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)