Post on 13-Sep-2020
Light and Matter Waves
λ =
h
p
M.C. Nemes
Escola Jorge Andre Swieca de Optica Quantica
February 11th 2008
SummaryPart I Part II Part III
M.C.Nemes 2 / 28
I. De Broglie: λ = h/p
◆ Questions raised
◆ Situation of answers today
SummaryPart I Part II Part III
M.C.Nemes 2 / 28
I. De Broglie: λ = h/p
◆ Questions raised
◆ Situation of answers today
II. Two aspects to consider:
◆ Interference: Classical Light waves ⇔ Quantum wavesCan we measure a difference?
SummaryPart I Part II Part III
M.C.Nemes 2 / 28
I. De Broglie: λ = h/p
◆ Questions raised
◆ Situation of answers today
II. Two aspects to consider:
◆ Interference: Classical Light waves ⇔ Quantum wavesCan we measure a difference?
◆ The Gouy phase:Classical Ligth Waves: well known (since 1890) in Optics
vsMatter Waves: is there an analog? An experiment?
SummaryPart I Part II Part III
M.C.Nemes 2 / 28
I. De Broglie: λ = h/p
◆ Questions raised
◆ Situation of answers today
II. Two aspects to consider:
◆ Interference: Classical Light waves ⇔ Quantum wavesCan we measure a difference?
◆ The Gouy phase:Classical Ligth Waves: well known (since 1890) in Optics
vsMatter Waves: is there an analog? An experiment?
III. Conclusions
Part I
Part I Part II Part III
IntroductionPart I Part II Part III
M.C.Nemes 4 / 28
λ =h
pL. de Broglie
Questions raised immediately:
IntroductionPart I Part II Part III
M.C.Nemes 4 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ is it true?Electrons diffraction
IntroductionPart I Part II Part III
M.C.Nemes 4 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ is it true?Electrons diffraction
IntroductionPart I Part II Part III
M.C.Nemes 4 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ why don’t we observe such phenomena with macroscopic objects?
IntroductionPart I Part II Part III
M.C.Nemes 4 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ why don’t we observe such phenomena with macroscopic objects?Bohr: Correspondence Principle
Quantum −→ Classical
Large quantum numbers
IntroductionPart I Part II Part III
M.C.Nemes 4 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ why don’t we observe such phenomena with macroscopic objects?Bohr: Correspondence Principle
Quantum −→ Classical
Large quantum numbers
... How large?
Experimental setupPart I Part II Part III
M.C.Nemes 5 / 28
Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, Gerbrand van der Zouw and
Anton Zeilinger, Nature 401, 680 (1999).
Experimental dataPart I Part II Part III
M.C.Nemes 6 / 28
Mass M ≈ 1.2 × 10−24 kgRadius R ≈ 3.5 × 10−10 mTemperature ΘF ≈ 900 KEnv. temperature ΘA ≈ 300 KAverage wave lenght λ ≈ 2.5 × 10−12 mAverage time of flight T ≈ 6 × 10−3 sGrating–screen distance L ≈ 1.25 mCollimator aperture a = 10−5 mEffective slit width b ≈ 3.6 × 10−8 mSlit spacing d = 10−7 m
Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, Gerbrand van der Zouw and
Anton Zeilinger, Nature 401, 680 (1999).
Experimental dataPart I Part II Part III
M.C.Nemes 6 / 28
Mass M ≈ 1.2 × 10−24 kgRadius R ≈ 3.5 × 10−10 mTemperature ΘF ≈ 900 KEnv. temperature ΘA ≈ 300 KAverage wave lenght λ ≈ 2.5 × 10−12 mAverage time of flight T ≈ 6 × 10−3 sGrating–screen distance L ≈ 1.25 mCollimator aperture a = 10−5 mEffective slit width b ≈ 3.6 × 10−8 mSlit spacing d = 10−7 m
Note that M ≈ 107 me.
Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, Gerbrand van der Zouw and
Anton Zeilinger, Nature 401, 680 (1999).
ResultsPart I Part II Part III
M.C.Nemes 7 / 28
I(x) =I02
sinc2
(
πbx
λL
)[
1 + cos
(
2πdx
λL
)]
sincx ≡ sinx
x
Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, Gerbrand van der Zouw and
Anton Zeilinger, Nature 401, 680 (1999).
IntroductionPart I Part II Part III
M.C.Nemes 8 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ How is the quantum–classical transition ?
IntroductionPart I Part II Part III
M.C.Nemes 8 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ How is the quantum–classical transition ?
“The transition occurs because of the action of the environment on theisolated quantum system under observation.”
D. Zeh, W. Zurek, J. P. Paz, ...
Losing coherencePart I Part II Part III
M.C.Nemes 9 / 28
Decoherence Diffusion factor, Λ (m2/s)
Scatteringthermal photons Λesp
ph ≈ 2,4 × 102
air molecules Λair . 3,2 × 1015
Photon emissionblack body radiation Λbb
ph ≈ 2,5 × 109
excitation decay Λvibph . 5,0 × 1013
Global effects Λ . 3,3 × 1015
Viale, A., Vicari, M. and Zanghi, N. Phys. Rev. A 68, 063610 (2003).
ResultsPart I Part II Part III
M.C.Nemes 10 / 28
As function of the distance:
Viale, A., Vicari, M. and Zanghi, N. Phys. Rev. A 68, 063610 (2003).
ResultsPart I Part II Part III
M.C.Nemes 10 / 28
As function of the pressure:
Viale, A., Vicari, M. and Zanghi, N. Phys. Rev. A 68, 063610 (2003).
ResultsPart I Part II Part III
M.C.Nemes 10 / 28
As function of the mass:
Viale, A., Vicari, M. and Zanghi, N. Phys. Rev. A 68, 063610 (2003).
IntroductionPart I Part II Part III
M.C.Nemes 11 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ Finally: “what is this wave?” (P. Debye)
IntroductionPart I Part II Part III
M.C.Nemes 11 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ Finally: “what is this wave?” (P. Debye)
Schrodinger equation (free particle):
i~∂
∂tΨ(~r, t) = − ~
2
2m∇2Ψ(~r, t)
⊕Interpretation
IntroductionPart I Part II Part III
M.C.Nemes 11 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ Finally: “what is this wave?” (P. Debye)
Schrodinger equation (free particle):
i~∂
∂tΨ(~r, t) = − ~
2
2m∇2Ψ(~r, t)
⊕Interpretation
Light wave in vacuum:
1
c2∂2
∂t2E(~r, t) = ∇2E(~r, t).
IntroductionPart I Part II Part III
M.C.Nemes 11 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ Finally: “what is this wave?” (P. Debye)
Schrodinger equation (free particle):
i~∂
∂tΨ(~r, t) = − ~
2
2m∇2Ψ(~r, t)
⊕Interpretation
Light wave in vacuum:
1
c2∂2
∂t2E(~r, t) = ∇2E(~r, t).
What are the similarities and differences between these waves?
IntroductionPart I Part II Part III
M.C.Nemes 11 / 28
λ =h
pL. de Broglie
Questions raised immediately:
■ Finally: “what is this wave?” (P. Debye)
Schrodinger equation (free particle):
i~∂
∂tΨ(~r, t) = − ~
2
2m∇2Ψ(~r, t)
⊕Interpretation
Light wave in vacuum:
1
c2∂2
∂t2E(~r, t) = ∇2E(~r, t).
What are the similarities and differences between these waves?Can one tell them apart experimentally?
Part II
Part I Part II Part III
Light vs matter wavesPart I Part II Part III
M.C.Nemes 13 / 28
It is well known the diffraction of a plane wave:
laserz
Light vs matter wavesPart I Part II Part III
M.C.Nemes 13 / 28
How is that for a matter wave?
t = z/v
C60
σ0
Light vs matter wavesPart I Part II Part III
M.C.Nemes 13 / 28
How is that for a matter wave?
t = z/v
C60
σ0
In principle, by changing σ0 one could change the interference pattern!
Simple ModelPart I Part II Part III
M.C.Nemes 14 / 28
û
3
x
z
L l
2σ0
2b
T t
d
Simple ModelPart I Part II Part III
M.C.Nemes 14 / 28
û
3
x
z
L l
2σ0
2b
T t
d
Ψ(x, 0) =
(
1
σ0√π
)1/2
exp
(
− x2
2σ20
)
Simple ModelPart I Part II Part III
M.C.Nemes 14 / 28
û
3
x
z
L l
2σ0
2b
T t
d
Ψ(x, T ) =
(
1
B(T )√π
)1/2
exp
[
− x2
2B2(T )
(
1 − iT
τ0
)
+ iφ(T )
]
Simple ModelPart I Part II Part III
M.C.Nemes 14 / 28
û
3
x
z
L l
2σ0
2b
T t
d
Ψ(x, T ) =
(
1
B(T )√π
)1/2
exp
[
− x2
2B2(T )
(
1 − iT
τ0
)
+ iφ(T )
]
τ0 = mσ20/~: matter wave characteristic!
Simple ModelPart I Part II Part III
M.C.Nemes 14 / 28
û
3
x
z
L l
2σ0
2b
T t
d
Ψ(x, T ) =
(
1
B(T )√π
)1/2
exp
[
− x2
2B2(T )
(
1 − iT
τ0
)
+ iφ(T )
]
B(T ) = σ0[1 + (T/τ)2]1/2
Simple ModelPart I Part II Part III
M.C.Nemes 14 / 28
û
3
x
z
L l
2σ0
2b
T t
d
Ψ(x, T ) =
(
1
B(T )√π
)1/2
exp
[
− x2
2B2(T )
(
1 − iT
τ0
)
+ iφ(T )
]
φ(T ) = arctan(T/τ)
“Aging” of the wave packetPart I Part II Part III
M.C.Nemes 15 / 28
We keep B2(T )σ2
0
fixed and artificially decrease σ0:
“Aging” of the wave packetPart I Part II Part III
M.C.Nemes 15 / 28
We keep B2(T )σ2
0
fixed and artificially decrease σ0:
Intensity
Position, x (µm)-40 -20 0 20 40
σ0 = 6 µm
“Aging” of the wave packetPart I Part II Part III
M.C.Nemes 15 / 28
We keep B2(T )σ2
0
fixed and artificially decrease σ0:
Intensity
Position, x (µm)-40 -20 0 20 40
σ0 = 0.02 µm
“Aging” of the wave packetPart I Part II Part III
M.C.Nemes 15 / 28
We keep B2(T )σ2
0
fixed and artificially decrease σ0:
Intensity
Position, x (µm)-40 -20 0 20 40
σ0 = 0.0175 µm
“Aging” of the wave packetPart I Part II Part III
M.C.Nemes 15 / 28
We keep B2(T )σ2
0
fixed and artificially decrease σ0:
Intensity
Position, x (µm)-40 -20 0 20 40
σ0 = 0.013 µm
The Gouy phasePart I Part II Part III
M.C.Nemes 16 / 28
Classical waves: in 1890, C.R. Gouy showed that a focused electromagneticbeam will acquire an additional π phase shift with respect to a plane wave asit evolves through its focus.
The Gouy phasePart I Part II Part III
M.C.Nemes 16 / 28
Classical waves: in 1890, C.R. Gouy showed that a focused electromagneticbeam will acquire an additional π phase shift with respect to a plane wave asit evolves through its focus.
Best known in the case of paraxial beams.
The Gouy phasePart I Part II Part III
M.C.Nemes 16 / 28
Classical waves: in 1890, C.R. Gouy showed that a focused electromagneticbeam will acquire an additional π phase shift with respect to a plane wave asit evolves through its focus.
Best known in the case of paraxial beams.
What is a paraxial beam?
The Gouy phasePart I Part II Part III
M.C.Nemes 16 / 28
Classical waves: in 1890, C.R. Gouy showed that a focused electromagneticbeam will acquire an additional π phase shift with respect to a plane wave asit evolves through its focus.
Best known in the case of paraxial beams.
What is a paraxial beam?
Wave equationPart I Part II Part III
M.C.Nemes 17 / 28
Helmholtz Equation:(
∇2 + k2)
E(~r) = 0 .
Wave equationPart I Part II Part III
M.C.Nemes 17 / 28
Helmholtz Equation:(
∇2 + k2)
E(~r) = 0 .
Electric Field:E(~r) = A(~r) eikz
A(~r) varies slowly within one laser wavelength λ = 2π/k.
Wave equationPart I Part II Part III
M.C.Nemes 17 / 28
Helmholtz Equation:(
∇2 + k2)
E(~r) = 0 .
Electric Field:E(~r) = A(~r) eikz
A(~r) varies slowly within one laser wavelength λ = 2π/k. Then
(
∂2A
∂x2+∂2A
∂y2
)
+∂2A
∂z2+ i2k
∂A
∂z= 0 .
Paraxial approximationPart I Part II Part III
M.C.Nemes 18 / 28
As A(~r) varies slowly in the direction of propagation z, then:
∂2A
∂z2� k
∂A
∂z
Paraxial approximationPart I Part II Part III
M.C.Nemes 18 / 28
As A(~r) varies slowly in the direction of propagation z, then:
∂2A
∂z2� k
∂A
∂z
Thus we find(
∂2
∂x2+
∂2
∂y2+ i4π
1
λL
∂
∂z
)
A(x, y, z) = 0 .
Paraxial approximationPart I Part II Part III
M.C.Nemes 18 / 28
As A(~r) varies slowly in the direction of propagation z, then:
∂2A
∂z2� k
∂A
∂z
Thus we find(
∂2
∂x2+
∂2
∂y2+ i4π
1
λL
∂
∂z
)
A(x, y, z) = 0 .
Compare with the Schrodinger equation for a free particle in 2D:
(
∂2
∂x2+
∂2
∂y2+ 2i
m
~
∂
∂t
)
ψ(x, y, t) = 0 .
Paraxial approximationPart I Part II Part III
M.C.Nemes 18 / 28
As A(~r) varies slowly in the direction of propagation z, then:
∂2A
∂z2� k
∂A
∂z
Thus we find(
∂2
∂x2+
∂2
∂y2+ i4π
1
λL
∂
∂z
)
A(x, y, z) = 0 .
Compare with the Schrodinger equation for a free particle in 2D:
OR
Paraxial approximationPart I Part II Part III
M.C.Nemes 18 / 28
As A(~r) varies slowly in the direction of propagation z, then:
∂2A
∂z2� k
∂A
∂z
Thus we find(
∂2
∂x2+
∂2
∂y2+ i4π
1
λL
∂
∂z
)
A(x, y, z) = 0 .
Compare with the Schrodinger equation for a free particle in 2D:
(
∂2
∂x2+
∂2
∂y2+ i4π
1
λp
∂
∂z
)
ψ(x, y, t = z/vz) = 0 .
Formally identical !
Guassian beamPart I Part II Part III
M.C.Nemes 19 / 28
Monocromatic beam in z = 0:
E(x′, y′, 0) = A0 exp
[
−x′2 + y′2
w20
]
Guassian beamPart I Part II Part III
M.C.Nemes 19 / 28
Monocromatic beam in z = 0:
E(x′, y′, 0) = A0 exp
[
−x′2 + y′2
w20
]
Propagating the field:
E(x, y, z) = A0w0
w(z)exp
[
−x2 + y2
w(z)2
]
exp
[
ikz + ik(x2 + y2)
2R(z)
]
exp[iζ(z)] .
Guassian beamPart I Part II Part III
M.C.Nemes 19 / 28
Monocromatic beam in z = 0:
E(x′, y′, 0) = A0 exp
[
−x′2 + y′2
w20
]
Propagating the field:
E(x, y, z) = A0w0
w(z)exp
[
−x2 + y2
w(z)2
]
exp
[
ikz + ik(x2 + y2)
2R(z)
]
exp[iζ(z)] .
where
w(z) = w0
[
1 +
(
z
z0
)2]1/2
and R(z) = z
[
1 +(z0z
)2]
ζ(z) = arctanz
z0is the Gouy phase and z0 =
kw20
2.
Guassian beamPart I Part II Part III
M.C.Nemes 19 / 28
Monocromatic beam in z = 0:
E(x′, y′, 0) = A0 exp
[
−x′2 + y′2
w20
]
Propagating the field:
E(x, y, z) = A0w0
w(z)exp
[
−x2 + y2
w(z)2
]
exp
[
ikz + ik(x2 + y2)
2R(z)
]
exp[iζ(z)] .
Recall the solution of the Schrodinger equation in 2D:
Ψ(x, y, T ) =1
B(T )√π
exp
[
−x2 + y2
B2(T )
]
exp
[
im(x2 + y2)
2~R(T )+ iφ(T )
]
AnalogiesPart I Part II Part III
M.C.Nemes 20 / 28
w(z) → B(T ) = σ0
[
1 +
(
T
τ0
)2]1/2
,
AnalogiesPart I Part II Part III
M.C.Nemes 20 / 28
w(z) → B(T ) = σ0
[
1 +
(
T
τ0
)2]1/2
,
R(z) → R(T ) = T
[
1 +
(
τ0T
)2]
,
AnalogiesPart I Part II Part III
M.C.Nemes 20 / 28
w(z) → B(T ) = σ0
[
1 +
(
T
τ0
)2]1/2
,
R(z) → R(T ) = T
[
1 +
(
τ0T
)2]
,
ζ(z) → ϕ(T ) = arctan(T/τ0) ,
AnalogiesPart I Part II Part III
M.C.Nemes 20 / 28
w(z) → B(T ) = σ0
[
1 +
(
T
τ0
)2]1/2
,
R(z) → R(T ) = T
[
1 +
(
τ0T
)2]
,
ζ(z) → ϕ(T ) = arctan(T/τ0) ,
z0 → τ0 = mσ20/~ .
AnalogiesPart I Part II Part III
M.C.Nemes 20 / 28
w(z) → B(T ) = σ0
[
1 +
(
T
τ0
)2]1/2
,
R(z) → R(T ) = T
[
1 +
(
τ0T
)2]
,
ζ(z) → ϕ(T ) = arctan(T/τ0) ,
z0 → τ0 = mσ20/~ .
Rayleigh length: “Aging” time:
z0 =πw2
0
λL⇔ τ0 =
mσ20
~
AnalogiesPart I Part II Part III
M.C.Nemes 20 / 28
w(z) → B(T ) = σ0
[
1 +
(
T
τ0
)2]1/2
,
R(z) → R(T ) = T
[
1 +
(
τ0T
)2]
,
ζ(z) → ϕ(T ) = arctan(T/τ0) ,
z0 → τ0 = mσ20/~ .
Rayleigh length: “Aging” time:
z0 =πw2
0
λL⇔ τ0 =
mσ20
~
What is the physical meaning ?
Uncertainty principlePart I Part II Part III
M.C.Nemes 21 / 28
■ Matter waves:
◆ Heisenberg’s: ∆x∆p ≥ ~
2.
Uncertainty principlePart I Part II Part III
M.C.Nemes 21 / 28
■ Matter waves:
◆ Heisenberg’s: ∆x∆p ≥ ~
2.
◆ Schrodinger-Robertson’s generalization:
Σ =
(
σMxx
12σ
Mxp
12σ
Mxp σM
pp
)
Uncertainty principlePart I Part II Part III
M.C.Nemes 21 / 28
■ Matter waves:
◆ Heisenberg’s: ∆x∆p ≥ ~
2.
◆ Schrodinger-Robertson’s generalization:
Σ =
(
σMxx
12σ
Mxp
12σ
Mxp σM
pp
)
σMxx =
⟨
(x− 〈x〉)2⟩
, σMpp =
⟨
(p− 〈p〉)2⟩
,
σMxp =
⟨
(x− 〈x〉)(p− 〈p〉) + (p− 〈p〉)(x− 〈x〉)⟩
Uncertainty principlePart I Part II Part III
M.C.Nemes 21 / 28
■ Matter waves:
◆ Heisenberg’s: ∆x∆p ≥ ~
2.
◆ Schrodinger-Robertson’s generalization:
detΣ = σMxxσ
Mpp − 1
4(σM
xp)2=
~2
4
quadratic evolution for any initial state!
Uncertainty principlePart I Part II Part III
M.C.Nemes 21 / 28
■ Matter waves:
◆ Heisenberg’s: ∆x∆p ≥ ~
2.
◆ Schrodinger-Robertson’s generalization:
detΣ = σMxxσ
Mpp − 1
4(σM
xp)2=
~2
4
quadratic evolution for any initial state!
■ Light waves: in analagous way we obtain
σLxxσ
Lkk − 1
4(σL
xk)2 =
1
4
Uncertainty principlePart I Part II Part III
M.C.Nemes 21 / 28
■ Matter waves:
◆ Heisenberg’s: ∆x∆p ≥ ~
2.
◆ Schrodinger-Robertson’s generalization:
detΣ = σMxxσ
Mpp − 1
4(σM
xp)2=
~2
4
quadratic evolution for any initial state!
■ Light waves: in analagous way we obtain
σLxxσ
Lkk − 1
4(σL
xk)2 =
1
4
σMxp =
T
τ0⇐⇒ σL
xk =z
z0
ThereforePart I Part II Part III
M.C.Nemes 22 / 28
By measuring σLxx and σL
kk we can obtain:
σLxk = ±
√
1
2
( w
w0
)2− 1
ThereforePart I Part II Part III
M.C.Nemes 22 / 28
By measuring σLxx and σL
kk we can obtain:
σLxk = ±
√
1
2
( w
w0
)2− 1
... and thenζ(z) = arctan(σL
xk)
ThereforePart I Part II Part III
M.C.Nemes 22 / 28
By measuring σLxx and σL
kk we can obtain:
σLxk = ±
√
1
2
( w
w0
)2− 1
... and thenζ(z) = arctan(σL
xk)
Varying beam waist:
wai
st(m
m)
z (mm)
Cor
rela
tion
σx
k
z − zc (mm)
ThereforePart I Part II Part III
M.C.Nemes 22 / 28
By measuring σLxx and σL
kk we can obtain:
σLxk = ±
√
1
2
( w
w0
)2− 1
... and thenζ(z) = arctan(σL
xk)
Varying beam waist:
Gouy’
sphas
e(r
ad)
z − zc (mm)
Matter wave diffractionPart I Part II Part III
M.C.Nemes 23 / 28
The apparatus:
Nairz, O., Arndt, M. and Zelinger, A. Phys. Rev. A 65, 032109 (2002).
Matter wave diffractionPart I Part II Part III
M.C.Nemes 23 / 28
Experimental data:
Matter wave diffractionPart I Part II Part III
M.C.Nemes 23 / 28
Experimental data: Theoretical predictions:
Matter wave diffractionPart I Part II Part III
M.C.Nemes 23 / 28
Experimental data: Theoretical predictions:
Part III
Part I Part II Part III
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
■ tell them apart in an interfer-ence experiment
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
■ tell them apart in an interfer-ence experiment
■ collimation dependence of thefinal pattern
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
■ tell them apart in an interfer-ence experiment
■ collimation dependence of thefinal pattern
■ not measured yet
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
■ tell them apart in an interfer-ence experiment
■ collimation dependence of thefinal pattern
■ not measured yet
■ learn from their similarity in theparaxial approximation
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
■ tell them apart in an interfer-ence experiment
■ collimation dependence of thefinal pattern
■ not measured yet
■ learn from their similarity in theparaxial approximation
■ Gouy’s phase for matter waves
ConclusionsPart I Part II Part III
M.C.Nemes 25 / 28
λ =h
p
■ tell them apart in an interfer-ence experiment
■ collimation dependence of thefinal pattern
■ not measured yet
■ learn from their similarity in theparaxial approximation
■ Gouy’s phase for matter waves
■ indirect indication of consis-tency
In progress:Part I Part II Part III
M.C.Nemes 26 / 28
Focusing lenses for matter interferometer:
EndingPart I Part II Part III
M.C.Nemes 27 / 28
λ =
h
p
Wave–particle duality
⇒ still a source of several fundamental questions
simple, nontrivial, intriguing physics!
CollaboratorsPart I Part II Part III
M.C.Nemes 28 / 28
■ I. G. da Paz
■ J.G. Peixoto de Faria
■ C. H. Monken
■ S. Padua