Quantum Advantage in Postselected Metrology
• Weak Values are Interference Phenomena
• The analogy between optical beam shifts and quantum weak measurements
• Sub-femtosecond determination of transmission delay times for a dielectric mirror (photonic bandgap) as a function of angle of incidence
• Quantum Averages of Weak Values
• Quantum theory of successive projective measurements
• Observing Dirac's classical phase space analog to the quantum state
• Extracting joint weak values with local, single-particle measurements
• On the propagation speed of evanescent modes
• Generalized shifts and weak values for polarization components of reflected light beams
• Superoscillations and tunneling times
• Negative quasiprobabilities enhance phase estimation in quantum-optics experiment
• Practical measurement of joint weak values and their connection to the annihilation operator
• Complex joint probabilities as expressions of determinism in quantum mechanics
• Measuring Non-Hermitian Operators via Weak Values
• Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables
• Strange Weak Values
• Conditions tighter than noncommutation needed for nonclassicality
• Demonstration of Weak Measurement Based on Atomic Spontaneous Emission
• Nonclassicality in Weak Measurements
• On the role of complex phases in the quantum statistics of weak measurements
• Nonclassical Properties of Coherent States
• Properties and Applications of the Kirkwood-Dirac Distribution
• Products of weak values: uncertainty relations, complementarity and incompatibility
• Tomography of many-body weak values: Mach-Zehnder interferometry
• Observation of the Decrease of Larmor Tunneling Times with Lower Incident Energy
• What is a quantum-mechanical 'weak value' the value of?
• Homodyne monitoring of post-selected decay
• Experimental simultaneous read out of the real and imaginary parts of the weak value
• Weak values of a quantum observable and the cross-Wigner distribution
• Quantum Mechanics of Successive Measurements with Arbitrary Meter Coupling
• An operational approach to indirectly measuring tunneling time
• Atom-optics knife-edge: Measuring narrow momentum distributions
• Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable
• Epistemically restricted phase space representation, weak momentum value, and reconstruction of quantum wave function
• Photon tunneling through absorbing dielectric barriers
• Kirkwood-Dirac classical pure states
• Quantum probability updating from zero prior (by-passing Cromwell's rule)
• Weak value of Dwell time for Quantum Dissipative spin-1/2 System
• Soliton-like solutions to the ordinary Schroedinger equation
• Possibility of the tunneling time determination
• How (Not) to Understand Weak Measurements of Velocities
• Interpreting weak value amplification with a toy realist model
• Superluminal tunneling times without superluminal signaling: Fading of the MacColl-Hartman effect at early times
• A strange weak value in spontaneous pair productions via a supercritical step potential
• Tunneling time from spin fluctuations in Larmor clock
• Measuring the time a tunnelling atom spends in the barrier
• A weak-value model for virtual particles supplying the electric current in graphene: the minimal conductivity and the Schwinger mechanism
• Weak value as an indicator of back-action
• The real and imaginary parts of a weak value appearing as back-actions via a post-selection
• Impact of a tunneling particle on the quantum state of a barrier: two-particle wave-packet model
• Field extension of real values of physical observables in classical theory can help attain quantum results
• New about the wave function,"Einstein's boxes'' and scattering a particle on a one-dimensional $\delta$-potential
• Space time symmetry in quantum mechanics