54. Jahrestagung der ÖPG Fachsitzung KTP 27.9.2004, Weyer.

54. Jahrestagung der ÖPGFachsitzung KTP27.9.2004, Weyer

29.9.2004, Weyer, OÖ

1.Dwell time operator2.Dwell time in Bohmian mechanics3.Comparison of the two models

29.9.2004, Weyer, OÖ

How does one define the time spentby a system in a certain spatial regionwithin quantum mechanics?

29.9.2004, Weyer, OÖ

1. Dwell time operator

A simple model system

State vector t as solution to the free Schrödinger equationin one spatial dimension

(x)ΨH(x)Ψx2m

(x)Ψt

i tt2

22

t

Projection operator onto the spatial interval [-L/2 L/2]

otherwise0

L/2]L/2,[xif(x)Ψx:Ψxχ t

tL/2L/2,

29.9.2004, Weyer, OÖ

Heuristic motivation for the average dwell time of a quantum system

Probability of finding the quantum system at a fixed time tin the spatial interval [-L/2 L/2]

Total mean time spent in [-L/2 L/2]

L/2

L/2

2

t

2

tL/2L/2, xΨdxΨxχ

L/2

L/2

2

t

2

tL/2L/2, xΨdxdtΨxχdt


29.9.2004, Weyer, OÖ

The associated dwell time operator of Damborenea etal. …

t/Hi

L/2]L/2,[t/Hi

D exχedtT

…in the Heisenberg picture


29.9.2004, Weyer, OÖ

…essentially self adjoint on a proper domain, commutes with the Hamiltonian

/tHiD

/tHi/τtHiL/2]L/2,[

/τtHi

/τHiL/2]L/2,[

/τtHiD

/tHi

eTeexχedτ

exχedτTe

0]T,H[ D


29.9.2004, Weyer, OÖ

Common set of improper eigenvectors…

cos(kx)2π

1(x)Ck

…spanning the subspace of even and odd wave functions (k>0)

sin(kx)2π

1(x)Sk

with

resp.

(x)SE(x)SH(x),CE(x)CH kkkkkk

k2

2

k ω:k2m

E


29.9.2004, Weyer, OÖ

Matrix elements of the dwell time operator

kxcosxkcosdx)ωδ(ω

Cxχ,Cedt

Cexχe,CdtCT,C

L/2

L/2

kk

kL/2]L/2,[ktωωi

kt/Hi

L/2]L/2,[t/Hi

kkDk

kk

kxcosxkcosdx)ωδ(ωCT,CL/2

L/2

kkkDk

with


kkδ2k

12mωωδ kk

29.9.2004, Weyer, OÖ

2kxcos12

1 cos(kx)xkcos :kk

xkkcosxkkcos2

1cos(kx)xkcos

with



kxcosxkcosdxk)kδ(k

mCT,C

L/2

L/2

kDk

29.9.2004, Weyer, OÖ


with

kLsink

1L2kxcos1dx

L/2

L/2-


L/2

L/2

kDk 2kxcos1dxk)kδ(k2

mCT,C

29.9.2004, Weyer, OÖ

kkδ2

1C,C kk


and with


kLsin

k

1Lk)kδ(

k2

mCT,C kDk

29.9.2004, Weyer, OÖ

kkkkD Ct:C

kL

kLsin1

k

mLCT


Therefore


kkkDk C,CkLsink

1L

k

mCT,C

29.9.2004, Weyer, OÖ

kkkkD St:S

kL

kLsin1

k

mLST


Therefore


kkkDk S,SkLsink

1L

k

mST,S

29.9.2004, Weyer, OÖ

Spectrum of the dwell time operator

q

qsin1

q

1τ

kL

kLsin1

k

mLt k

with

2mL:τ kL:q and


29.9.2004, Weyer, OÖ

Spectrum of the dwell time operator


29.9.2004, Weyer, OÖ

How is this notion of dwell timerelated to a corresponding notionwithin Bohmian mechanics?

29.9.2004, Weyer, OÖ

2. Dwell time in Bohmian mechanics

Mathematical framework of the theory

In Bohmian mechanics the complete description of the system is not onlygiven by the state vector t as solution to

(x)ΨH(x)Ψx2m

(x)Ψt

i tt2

22

t

but also by a trajectory in configuration space

tQ

which is assumed to represent the positions of an actual particle

29.9.2004, Weyer, OÖ

Pointer in a Schrödinger cat state

)y,,(yZ(x))y,,(yZ(x) n12n11


29.9.2004, Weyer, OÖ

Dynamics of the particle trajectories

Equation of motion:

)(Qρ

jQ

dt

dt

t

tt

xΨx

xΨm

xj t*

tt

xx *ttt

with


29.9.2004, Weyer, OÖ

Probability in Bohmian mechanics

In an ensemble of quantum systems with wave function t, the positionsof the particles are distributed according to

dxxΨdxxρ2

tt

t0t Q)(QΦ

(X)Φ

t

X

0

t

dxxρdxxρ

At every time t, t delivers a probability measure on configuration space.This measure is transported by the flux of the Bohmian vector field

in the following way:


29.9.2004, Weyer, OÖ

The definition of dwell time

With the existence of world lines, the dwell time inside the spatial interval[-L/2 L/2] finds a straightforward definition within Bohmian mechanics:

The Bohmian dwell time D(Q0) is the duration,the Bohmian particle with initial condition Q0

stays inside [-L/2 L/2].


29.9.2004, Weyer, OÖ

Freely moving Gaussian wave packet in one spatial dimension

D


-L/2 L/2

29.9.2004, Weyer, OÖ

Calculation of Bohmian dwell time statistics

● Picking a relevant sample of initial configurations

● Calculating the Bohmian trajectory to each initial data

● Calculating the dwell time for each trajectory

● Weighing each trajectory according to the Bohmian probability measure


29.9.2004, Weyer, OÖ

Average Bohmian dwell time

L/2

L/2

2

tD xΨdxdtτ


29.9.2004, Weyer, OÖ

The same average doesnot imply the same statistics!

29.9.2004, Weyer, OÖ

Bohmian dwell time probability distribution

Distribution of Bohmian dwell times:

with

3. Comparison of the two models

t0 ΧQ

0

2

00D )(QΨdQt)P(τ

t)(QτQΧ 0D0t

29.9.2004, Weyer, OÖ

0

kk

0

ikx0

xSixCkdk

ekdk2π

1xΨ

Probability distribution for TD

The system‘s wave function


29.9.2004, Weyer, OÖ

Probability distribution for TD

Distribution function:


with

otherwise0

tpif1(p)χ t

ktkktk

2

D tχttχt2

1kdkt)P(T

29.9.2004, Weyer, OÖ


Gaussian wave packet with <q>=2

29.9.2004, Weyer, OÖ

Dwell time probability distribution


29.9.2004, Weyer, OÖ

Gaussian wave packet with <q>=2


29.9.2004, Weyer, OÖ

Dwell time probability distribution


29.9.2004, Weyer, OÖ

● J.A. Damborenea, I.L. Egusquiza, J.G. Muga and B. Navarro (2004), preprint: quant-ph/0403081

● A. M. Steinberg in Time in Quantum Mechanics J.G. Muga, R.Sala Mayato, I.L. Egusquiza (Eds.), Springer-Verlag, Berlin (2002)

http://bohm-mechanik.uibk.ac.athttp://bohm-mechanics.uibk.ac.at

54. Jahrestagung der ÖPG Fachsitzung KTP 27.9.2004, Weyer.

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