Wave-Particle DualityPhotons behave like particlesParticles behave like photons?
Luis de Broglie (1924)Particle with momentum p=mv
should behave like a wave with wavelength
Experimental confirmation: Davisson, Germer (1927-1928): Diffraction of electrons off metal shows wave-like interference patterns
dBh hp mv
λ = = de Broglie wavelength
De Broglie wavelength and the classical limit
Plane Wave
( ) ( ) ( )ph
2 1( ) cos 2 cos v cos( ) Re . .2
i kx t i kx txf x A t A x t A kx t A e Ae c cω ωππ ν ωλ λ
− − = − = − = − = = +
A - amplitude
λ - wavelength
- wave vector2k πλ
=
2ω πν=
ν - cyclic frequency
- angular frequency
Complex conjugate, […]*
2h hkp kλ π
= = = Momentum (per de Broglie):
However, the plane wave is completely delocalized in space – how can it describe a particle?
Consider a sum (superposition) of several plane waves!
Two Planck constants:/ 2h π=
phvkωλν= = - phase velocity
1( ) cos( )
n
i i ii
f x A k x tω=
= −∑
Wavepacket: a Superposition of Plane WavesSuperposition (sum) of several plane waves:
Note: (Almost) any function f(x) can be represented as a sum of cosine (or sine) waves with different ki – a.k.a. Fourier series.
One plane wave n=1 1 1.5 (a.u.)p k= =
1( ) cos( )
n
i i ii
f x A k x tω=
= −∑
Wavepacket: a Superposition of Plane WavesSuperposition (sum) of several plane waves:
Note: (Almost) any function f(x) can be represented as a sum of cosine (or sine) waves – a.k.a. Fourier series.
Two plane waves n=2 1
2
1.5 (a.u.) 1.6 (a.u.) p k
k= == =
Δx Δp
1( ) cos( )
n
i i ii
f x A k x tω=
= −∑
Wavepacket: a Superposition of Plane WavesSuperposition (sum) of several plane waves:
Note: (Almost) any function f(x) can be represented as a sum of cosine (or sine) waves – a.k.a. Fourier series.
Three plane waves n=3 1
2
3
1.5 (a.u.) 1.6 (a.u.) 1.7 (a.u.)
p kkk
= == == =
ΔxΔp
1( ) cos( )
n
i i ii
f x A k x tω=
= −∑
Wavepacket: a Superposition of Plane WavesSuperposition (sum) of several plane waves:
Note: (Almost) any function f(x) can be represented as a sum of cosine (or sine) waves – a.k.a. Fourier series.
Four plane waves n=41
2
3
4
1.5 (a.u.) 1.6 (a.u.) 1.7 (a.u.) 1.8 (a.u.)
p kkkk
= == == == =
Δx
Δp
1( ) cos( )
n
i i ii
f x A k x tω=
= −∑
Wavepacket: a Superposition of Plane WavesSuperposition (sum) of several plane waves:
Note: (Almost) any function f(x) can be represented as a sum of cosine (or sine) waves – a.k.a. Fourier series.
Five plane waves n=51
2
3
4
5
1.4 (a.u.) 1.5 (a.u.) 1.6 (a.u.) 1.7 (a.u.) 1.8 (a.u.)
p kkkkk
= == == == == =
Δx
Δp
1( ) cos( )
n
i i ii
f x A k x tω=
= −∑
Wavepacket: a Superposition of Plane WavesSuperposition (sum) of several plane waves:
Note: (Almost) any function f(x) can be represented as a sum of cosine (or sine) waves – a.k.a. Fourier series.
Ten plane waves n=10 1
2
3
4
5
6
7
8
9
10
1.3 (a.u.) 1.4 (a.u.) 1.5 (a.u.) 1.6 (a.u.) 1.7 (a.u.) 1.8 (a.u.) 1.9 (a.u.) 2.0 (a.u.) 2.1 (a.u.) 2.2 (a.u.)
p kkkkkkkkkk
= == == == == == == == == == =
Δx
Δp
Wavepacket is a Superposition of Plane WavesAs we add more and more waves: The wavepacket becomes more localized in space, but the momentum (i.e., wavevector and/or wavelength) becomes less well defined.
Δx~50
p
Δp~0.1n=2
n=10
p
Δp~1
Δx~5
The Heisenberg’s Uncertainty Principle
22
22
x
p
x x x
p p p
σ
σ
∆ = = −
∆ = = −
Mathematical definition of uncertainty:(Standard deviation)
Werner Heisenberg, 1927
Fifth Solvay Conference (1927)Einstein (disenchanted with Heisenberg's Uncertainty Principle): "God does not play dice" Bohr: "Einstein, stop telling God what to do"
2x p∆ ⋅∆ ≥
341.054572 10 J s2hπ
−≡ = × ⋅
- Planck’s constantPosition and momentum are conjugated variables
Another pair of conjugated variables: Energy and time
– also connected by an uncertainty principle
2E t∆ ⋅∆ ≥
The Wavefunction (in coordinate space)
( , )x tΨIn quantum mechanics, the wavefunction replaces classical mechanical trajectory x(t).
Ψ(x,t) is a complex-valued function (to conveniently describe the wave-like behavior).
Born interpretation:
Max Born
2( , ) ( , ) ( , ) ( , )P x t x t x t x t∗= Ψ = Ψ ΨP(x,t) is the probability density of finding the particle at position x at time t
The wavefunction Ψ(x,t) contains complete knowledge about the quantum-mechanical system.
Ψ(x,t) obeys quantum-mechanical Equation of Motion (EOM) –the Schrödinger Equation
How does one find Ψ(x,t) ?
Normalization:Because ( , ) 1P x t dx
+∞
−∞
=∫ , the wavefunction must satisfy the normalization condition:
2( , ) ( , ) ( , ) 1x t d x x t x t d x+∞ +∞
∗
−∞ −∞
Ψ = Ψ Ψ =∫ ∫
Born conditions for an acceptable wavefunction:
(1) Must be finite everywhere(2) Must be single-valued(3) Must be continuous(4) Derivative must be continuous*
*except for points where potential V(x) is infinitex
( )xψ
x
( )xψddxψ
x
(2)
(3)(4)
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