Tokyo Tech. Akio HOSOYA. March 10, @KEK. A Pedagogical Introduction to Weak Value and Weak Measurement A rephrase of three box model Resch, Lundeen, and Steinberg, Phys. Lett. A 324, 125 (200 0). r. Double Slit Experiment. 1. Standard description
Tokyo Tech. Akio HOSOYA
March 10, @KEK
A Pedagogical Introduction to Weak Value and Weak MeasurementA rephrase of three box model Resch, Lundeen, andSteinberg, Phys. Lett. A324, 125 (2000)
1. Standard description
|Ψ〉 = λ|L〉 + ρ|R〉, λ, ρ ∈ R
positions of slits
rL= (d/2, 0)
rR= (−d/2, 0)
The probability amplitude to find a particle at r:
Ψ(r):=<r|Ψ>= λ〈r|L>+ ρ〈r|R〉
= λexp[ik|r−rL|] + ρexp[ik|r−rR|]
≈ λexp[ikr−iξ] + ρexp[ikr+iξ]
.where ξ =kxd/2r
P(r) = |Ψ(r)|2
= 1 + 2λρ cos2ξ
By the Born rule, the probability to find
a prticle at r is
２. Slit with width
Suppose the slits have length ℓ.
Let the (x,y) coordinates of the slits be
rL = (d/2, η, 0)
rR = (−d/2, η, 0)
Then the wave
≈ λexp[ikr−ikyη/r-iξ] + ρexp[ikr-ikyη/r+iξ]
fromrLandrRis superposed for
−ℓ/2 ≤ η ≤ ℓ/2 to give
where ϕ(y)=sin(yℓk/2r)/(yℓk/2r )
The probability distribution is
P(r) = |Ψ(r)|2
is product of the previous
the y-distribution of diffraction
|ϕ(y)|2=|sin(yℓk/2r)/(yℓk/2r ) |2
Suppose the left slit is slightly tilted by a small
angle so that the optical axis is shifted by α ,
while the right slit remains as before.
Then the probability amplitude to find a particle
at (x,y) becomes
Ψ(r) ≈ eikr[λe−iξφ(y − α) + ρeiξφ(y)]
The probability is
P(x, y) = |Ψ(r)|2
≈ λ2φ2(y − α) + ρ2φ2(y)
+2λρφ(y − α)φ(y) cos 2ξ
The tilt of the left slide slightly changes
the interference pattern in the x-y plane schematically as
For weak interaction i.e., small αthe intertference
pattern is only slightly modified.
Since the initial superposition shows up soley
through the interference pattern in
the x-direction, we can say that our weak
measurement changes the initial state only
The average of the y-coordinate for a fixed
x is gives by
< y >=∫dyyP(x,y)/∫dyP(x,y)
≈ α (λ2+λρ cos 2ξ)/(λ2+ρ2+2λρ cos 2ξ)
We have chosen |Ψ〉 = λ|L〉 + ρ|R〉
as the pre-selected state. The eigen state
of the position x <x| is post-selected.
The weak value of an observable A is defined
in general by
In particular, the weak value of the projection
operator to the left slit
PL := |L><L|
<PL >w= <x|L><L|Ψ>/<x|Ψ〉
We can see that the shift of the interference pattern
< y >
≈ α (λ2+λρ cos 2ξ)/(λ2+ρ2++2λρ cos 2ξ)
We can extract the real part of weak value
Re[<PL >w] by the average of the shift
< y > in the y-direction in the weak measurement.
This gives an information of
the initial state |Ψ〉only slightly changing the interference pattern in the x-direction,
i.e., characteristic feature of the initial state
Prepare the initial state |Ψ〉and post-select the state <x| for the observed system.
To get the weak value:
of an observable A in the system,
introduce the probe observable yand its eigen function ϕ(y) as a new degree of freedom.
For the system and probe, introduce a
H= gδ(t) A⊗Py
Pyshifts the y-coordinate.
In the previous DS model, g=α, A=|L><L|since only the left slit is tilted.
Aharonov, Albert and Vaidman, PRL 60,1351
Aharonov and Rohlich “Quantum Paradox”.
Ψ(r) =<r|exp[-i∫Hdt] |Ψ〉⊗|φ>
=eikr[λe−iξφ(y − α) + ρeiξφ(y)]
How can we interpret the weak value?
Consider the DW model in which the
<PL >w= <x|L><L|Ψ>/<x|Ψ〉
can be extracted from the interference
A general tendency is;
The positively (negatively )larger the <PL >wis, the more upwards (downwards) shifted. This suggests the more likely the particle come from the left (right) slit.
<PL >wis a measure of tendency coming from the left slit L which we retrospectively infer when the particle is found at x for a given initial state |Ψ〉.
The probabilistic interpretation
that <PL >wis the conditional probability
for the weak value is debatable.
However, it is consistent with the Kolomgorov
measure theoretical approach dropping the
positivity from the axioms but keeping the
Note that this interpretation is equivalent
is the joint probability for a particle to pass
through the left slit and then arrive at x by
the Beysian rule.
However, consistency does not imply
that it is compulsory.
In the double slit model ,we show the essential
feature of the weak measurement and how the
information of the initial state as the weak
value is extracted only slightly disturbing the
Point: introduction of new degree of freedom
(y-coordinate) and its interaction with the
system (tilted glass)