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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
1. Dwell time operator
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
1. Dwell time operator
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
1. Dwell time operator
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
1. Dwell time operator
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
1. Dwell time operator
kkδ2k
12mωωδ kk
29.9.2004, Weyer, OÖ
2kxcos12
1 cos(kx)xkcos :kk
xkkcosxkkcos2
1cos(kx)xkcos
with
Matrix elements of the dwell time operator
1. Dwell time operator
kxcosxkcosdxk)kδ(k
mCT,C
L/2
L/2
kDk
29.9.2004, Weyer, OÖ
Matrix elements of the dwell time operator
with
kLsink
1L2kxcos1dx
L/2
L/2-
1. Dwell time operator
L/2
L/2
kDk 2kxcos1dxk)kδ(k2
mCT,C
29.9.2004, Weyer, OÖ
kkδ2
1C,C kk
Matrix elements of the dwell time operator
and with
1. Dwell time operator
kLsin
k
1Lk)kδ(
k2
mCT,C kDk
29.9.2004, Weyer, OÖ
kkkkD Ct:C
kL
kLsin1
k
mLCT
Matrix elements of the dwell time operator
Therefore
1. Dwell time operator
kkkDk C,CkLsink
1L
k
mCT,C
29.9.2004, Weyer, OÖ
kkkkD St:S
kL
kLsin1
k
mLST
Matrix elements of the dwell time operator
Therefore
1. Dwell time operator
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
1. 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
2. Dwell time in Bohmian mechanics
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
2. Dwell time in Bohmian mechanics
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:
2. Dwell time in Bohmian mechanics
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].
2. Dwell time in Bohmian mechanics
29.9.2004, Weyer, OÖ
Freely moving Gaussian wave packet in one spatial dimension
D
2. Dwell time in Bohmian mechanics
-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
2. Dwell time in Bohmian mechanics
29.9.2004, Weyer, OÖ
Average Bohmian dwell time
L/2
L/2
2
tD xΨdxdtτ
2. Dwell time in Bohmian mechanics
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
3. Comparison of the two models
29.9.2004, Weyer, OÖ
Probability distribution for TD
Distribution function:
3. Comparison of the two models
with
otherwise0
tpif1(p)χ t
ktkktk
2
D tχttχt2
1kdkt)P(T
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