Download Wave Collapse Doesn`t Matter

Wave Collapse Doesn’t Matter
Chris Stucchio, Courant Institute
Joint work with Avy Soffer, Juerg Frohlich and Michael Sigal.
Quantum Mechanics - consensus
Wavefunction
State of the universe
ψ(x1 , . . . , xN , t)
xi
=
position of i’th particle
t
=
time
Probability Theory
Probability distribution of particle configurations
2
|ψ(x1 , . . . , xN , t)| dx1 . . . dxN
Evolution
i∂t ψ = Hψ
Schrodinger Equation
Physical Facts
• Suppose we allow the wavefunction evolves to a “split” state
ψ(x, T ) =
√
λφ(x − L) +
√
1 − λφ(x + L)
• Meaning of this wavefunction:
particle near x = L
particle near x = −L
with probability λ
with probability 1 − λ
Repeated “measurements” of particle
position will yield same result
Physical Facts
• In the absence of measurement, interference effects are observed.
• Split, recombine than measure:
P (x) = |φ1 (x)| + |φ2 (x)| + 2!φ1 (x)φ2 (x)
2
2
• Split, measure, recombine, then measure again:
P (x) = |φ1 (x)| + |φ2 (x)|
2
2
interference
term
Copenhagen Interpretation and Wave Collapse
“Textbook Quantum Mechanics”
How to predict outcome of experiments
• “Prepare” initial wavefunction.
ψ(x, 0) =
√
λφ(x − L) +
√
1 − λφ(x + L)
• Allow it to evolve under Schrodinger equation.
• “Measure” the position of the particle.
How to predict outcome of experiments
√
λφ(x − L) +
√
1 − λφ(x + L)
probability = λ
measurement
φ(x − L)
probability = 1 − λ
φ(x + L)
Problems with this interpretation
• Why do certain states of the universe constitute a measurement?
• Why are measurements special?
• Are there experiments which are not measurements?
Are all wavefunctions possible?
(Not normalized)
Decoherence
Dynamics in configuration space
Configuration space is really big.
Many particles moving a small distance adds up.
|(1, 1, . . . , 1) − (0, 0, . . . , 0)| =
√
3N
Measurements are not special
• Between measurements, the system remains on a low-dimensional
submanifold of configuration space.
• Measurements are interactions in which many degrees of freedom become
relevant.
• After measurement, different states are a distance
implies no interference, since:
2!ψ̄1 (x)ψ2 (x) ≈ 0
√
O( N ) apart. This
An unmeasured
interaction
An unmeasured
interaction
The measurement
process
The measurement
process
Interaction Switched On
A realistic example
The measurement process
• Want to measure the position of
a quantum particle.
• Measurement apparatus is a
many-body quantum fluid
(BEC), which interacts with
particle.
• Use conventional methods to
measure position of the splash.
The measurement process
• Want to measure the position of
a quantum particle.
• Measurement apparatus is a
many-body quantum fluid
(BEC), which interacts with
particle.
• Use conventional methods to
measure position of the splash.
The particle
is here.
Many Body Schrodinger equation


#
#
−∆
−∆
1
x
y
i∂t ψ(x, #y , t) = 
+
+
VpN (x − yj ) +
VsN (yi − yj ) ψ(x, #y , t)
2M
2m
2
j
i!=j
ψ0 (x, y) = φ(x)
N
!
χ(yj )
j=1
x =
yj
VPN (x − yj )
N
VS (yi
− yj )
Position of particle to be measured
= Position of j-th fluid particle
= Interaction between particle and fluid
= Internal fluid interaction
Pitaevskii Equation
t from
the many body
Schrodinger equation in the hydrodynam
Hydrodynamic
Formulation
n:
July 10, 2008
∂t ρ(x, "y , t) + ∇ · [ρ(x, "y , t)v(x, "y , t)] = 0
!hydrodynamic
"v(x,body
"y , the
t) +many
"y , t)Schrodinger
· ∇"v(x, "yequation
, t) = in −
(x, "y )
∂t "v(x,
Start
from
the∇V
for
mulation:
∂t ρ(x, "y , t) + ∇ · [ρ(x, "y , t)v(x, "y , t)] 2= 0
(1a)
ρ(x, "y , t) = |ψ(x, "y , t)|
! (x, "y )
"
"
v
(x,
"
y
,
t)
+
v
(x,
"
y
,
t)
·
∇"
v
(x,
"
y
,
t)
=
−
∇V
(1b)
∂
t
N
"
" −1
−1
−1
1
!
N∇ = (M
N
∇
,
m
∇
,
.
.
.
,
m
∇
),
with
x
!
y
!
y
1
N
"y ) =with Vp (x − "yj ) +
Vs ("yi − "yj )
2
vy particle
and
the
mass
j=1
i!=j of the light particle.
N
"
"
#
#
1
V (x, "y ) =
VpN (x − "yj ) +
VsN∆
("yxi − "yρ(x,
j)
"
y
,
t)
∆
ρ(x,
"
y
,
t)
y
2
#
+ #
j=1
i!=j+
#
M #ρ(x, "y , t)
m ρ(x, "y , t)
∆x ρ(x, "y , t) ∆y ρ(x, "y , t)
#
+
+ #
M−1ρ(x, "y , t)
m ρ(x, "y , t)
! = (M −1 ∇x , m−1 ∇!y , . . . , m
erator ∇
1
(1c)
∇!yN ), with M the mass
! = (M −1 ∇x , m−1 ∇!y , . . . , m−1 ∇!y ), with M the mass
he operator ∇
1
N
eavy particle and m the mass of the light particle.
Multiconfiguration Reduction
Ansatz
• Many-Body Schrodinger equation is hard. Solution: derive reduced equation.
We substitute
into (1) ρ(x,
the yfollowing
y1 , t) and vy (x, y1 , t)
• Reduced variables
1 , t), vx (x,ansatz:
ρ(x, "y , t) =
N
$
ρ(x, "yj , t)
j=1


N
"
N
#
"v(x, "y , t) =
Pj "v (x, "yj , t)

!v(x, , t) =
vx (x, yj , t), vy (x, y1 , t), . . . , vy (x, yN , t)
j=1
j=1
here "v : R3x ×R3y ×R+ → R3x ×R3y , and Pj is a matrix mapping the x comp
"v to the x component of "v and the y component of "v to the "yj compon
. To simplify notation, let "vx (x, y, t) be the x component of the velocit
+
N
Derivation of
k!=j
ρ(x, "yk , t) ρ(x, "yj , t)∇x ·


%
"vx (x, "yl , t)
l=1
*

 ∇yj · [ρ(x, "yj , t)"vy (x, "yj , t)]
+
ρ(x,
"
y
,
t)
k
reduced equation
k!=j
=0
(3)
or factoring out the product term and considering each individual term of the
sum separately,
"
(
)
N
!
∂t ρ(x, "yj , t) + ρ(x, "yj , t)
"vx (x, "yl , t) ∇x · ρ(x, "yj , t)
l=1
(
1
+ ρ(x, "yj , t)∇x · ρ(x, "yj , t)
N
N
!
)
"vx (x, "yl , t)
l=1
*
+ ∇yj · [ρ(x, "yj , t)"vy (x, "yj , t)]
The velocity equation can be rewritten as follows:
(
"N
*
N
!
!
∂t"vx (x, "yj , t) +
"vx (x, "yk , t) · ∇x"vx (x, "yj , t)
j=1
k=1
)
+ "vy (x, "yj , t) · ∇y "vx (x, "yj , t) = −
∇x
=0
V (x, "y )
(4)
(5a)
+
N
Derivation of
k!=j
ρ(x, "yk , t) ρ(x, "yj , t)∇x ·


%
"vx (x, "yl , t)
l=1
*

 ∇yj · [ρ(x, "yj , t)"vy (x, "yj , t)]
+
ρ(x,
"
y
,
t)
k
reduced equation
k!=j
=0
(3)
or factoring out the product term and considering each individual term of the
sum separately,
"
(
)
N
!
∂t ρ(x, "yj , t) + ρ(x, "yj , t)
"vx (x, "yl , t) ∇x · ρ(x, "yj , t)
l=1
(
1
+ ρ(x, "yj , t)∇x · ρ(x, "yj , t)
N
N
!
)
"vx (x, "yl , t)
l=1
*
+ ∇yj · [ρ(x, "yj , t)"vy (x, "yj , t)]
We equation
will reduce
this as follows:
The velocity
can be rewritten
(
"N
*
N
!
!
∂t"vx (x, "yj , t) +
"vx (x, "yk , t) · ∇x"vx (x, "yj , t)
j=1
k=1
)
+ "vy (x, "yj , t) · ∇y "vx (x, "yj , t) = −
∇x
=0
V (x, "y )
(4)
(5a)
∂t ρ(x, "yj , t) + ρ(x, "yj , t)
l=1
"vx (x, "yl , t) ∇x · ρ(x, "yj , t)
(
1
+ ρ(x, "yj , t)∇x · ρ(x, "yj , t)
N
Derivation of reduced equation
N
!
"vx (x, "yl , t)
l=1
*
+ ∇yj · [ρ(x, "yj , t)"vy (x, "yj , t)]
• Equation for velocities:
The velocity equation can be rewritten as follows:
(
"N
*
N
!
!
∂t"vx (x, "yj , t) +
"vx (x, "yk , t) · ∇x"vx (x, "yj , t)
j=1
)
k=1
= 0 (4)
)
∇x
+ "vy (x, "yj , t) · ∇yj "vx (x, "yj , t) = −
V (x, "y )
M
∂t"vy (x, "yj , t) +
"
N
!
k=1
2
*
"vx (x, "yk , t)
(5a)
· ∇x"vy (x, "yj , t)
∇!yj
+ "vy (x, "yj , t) · ∇y "vy (x, "yj , t) = −
V (x, "y )
m
Mean Field Approximation
(5b)
∂t ρ(x, "yj , t) + ρ(x, "yj , t)
l=1
"vx (x, "yl , t) ∇x · ρ(x, "yj , t)
(
1
+ ρ(x, "yj , t)∇x · ρ(x, "yj , t)
N
Derivation of reduced equation
N
!
"vx (x, "yl , t)
l=1
*
+ ∇yj · [ρ(x, "yj , t)"vy (x, "yj , t)]
• Equation for velocities:
The velocity equation can be rewritten as follows:
(
"N
*
N
!
!
∂t"vx (x, "yj , t) +
"vx (x, "yk , t) · ∇x"vx (x, "yj , t)
j=1
)
k=1
= 0 (4)
)
∇x
+ "vy (x, "yj , t) · ∇yj "vx (x, "yj , t) = −
V (x, "y )
M
∂t"vy (x, "yj , t) +
"
N
!
k=1
2
*
"vx (x, "yk , t)
(5a)
· ∇x"vy (x, "yj , t)
∇!yj
+ "vy (x, "yj , t) · ∇y "vy (x, "yj , t) = −
V (x, "y )
m
Mean Field Approximation
(5b)
Mean field for velocity
• We need not consider a general point in configuration space, only typical
ones.
• Probability distribution of fluid particle:
• Central limit theorem:
N
!
ρ(x, yj , t)
!
dyj
ρ(x, yj , t)dyj
"
vx (x, y, t)ρ(x, y, t)dy
"
vx (x, yj , t) → (N − 1)
ρ(x,
y,
t)dy
j=2
∂t ρ(x, "y , t) + ∇ · [ρ(x, "y , t)v(x, "y , t)] = 0
! (x, "y )
= −∇V
∂t "v(x, "y , t) + "v(x, "y , t) · ∇"v(x, "y , t)
Mean Field for Potential
with
(1a)
(1b)
N
"
"
1
V (x, "y ) =
VpN (x − "yj ) +
VsN ("yi − "yj )
2
j=1
i!=j
#
#
∆x ρ(x, "y , t) ∆y ρ(x, "y , t)
#
+
+ #
M ρ(x, "y , t)
m ρ(x, "y , t)
(1c)
! = (M −1 ∇x , m−1 ∇!y , . . . , m−1 ∇!y ), with M the mass of the
The operator ∇
1
N
• Similar tricks can be used for the potential:
heavy particle and m the mass of the light particle.
1 !
Ansatz
N
Vs (y1 − yj ) → (N − 1)
"
N
Vs (y1
"
We substitute into (1) the following ansatz:
j!=1
ρ(x, "y , t) =
N
$
j=1
− y)ρ(x, y, t)dy
ρ(x, y, t)dy
ρ(x, "yj , t)
(2a)
Probably Approximately Correct
• How accurate is this?
• Mcdiarmid’s inequality. For a vector of i.i.d. variables, if
sup |f (x1 , . . . , xi−1 , xi , xi+1 , xN ) − f (x1 , . . . , xi−1 , xi , x̂i+1 , xN )| < C
x,x̂i
• then:
!
2"
P (|f (!x) − E[f (!x)]| > ") ≤ 2 exp −
N C2
2
"
Probably Approximately Correct
• Implication:

N
#
$

N
V
(x − y)ρ(x, y)dy
p
N
$
Vp (x − yj ) − (N − 1)
P |
| ≥ N "
ρ(x,
y)dy
j=1
!
"
2!2 N
≤ 2 exp − N
||Vp (x)||L∞
• The probability distribution is w.r.t. conditional distribution of fluid particle:
ρ(x, y)
P (y) = !
ρ(x, y)dy
N
#
!vx (x, !yl , t) ≈ (N − 1)
!
!vx (x, y " , t)dP (y " )
Y-indepence of equations
l=2
"
!vx (x, y " , t)ρ(x, y " , t)dy "
"
= (N − 1)
ρ(x, y " , t)dy "
• Equations depend (to order O(N)) not on vx (x, y, t) but on it’s expected
Substituting this into (4) and considering only one term in the sum yields:
value:
$"
%
"
"
"
!vx (x, y , t)ρ(x, y , t)dy
"
∂t ρ(x, y, t) + (N − 1)(∇x · ρ(x, y, t))
d!yl
"
"
ρ(x, y , t)dy
%
$"
"
"
"
!vx (x, y , t)ρ(x, y , t)dy
"
+ (N − 1)ρ(x, y, t)∇x ·
d!yl
"
"
ρ(x, y , t)dy
+ ∇x · [ρ(x, y, t)!v (x, y, t)] + ∇y [ρ(x, y, t)!vy (x, y, t)] = 0
(7)
Thus, ρ(x, y, t) obeys a mean field equation depending only on itself and on
!v (x, y, t). We now derive a similar equation for !v (x, y, t). By the same argument
as above, the equation for !vx (x, y, t) becomes:
N
#
j=1
∂t!vx (x, !yj , t) + (N − 1)
$"
!vx (x, y , t)ρ(x, y , t)dy
"
ρ(x, y " , t)dy "
"
"
"
%
· ∇x!vx (x, !yj , t)
+ !vx (x, !yj , t) · ∇x!vx (x, !yj , t) + !vy (x, !yj , t) · ∇y !vx (x, !yj , t) = −∇x V (x, !y ) (8)
Y-indepence of equations
• Equations
depend
(to order
N on vx (x, y, t) but on it’s expected
Thus, (4)
becomes
(sinceO(N))
∇x Vnot
yi − !yj ) = 0):
s (!
value:
N
!
j=1
"
∂t!vx (x, !yj , t) + (N − 1)
#$
!vx (x, y , t)ρ(x, y , t)dy
$
ρ(x, y ! , t)dy !
!
!
!
%
· ∇x!vx (x, !yj , t)
&
+ !vx (x, !yj , t) · ∇x!vx (x, !yj , t) + !vy (x, !yj , t) · ∇y !vx (x, !yj , t)
=
'
N
!
j=1
(
∇x N
∇x
−
Vp (x − !yj ) −
Vq
M
M
In (9), Vq is the quantum pressure term.
3
Quantum Pressure
The only quantity left to evaluate is the Laplacian term:
(9)
5
Quasi-classical equations
Note
that inY-indepence
the MCGP, the mangling
terms "vx (x, y, t) · ∇x ( . . . ) are a factor
Partial
of equations
of N smaller than the average advection terms,
"
"vx (x, y ! , t)ρ(x, y ! , t)dy !
• Equations depend (to order
" O(N)) not on vx (x,
· ∇y,
. . but
. ). on it’s expected
x ( t)
!
!
ρ(x, y , t)dy
value.
To solve the MCGP, we can drop these small terms. Let "v˜x (x, t) denote the
average
velocity,
i.e.:consider expected value of x-velocity a reduced variable:
• We can
therefore
"
!
!
!
"
v
(x,
y
,
t)ρ(x,
y
,
t)dy
x
˜
"
"vx (x, t) = (N − 1)
(18)
!
!
ρ(x, y , t)dy
Now substitute ρ(x, y, t) = P 1/N (x, t)f (x, y, t). Differentiating the density equa• Derive
equation for this reduced variable by multiplying velocity equation by
tion
(7) yields:
probability distribution, and integrate by parts.
[∂t P (x, t)1/N ]f (x, y, t) + P (x, t)1/N [∂t f (x, y, t)]
1
+ [∇x · "v˜x (x, t)]P (x, t)1/N f (x, y, t) + "v˜x (x, t) · [∇x P (x, t)1/N ]f (x, y, t)
N
+ "v˜x (x, t) · [∇x f (x, y, t)]P (x, t)1/N + P (x, t)1/N ∇y [f (x, y, t)"vy (x, y, t)] = 0
(19)
Plug and chug
Plug and chug
Plug and chug
Plug and chug
To solve the MCGP, we can drop these small terms. Let "v˜x (x, t) d
average velocity, i.e.:
"
!
!
!
"
v
(x,
y
,
t)ρ(x,
y
,
t)dy
x
Reduced Equations
˜
"
"vx (x, t) = (N − 1)
ρ(x, y ! , t)dy !
1/N
Now
substitute
ρ(x,
y,
t)
=
P
(x, t)f (x, y, t). Differentiating the den
• One more substitution:
tion (7) yields:
•
1/N distribution of particle
1/Nposition
P (x, t)[∂t--PProbability
(x, t)
]f (x, y, t) + P (x, t)
[∂t f (x, y, t)]
1
1/N
1/N
˜
˜
+
[∇
·
"
v
(x,
t)]P
(x,
t)
f
(x,
y,
t)
+
"
v
(x,
t)
·
[∇
P
(x,
t)
]f (
x
x
x
• f (x, y, t) -- Fluidx distribution
assuming particle is at x.
N
+ "v˜x (x, t) · [∇x f (x, y, t)]P (x, t)1/N + P (x, t)1/N ∇y [f (x, y, t)"vy (x, y
• Note:
ρ(x, y, t)
f (x, y, t) = !
ρ(x, y, t)dy
7
N
Adding these equations together shows:
= ∂t"v˜x (x, t) + "v˜x (x, t) · ∇1
v˜x (x, t) 1/N
!
x"
˜x (x, t)]
+
P
(x,
t)
[∇
·
"
v
x
!
[∂t"vx (x, y, t) + "v˜x (x, t) · ∇x"vx (x, y, t) + "vy (x, y, t) · ∇
Ny"vx (x, y, t)]f (x, y, t)dy
+
=0
"vx (x, y, t)
Reduced
Equations
= ∂ "v˜ (x, t) + "v˜ (x, t) · ∇ "v˜ (x, t)
˜
"∂t f (x, y, t)
+ "vx (x, t) · ∇! x f (x, y, t) + ∇y [f (x, y, t)"vy (x, y, t)] = 0
#
× ∂t f (x, y, t) + ["v˜x (x, t) · ∇x f (x, y, t)] + [∇y · "vy (x, y, t) + "vy (x, y, t) · ∇y f (x, y, t)] d
t x
or (simplifying)
"
x
+
x x
"vx (x, y, t)
= ∂t"v˜x (x, t) + "v˜x (x, t) · ∇x"v˜x#(x, t)
× ∂t f (x, y, t) + ["v˜x (x, t) · ∇x f (x, y, t)] + [∇y · "vy (x, y, t) + "vy (x, y, t) · ∇y f (x, y, t)] dy
˜
∂
P
(x,
t)
+
∇
·
[
"
v
(x,
t)P
(x,
t)]
=
0
(20a)
x
x
The term inside the integral on thet second
to
last
line
is
zero
by
(20b).
Thus,
= ∂t"v˜x (x, t) + "v˜x (x, t) · ∇x"v˜x (x, t)
˜
we
find
that
multiplying
(21)
by
f
(x,
y,
t)
and
integrating
over
y
yields:
∂
f
(x,
y,
t)
+
"
v
(x,
t)
·
∇
f
(x,
y,
t)
+
∇
·
["
v
(x,
y,
t)f
(x,
y,
t)] = 0(20b)
t
x
x
y
y
The term inside the integral on the second to last line is zero by (20b). Thus,
!
" that multiplying (21) by f (x, y, t)#and integrating
we find
over
(N − 1)∇x y yields:
˜
˜
˜
By further
(x,
0)"
x,(xthen
y,y,t)"
∂t"assuming
vx (x, t) + "vxthat
(x, t)"f
·∇
vxy,
(x,
t) L2=(R,dy)
− = 1 for all[V
− y)"f
+ (x,
Vq (x,
t)]Lf2(x,
y
x"
(R,dy
!
M
"
#
(N
−
1)∇
1 for all t.˜
x
∂t"vx (x, t) + "v˜x (x, t) · ∇x"v˜x (x, t) = −
[V (x − y) + Vq (x, y, t)] f (x, y, t)dy
M velocity equation:
Now, consider the x-component of the
(22)
And lastly the equation for the y-velocity:
And lastly the equation for the y-velocity:
∂t"vy (x, y, t)˜ + "v˜x (x, t) · ∇x"vy (x, y, t) + "vy (x, y, t) · ∇y "vy (x, y, t)
∂t"vx∂(x,
y,
t)
+
"
v
(x,
t)
·
∇
"
v
(x,
y,
t)
+
"
v
(x,
y,
t)
·
∇
"
v
(x,
y,
t)
!
x
x
x
y
y
x
˜
vy (x,∇
y,yt) + "vx (x, t) · ∇x∇
"vyy(x, y, t) + "vy (x, y, t) · ∇y "vy (x, y, t)
t"
∇y
N
N
!
!
!
!
=∇
−y ∇Vp (x − y)∇+
(N∇− 1)
V1/2
−y,y t)
)f (x, y∇
,∇t)dy
−ρ1/2V
y,t)t)
q (x,
s ! (y
∆
ρ
(x,
∆
(x,
y,
y
y
N
N
!
!
x −N
x − y )f (x, y , t)dy − x V (x,
y y,m
m
m− 1) xVs (y
= −= −
Vp (x
y)
+
(N
t)
q
V
(x
−
y)
−
−
p
m
m
M
M 2 ρ1/2 (x, y, t)
Mm
m ρ1/2 (x, y, t) (23)
!
(23)
1/2
"
1/2
"
"
1/2
Note
we have
not
dropped
the
quantum
pressure.
ρ
(x,
y
,
t)∇
ρ
(x,
y
,
t)dy
2(Nthat
− 1)∇
∇
ρ
(x,
y,
t)
x
x
x
Note
pressure.
!
− that we have not dropped the quantum
·
M2
ρ1/2 (x, y, t)
ρ(x, y "" , t)dy ""
Scaling
• Work on finite box with fixed particle density, let box get bigger.
N/|Λ| = ρ0 , Λ ↑ R
3
• Scale particle and two-body fluid force with N:
M ∼ M N,
VsN (y − y ! ) = N −1 Vs (y − y ! )
• With this scaling, a long calculation shows that X-components of quantum
pressure also vanish.
• Scaling reasonable: physical examples have M=235 or M=720, m=4.
N
1
+ P (x, t)1/N [∇x · "v˜x (x, t)] = 0
N
Scaling
∂t f (x, y, t) + "v˜x (x, t) · ∇x f (x, y, t) + ∇y [f (x, y, t)"vy (x, y, t)] = 0
or (simplifying)
∂t P (x, t) + ∇x · ["v˜x (x, t)P (x, t)] = 0 (20a)
∂t f (x, y, t) + "v˜x (x, t) · ∇x f (x, y, t) + ∇y · ["vy (x, y, t)f (x, y, t)] = 0(20b)
By further assuming that "f (x, y, 0)"L2 (R,dy) = 1 for!all x, then "f (x, y, t)"L2 (R,dy
1 for all t.
∇x
˜
˜
˜
(∂Now,
vx (x,
t) + "vthe
t) · ∇x"vx (x,oft))
−
V (x − y)f (x, y, t)dy
t"
x (x,x-component
consider
the=velocity
equation:
M
˜x (x, t) ˜· ∇x"vx (x, y, t) + "vy (x, y, t) · ∇y "vx (x, y, t)
∂t"vx (x,∂y,
t)
+
"
v
vy (x, y, t) + "v (x, t) · ∇x"vy (x, y, t) + "vy (x, y, t) · ∇y "vy (x, y, t)
t"
!
1/2
1/2
∇
∇
∆
ρ
(x,
y,
t)
∇
∆
ρ
∇y x N
∇y x x
x ! y ∇y (x, y, t)
!
!
Vp−(x
−+y) − 2Vs (y1/2
−, t)dy − 1/2 Vq (x, y, t)
= −= − V
(x
y)
−
y
)f
(x,
y
M m ρ m(x, y, t)
mM
m M ρ !(x, y, t)
ρ1/2 (x, y " , t)∇x ρ1/2 (x, y " , t)dy "
2(N − 1)∇x ∇x ρ1/2 (x, y, t)
!
−
·
M2
ρ(x, y "" , t)dy ""
ρ1/2 (x, y, t)
Bohmian Coordinates
• Equation of characteristics:
q (x, t) = !v˜x (x, t)
!
q(x, 0) = x
• Result along characteristic:
∂t f (q(x, t), y, t) + ∇y · [f (q, y, t)"vy (q, y, t)] = 0
∂t"vy (q, y, t) + "vy (q, y, t) · ∇y "vy (q, y, t)
!
∇y
∇y
!
!
=−
V (q(x, t) − y) + Vs (y − y )f (x, y , t)dy −
Vq (q, y, t)
m
m
−∇y
q (x, t) =
M
!!
!
V (q(x, t) − y)f (q(x, t), y, t)dy
Equivalent Schrodinger Equation
• Equivalent to NLS coupled to a classical particle.
!
−1
i∂t Ψ(y, t) =
∆y + V (y − q(x, t)) +
2m
∇y
q (x, t) = −
M
!!
!
"
#
Vs (y − y ! )|Ψ(y ! , t)|2 dy ! Ψ(y, t)
V (y − q(x, t))|Ψ(y, t)|2 dy
Dynamics: Friction and stopping
Friction by Cerenkov Radiation
• Particle moves in fluid, and generates a wake behind it. Loss of energy to
wake slows the particle down, and is a frictional force.
• If the nonlinear forces are zero, we can prove rigorously that the particle stops
in the absence of nonlinear fluid forces. Numerical results confirm result is
true for nonlinear fluids.
• Decay rate:
|q (x, t)| ≤ C"t#
!
||∇y f (x, y, t) − ∇y f (x, y, t = ∞)||L3
−3/2
−1/2
≤ C"t#
)
Numerical Results
Repulsive Potential
Numerical results
• Particle eventually stops, but
oscillates around it’s stopping
point.
• Oscillation frequency and
decay rate can be calculated
(to leading order) by Laplace
transforms.
Attractive interactions
• The mass held by an attractive potential will grow without bound, unless
arrested by a repulsive nonlinearity.
• Regardless of M, the particle combined with the cloud of particles it attracts
will be O(N ).
• Semiclassical dynamics are achieved regardless of the mass of the particle!
Numerical Results
Attractive Potential
Key ideas of proof
q ! (x, t) to leading order as an linear integral equation
• Write equation for
(which is history dependent):
q !! (x, t)
= −
K(t, s)
=
!
t
K(t, s)q ! (x, s)ds + remainder
0
"
#
i∆(t−s)/2m
2ρ0
e
∂yz
" ∂yz V (y)|
V (y)
M
−∆/2m + V (y)
!
q
• Use dispersive estimates to show that (x, t) vanishes, and show remainder
does not cause problems.
• Transients appear to leading order in this framework. They can be calculated
by dropping the remainder, taking the Laplace transform and searching for
poles.
Decoherence
Bringing it back to the wavefunction
• Fix an initial state for the particle, with L larger than the stopping distance.
φ0 (x) =
√
λφ(x − L) +
√
1 − λφ(x + L)
• Initial wavefunction:
ψ0 (x, "y ) = φ0 (x)
N
!
j=1
χ0 (yj )
Bringing it back to the wavefunction
• Final wavefunction:
ψ(x, "y , t ≈ ∞)
=
√
λφ̃(x − L)
N
!
j=1
χ∞ (yj − L) +
• A “schrodingers cat” wavefunction.
√
1 − λφ̃(x + L)
N
!
j=1
χ∞ (yj + L)
A model for measurement
• Measurement consists of determining the state of the macroscopic system, in
y (or at least some function F (!y ) ).
this case a vector !
• From
this?
!y we infer a value for x. But with what statistical significance can we do
• This framework covers the instrumentalist picture, Bohmian Mechanics, and
most particle-based ontologies.
Statistical Significance
y , determine value of x.
• Consider measurement process: given knowledge of !
With what statistical significance can we answer this question?
• Partition configuration space
for !
y ∈ Ω1 and vice versa.
• Confidence level: P1 (!
y
dP1
R3N = Ω1 ∪ Ω2, and use the rule x ≈ −L
∈ Ω1 ) + P2 (!y ∈ Ω2 ) , where
=
N
!
j=1
dP2
=
N
!
j=1
|χ∞ (yj + L)|2 d"y
|χ∞ (yj − L)| d"y
2
Statistical Significance
• Choose Ω1 , Ω2 so that
an unbiased estimator.
P1 (!y ∈ Ω1 ) = P2 (!y ∈ Ω2 ) = p/2 to get
• This gives best possible decision procedure.
y ) rather than
• In the event we know only F (!
can only go down.
•
!y , our statistical confidence
F (!y ) models deterministic experimental errors, e.g. differences in !y which
are experimentally invisible.
Bounds on the interference:
• Interference term: 2!
N
!
j=1
• Bounds:
≤ ||
+||
χ̄∞ (yj + L)d"y
N
!
j=1
χ∞ (yj − L)
"
"
"
! "#
N
N
#
"
"
"
" d"y
χ
(y
+
L)
χ
(y
−
L)
∞
j
∞
j
"
"
"j=1
"
j=1
N
#
j=1
N
#
j=1
χ∞ (yj + L)||Ω1 ||
χ∞ (yj + L)||Ω2 ||
N
#
j=1
N
#
j=1
χ∞ (yj − L)||Ω1
χ∞ (yj − L)||Ω2
$
$
√
≤ p/21 + 1 p/2 = O( statistical confidence)
Statistical Significance and Interference
• The p-value of the experiment provides an upper bound on the size of the
interference term.
• Good experiments (statistically significant ones) destroy interference.
• Experimental prediction: “fractional measurements” are possible. A “fractional
measurement” is an experiment with large p-values which only partially
destroys interference.
The One-Pixel Camera
• Consider an experimental measurement consisting of counting the number of
fluid particles in a fixed region (the “pixel”).
• If splash is contained within pixel, average number of fluid particles observed
is different than if not. This provides a means of determining whether the
particle is within the pixel.
• Statistical significance: p=0.1% requires splash to involve 47 particles for
repulsive particle previously simulated.
• Thus, 47 fluid particles is sufficient to reduce interference to about 5% of the
total wavefunction.
The Wave Collapse
Approximation
• Suppose we make a
measurement, and the particle
is observed to be on the right.
• To simplify calculations, set left
wavepacket equal to zero.
• This is computationally simpler
than tracking both
wavepackets, and equally
accurate.
The Wave Collapse
Approximation
• Suppose we make a
measurement, and the particle
is observed to be on the right.
• To simplify calculations, set left
wavepacket equal to zero.
• This is computationally simpler
than tracking both
wavepackets, and equally
accurate.
Interpreting the results
• Instrumentalist picture: particles exist at the moment of measurement,
distributed according to the probability distribution. Statistical distribution of
configurations is consistent with wave collapse, regardless of whether or not
it occurs.
• Bohmian picture: particles exist for all time; in particular the particle we
measure follows the trajectory q(x,t). The wave collapse approximation does
not significantly alter q(x,t).
• GRW/Objective (Stochastic) Collapse: No comment.
Conclusion
• Derived multiconfiguration mean field model for quantum system consisting
of a particle interacting with a Bose gas.
• Reduced model to classical particle coupled to a Bose gas.
• Derived quantum friction, showing that the particle eventually stops.
• Showed that statistical significance of experimental outcomes provides upper
bound on quantum interference.
• Suggested possibilities for fractional measurements.
Thank you