Quantum-mechanical backflow and scattering Gandalf Lechner joint work with Henning Bostelmann and Daniela Cadamuro
• 1969: First description of the effect [Allcock] [Bracken/Melloy] [Eveson/Fewster/Verch] [Palmero et. al.] Backflow • Quantum physics has some “strange”, counter-intuitive effects (tunneling, uncertainty relations ....) • “Backflow” is a (less known) quantum effect of this sort. History: • 1998: First quantitative analysis of backflow • 2003: Backflow as a quantum inequality • 2013: Suggestion for experimental observation 1
• 1969: First description of the effect [Allcock] [Bracken/Melloy] [Eveson/Fewster/Verch] [Palmero et. al.] Backflow • Quantum physics has some “strange”, counter-intuitive effects (tunneling, uncertainty relations ....) • “Backflow” is a (less known) quantum effect of this sort. History: • 1998: First quantitative analysis of backflow • 2003: Backflow as a quantum inequality • 2013: Suggestion for experimental observation 1
Backflow • Quantum physics has some “strange”, counter-intuitive effects (tunneling, uncertainty relations ....) • “Backflow” is a (less known) quantum effect of this sort. History: • 1998: First quantitative analysis of backflow • 2003: Backflow as a quantum inequality • 2013: Suggestion for experimental observation 1 • 1969: First description of the effect [Allcock] [Bracken/Melloy] [Eveson/Fewster/Verch] [Palmero et. al.]
• In classical physics, L t is non-increasing with growing t . Momentum and probability currents Setting: A force-free single particle in one dimension. • In a probabilistic description, suppose that at time t 0, the particle has momentum 0 (“right-mover”) with probability 1. • What is the time-dependence of the probability L t that the position of the particle is 0? • Analogy (Bracken/Melloy): “Waiting for the bus” • In quantum physics, not necessarily! • In this talk, only non-relativistic quantum mechanics 2
• In classical physics, L t is non-increasing with growing t . Momentum and probability currents Setting: A force-free single particle in one dimension. probability 1. • What is the time-dependence of the probability L t that the position of the particle is 0? • Analogy (Bracken/Melloy): “Waiting for the bus” • In quantum physics, not necessarily! • In this talk, only non-relativistic quantum mechanics 2 • In a probabilistic description, suppose that at time t = 0, the particle has momentum > 0 (“right-mover”) with
• In classical physics, L t is non-increasing with growing t . Momentum and probability currents Setting: A force-free single particle in one dimension. probability 1. • Analogy (Bracken/Melloy): “Waiting for the bus” • In quantum physics, not necessarily! • In this talk, only non-relativistic quantum mechanics 2 • In a probabilistic description, suppose that at time t = 0, the particle has momentum > 0 (“right-mover”) with • What is the time-dependence of the probability L ( t ) that the position of the particle is < 0?
• In classical physics, L t is non-increasing with growing t . Momentum and probability currents Setting: A force-free single particle in one dimension. probability 1. • Analogy (Bracken/Melloy): “Waiting for the bus” • In quantum physics, not necessarily! • In this talk, only non-relativistic quantum mechanics 2 • In a probabilistic description, suppose that at time t = 0, the particle has momentum > 0 (“right-mover”) with • What is the time-dependence of the probability L ( t ) that the position of the particle is < 0?
• In this talk, only non-relativistic quantum mechanics Momentum and probability currents Setting: A force-free single particle in one dimension. probability 1. • Analogy (Bracken/Melloy): “Waiting for the bus” 2 • In a probabilistic description, suppose that at time t = 0, the particle has momentum > 0 (“right-mover”) with • What is the time-dependence of the probability L ( t ) that the position of the particle is < 0? • In classical physics, L ( t ) is non-increasing with growing t . • In quantum physics, not necessarily!
Momentum and probability currents Setting: A force-free single particle in one dimension. probability 1. • Analogy (Bracken/Melloy): “Waiting for the bus” 2 • In a probabilistic description, suppose that at time t = 0, the particle has momentum > 0 (“right-mover”) with • What is the time-dependence of the probability L ( t ) that the position of the particle is < 0? • In classical physics, L ( t ) is non-increasing with growing t . • In quantum physics, not necessarily! • In this talk, only non-relativistic quantum mechanics
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: 3
Plots of probability distribution of position with increasing time: . 3
Plots of probability distribution of position with increasing time: . 3
4 bus has not passed already can increase with waiting time t ! In “bus picture”, this means that the probability L ( t ) that the
j H t x . fixed, write j e itH Mathematical setup 2 p 2 0 for all x x b) j contains only positive momenta, i.e. supp a) b , where Backflow is the fact that a p . p 1 • For example free Hamiltonian H 0 x is purely kinematical. When dynamics with Hamiltonian H is • j 5 • Normalized wave function ψ ∈ L 2 ( R ) has probability density ∣ ψ ( x )∣ 2 (for position) and probability current density j ψ ( x ) = i 2 ( ψ ′ ( x ) ψ ( x ) − ψ ( x ) ψ ′ ( x )) .
Mathematical setup Backflow is the fact that a 0 for all x x b) j contains only positive momenta, i.e. supp a) b , where 5 • Normalized wave function ψ ∈ L 2 ( R ) has probability density ∣ ψ ( x )∣ 2 (for position) and probability current density j ψ ( x ) = i 2 ( ψ ′ ( x ) ψ ( x ) − ψ ( x ) ψ ′ ( x )) . • j ψ is purely kinematical. When dynamics with Hamiltonian H is fixed, write j e − itH ψ ( x ) = ∶ j H ψ ( t , x ) . • For example free Hamiltonian ̃ ( H 0 ψ )( p ) = 1 ψ ( p ) . 2 p 2 ˜
Mathematical setup 5 • Normalized wave function ψ ∈ L 2 ( R ) has probability density ∣ ψ ( x )∣ 2 (for position) and probability current density j ψ ( x ) = i 2 ( ψ ′ ( x ) ψ ( x ) − ψ ( x ) ψ ′ ( x )) . • j ψ is purely kinematical. When dynamics with Hamiltonian H is fixed, write j e − itH ψ ( x ) = ∶ j H ψ ( t , x ) . • For example free Hamiltonian ̃ ( H 0 ψ )( p ) = 1 ψ ( p ) . 2 p 2 ˜ Backflow is the fact that a ) / ⇒ b ) , where � a) ψ contains only positive momenta, i.e. supp ˜ ψ ⊂ R + b) j ψ ( x ) > 0 for all x ∈ R
dtf t j H t 0 temp f • Current quantum field (quadratic form) subspace of momentum P (“right/left-movers”) superposition of high and low (positive) momentum. • Unboundedness below: take “backflow states” as • Unboundedness above: high momentum effect. below. , the quadratic form E J x E is unbounded above and For any x Lemma • Notation: E projection onto positive/negative spectral exist as operators. J H x dxf x j J f • But for test functions f , 6 ⟨ ψ, J ( x ) ψ ⟩ ∶ = j ψ ( x ) = i 2 ( ψ ′ ( x ) ψ ( x ) − ψ ( x ) ψ ′ ( x ))
• Current quantum field (quadratic form) subspace of momentum P (“right/left-movers”) superposition of high and low (positive) momentum. • Unboundedness below: take “backflow states” as • Unboundedness above: high momentum effect. below. , the quadratic form E J x E is unbounded above and For any x Lemma projection onto positive/negative spectral • Notation: E exist as operators. • But for test functions f , 6 ⟨ ψ, J ( x ) ψ ⟩ ∶ = j ψ ( x ) = i 2 ( ψ ′ ( x ) ψ ( x ) − ψ ( x ) ψ ′ ( x )) ⟨ ψ, J ( f ) ψ ⟩ ∶ = ∫ dxf ( x ) j ψ ( x ) , ⟨ ψ, J H temp ( f ) ψ ⟩ ∶ = ∫ dtf ( t ) j H ψ ( t , 0 )
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