Download [-L/2 L/2].

54. Jahrestagung der ÖPG
Fachsitzung KTP
27.9.2004, Weyer
1. Dwell time operator
2. Dwell time in Bohmian mechanics
3. Comparison of the two models
29.9.2004, Weyer, OÖ
How does one define the time spent
by a system in a certain spatial region
within quantum mechanics?
29.9.2004, Weyer, OÖ
1. Dwell time operator
A simple model system
State vector Yt as solution to the free Schrödinger equation
in one spatial dimension

2 2
i Ψ t (x)  
Ψ t (x)  Ĥ Ψ t (x)
2
t
2m x
Projection operator onto the spatial interval [-L/2 L/2]
Ψ t (x) if x  [L/2, L/2]
χ L/2,L/2 x̂  Ψ t : x  
otherwise
 0
29.9.2004, Weyer, OÖ
1. Dwell time operator
Heuristic motivation for the average dwell time of a
quantum system
Probability of finding the quantum system at a fixed time t
in the spatial interval [-L/2 L/2]
χ L/2,L/2 x̂  Ψ t
2
L/2

 dx Ψ t x 
2
 L/2
Total mean time spent in [-L/2 L/2]

 dt χ L/2,L/2 x̂  Ψ t

2

  dt

L/2
 dx Ψ t x 
2
 L/2
29.9.2004, Weyer, OÖ
1. Dwell time operator
The associated dwell time operator of Damborenea etal. …
…in the Heisenberg picture


iĤt/
iĤt/
TD   dt e χ [  L/2,L/2]x̂  e

29.9.2004, Weyer, OÖ
1. Dwell time operator
…essentially self adjoint on a proper domain,
commutes with the Hamiltonian

e iĤt /  T̂D   dτ e iĤ  t  τ /  χ [  L/2,L/2]x̂  e iĤτ /  


iĤ  t  τ  / 
iĤ  t  τ  /  iĤt / 
iĤt / 


dτ
e
χ
x̂
e
e

T̂
e
[  L/2,L/2]
D


 [Ĥ, T̂D ]  0
29.9.2004, Weyer, OÖ
1. Dwell time operator
Common set of improper eigenvectors…
…spanning the subspace of even and odd wave functions (k>0)
1
Ck (x) 
cos(kx)
2π
resp.
1
Sk (x) 
sin(kx)
2π
with
Ĥ C k (x)  E k C k (x), Ĥ Sk (x)  E k Sk (x)
2 2
Ek 
k :  ωk
2m
29.9.2004, Weyer, OÖ
1. Dwell time operator
Matrix elements of the dwell time operator

L/2
iĤt/

 L/2
Ck , T̂D Ck  δ(ω
 dtk Cωk ,ke)
iĤt/



kxC k 
dx
cos
k
x
cos


χ
x̂
e
 [  L/2,L/2]

i  ω k  ω k  t
dt
e
C k , χ [  L/2,L/2]x̂  C k

with


L/2
2m
1) dx cosk x  coskx 
δ(ω

ω

k
k
δ ω k  ωk  
δk  k 
 2k  L/2
29.9.2004, Weyer, OÖ
1. Dwell time operator
Matrix elements of the dwell time operator
L/2
Ck , T̂D Ck
m

δ(k  k)   dx coskx  coskx 
k
 L/2
with
1
 cosk  k  x   cosk  k  x  
2
1


k  k : cosk x  cos(kx)  1  cos2kx 
2
coskx  cos(kx) 
29.9.2004, Weyer, OÖ
1. Dwell time operator
Matrix elements of the dwell time operator
L/2
Ck , T̂D Ck
m

δ(k  k)   dx 1  cos2kx 
2k
 L/2
with
L/2
1
-L/2dx 1  cos2kx   L  k sin kL
29.9.2004, Weyer, OÖ
1. Dwell time operator
Matrix elements of the dwell time operator
C k , T̂D C k
m
1



δ(k  k)  L  sin kL  
2k
k


and with
C k , C k
1
 δk  k 
2
29.9.2004, Weyer, OÖ
1. Dwell time operator
Matrix elements of the dwell time operator
C k , T̂D C k
m

k
1




L

sin
kL

  C k , C k
k


Therefore
mL
T̂D C k 
k
 sin kL  

1  kL  C k : t k C k
29.9.2004, Weyer, OÖ
1. Dwell time operator
Matrix elements of the dwell time operator
Sk , T̂DSk
m

k
1




L

sin
kL

  Sk  , Sk
k


Therefore
mL
T̂D Sk 
k
 sin kL  

1  kL  Sk : t k Sk
29.9.2004, Weyer, OÖ
1. Dwell time operator
Spectrum of the dwell time operator
mL
t 
k

k
1  sin q 
 sin kL 
1  kL   τ q 1  q 


with
q : kL
and
mL2
τ :

29.9.2004, Weyer, OÖ
1. Dwell time operator
Spectrum of the dwell time operator
29.9.2004, Weyer, OÖ
How is this notion of dwell time
related to a corresponding notion
within 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 only
given by the state vector Yt as solution to

2 2
i Ψ t (x)  
Ψ t (x)  Ĥ Ψ t (x)
2
t
2m x
but also by a trajectory in configuration space
Qt
which is assumed to represent the positions of an actual particle
29.9.2004, Weyer, OÖ
2. Dwell time in Bohmian mechanics
Pointer in a Schrödinger cat state
1 (x)  Z (y1 ,, yn )  2 (x)  Z (y1 ,, yn )
29.9.2004, Weyer, OÖ
2. Dwell time in Bohmian mechanics
Dynamics of the particle trajectories
Equation of motion:
 jt 
d
Q t    (Q t )
dt
 ρt 
with
  *


jt x    Ψ t x  Ψ t x 
m 
x



t x   Yt *Yt x 
29.9.2004, Weyer, OÖ
2. Dwell time in Bohmian mechanics
Probability in Bohmian mechanics
In an ensemble of quantum systems with wave function Yt, the positions
of the particles are distributed according to
ρ t x  dx  Ψ t
2
x dx
At every time t, t delivers a probability measure on configuration space.
This measure is transported by the flux of the Bohmian vector field
Φ t (Q 0 )  Q t
in the following way:
 ρ x dx   ρ x dx
0
X
t
Φ t (X)
29.9.2004, Weyer, OÖ
2. Dwell time in Bohmian mechanics
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 tD(Q0) is the duration,
the Bohmian particle with initial condition Q0
stays inside [-L/2 L/2].
29.9.2004, Weyer, OÖ
2. Dwell time in Bohmian mechanics
Freely moving Gaussian wave packet in one spatial dimension
tD
-L/2
L/2
29.9.2004, Weyer, OÖ
2. Dwell time in Bohmian mechanics
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
2. Dwell time in Bohmian mechanics
Average Bohmian dwell time

τ D   dt

L/2
dx
Ψ
t

2
x 
 L/2
29.9.2004, Weyer, OÖ
The same average does
not imply the same statistics!
29.9.2004, Weyer, OÖ
3. Comparison of the two models
Bohmian dwell time probability distribution
Distribution of Bohmian dwell times:
P(τ D  t) 
 dQ
2
0
Ψ0 (Q0 )
Q0 Χ t
with
Χt  Q0 τ D (Q0 )  t
29.9.2004, Weyer, OÖ
3. Comparison of the two models
Probability distribution for TD
The system‘s wave function

1
ikx


Ψ 0 x  
dk

k
e


2π 0

 dk  k  C x   i S x 
k
k
0
29.9.2004, Weyer, OÖ
3. Comparison of the two models
Probability distribution for TD
Distribution function:

 
 
1 





P(TD  t)   dk  k
tk  χ t tk  tk  χ t tk
2
2
with
if p  t
1
χ t (p)  
0 otherwise
29.9.2004, Weyer, OÖ
3. Comparison of the two models
Gaussian wave packet with <q>=2
29.9.2004, Weyer, OÖ
3. Comparison of the two models
Dwell time probability distribution
29.9.2004, Weyer, OÖ
3. Comparison of the two models
Gaussian wave packet with <q>=2
29.9.2004, Weyer, OÖ
3. Comparison of the two models
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.at
http://bohm-mechanics.uibk.ac.at
29.9.2004, Weyer, OÖ