Engel & Reid
299-300
From Classical to
Quantum mechanics
vrije Universiteit amsterdam
Classical wave behaviour
Light is a wave
Two-slit experiment with photons (281-285)
1
One source
http://www.falstad.com/mathphysics.html
Interference 2 sources
http://www.falstad.com/mathphysics.html
One narrow slit
One wide slit
http://www.falstad.com/mathphysics.html
http://www.falstad.com/mathphysics.html
2 narrow slits
Two-slit experiment with light
http://www.falstad.com/mathphysics.html
2
Non-classical behaviour of light
Non-classical
behaviour of light
Two-slit experiment with photons (281-285)
Black body radiation (277- 278)
Energy density low temperature
Spectral energy density dE/df
Hot Steel and Pyrometer
dEω
ω2
hω
= 2 3 hω
dω π c e k BT − 1
1.5
P=5.67 x 10-8 T4 [W/m2]
1.0
300 K
600 K
900 K
1200 K
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Photon energy [eV]
1 eV energy is 2.417970×1014 Hz
The energy of photons is quantized
E = hf = hω
dEω
ω2
hω
= 2 3 hω
dω π c e k T − 1
Background blackbody radiation
ω2
hω
dEω
= 2 3 hω
dω π c e k B T − 1
Y Axis Title
B
T= 2.726 K
X A x is T itle
http://www.unidata.ucar.edu/staff/blynds/tmp.html#Uni
3
0.00001 K
variations
George F. Smoot
Non-classical behaviour of light
Two-slit experiment with photons (281-285)
Black body radiation (277- 278)
Photoelectric effect (279 - 280)
John C. Mather
Photoelectric effect 1
Photoelectric effect 2
0
0
Photoelectric effect 3
Photoelectric effect 4
0
Weak
negative
voltage
0
4
Experimental results
Photoelectric effect 5
+
Large
negative
voltage
Stopping potential (V)
20
_
Potassium
Tungsten
Potassium
Tungsten
10
h/e
0
h/e
0
-10
0
2x10
15
4x10
15
6x10
15
-1
Frequency of light (s )
Non-classical behaviour of light
Two-slit experiment with photons (281-285)
Black body radiation (277- 278)
Photoelectric effect (279 - 280)
X-ray production
Wilhelm Roentgen’s laboratory
X-rays
+
_
Wilhelm Roentgen’ photographs
http://www.fh-wuerzburg.de/roentgen/
http://www.fh-wuerzburg.de/roentgen/
5
Modern X-ray tube
Beryllium
window
Intensity
Target
X-ray spectra
Focus cup
Tungsten
filament
Photon energy (keV)
Non-classical behaviour of light
The positive electron
Two-slit experiment with photons (281-285)
Black body radiation (277- 278)
Photoelectric effect (279 - 280)
X-ray production
Pair production
Pair creation discovery
Bubble chamber
6
Pair creation simulation
Pair production
nucleus
positron
photon
Non-classical behaviour of light
electron
Compton’s paper
Two-slit experiment with photons (281-285)
Black body radiation (277- 278)
Photoelectric effect (279 - 280)
X-ray production
Pair production
Compton effect
Compton effect
λ′ − λ =
Assumption: photon has momentum h/λ
h
(1 − cos Θ)
me c
h
λ
h
λ
y
hc
λ
Θ
ϕ
x
=
=
h
λ'
h
λ'
λ′ − λ =
cos Θ + γmu cos ϕ
sin Θ − γmu sin ϕ
+ mc =
2
hc
λ'
+ γmc
h
(1 − cos Θ)
me c
y
2
Θ
x
ϕ
7
Non-classical behaviour of light
Two-slit experiment with one photon at a time
Two-slit experiment with photons (281-285)
Black body radiation (277- 278)
Photoelectric effect (279 - 280)
X-ray production
Pair production
Compton effect
Two-slit experiment
with one photon
(281-285)
Non-classical behaviour of light
Two-slit experiment with photons (281-285)
E = hf = hω
Black body radiation (277-278)
E = hf = hω
Photoelectric effect (279 - 280)
X-ray production
E = hf = hω
E = hf = hω
Pair production
h
p = mv = = hk
Compton effect
λ
Two-slit experiment
Single photon
with one photon
(281-285)
has still a wave
character
Conclusions for light
Depending on the type of
experiment light can be
described as a wave or as
particles
The light particles are
photons.
The energy of a photon with
an angular frequency ω is
The momentum of a photon
with wave vector k is
E = hω
p = hk
Non-classical behaviour of particles
Non-classical
behaviour of
particles
Two-slit experiment with
electrons (281-285)
Davisson and Germer
experiment (281)
Electron microscope
(lecture)
Neutron diffraction
(lecture)
8
Non-classical behaviour of particles
Two-slit experiment for electrons
Two-slit experiment with
electrons (281-285)
Two-slit : quantitative analysis
Two-slit : interference
Two-slit : destructive interference
Two-slit : constructive interference
Amplitude
9
Electrons+photons
Non-classical behaviour of particles
electrons
Ψ 2 ∝n
Two-slit experiment with
electrons (281-285)
Davisson and Germer
experiment (281)
photons
Davisson original paper
Detail of the original paper
Davisson-Germer experiment
Experimental results
θ
d
10
Davisson-Germer experiment 2
Davisson-Germer experiment 3
X
θ
θ
d
d
Electron scattering at the surface
Non-classical behaviour of particles
Two-slit experiment with
electrons (281-285)
Davisson and Germer
experiment (281)
Electron microscope
θ
(lecture)
d
d sin θ = nλ
Scanning electron microscope
Non-classical behaviour of particles
Two-slit experiment with
electrons (281-285)
Davisson and Germer
experiment (281)
Electron microscope
(lecture)
Neutron diffraction
λ=
h
2meV
(lecture)
11
Electron scattering in the bulk
Neutron diffraction
Neutrons have
also a wave
character
θ
D
2 D cos θ = nλ
Conclusions for particles
Two-slit experiment with
electrons (281-285)
Davisson and Germer
experiment (281)
Electron microscope
(lecture)
Neutron diffraction
(lecture)
E
ω=
h
p
k=
h
Guessing
Schrödinger’s
equation for a free
particle
What have we learned so far ?
Wave-particle duality
From wave to particle
p = mv =
Schrödinger
equation
h
λ
From particle to wave
= hk
E = hν = hω
λ=
h
p
k=
p
h
ν=
E
h
ω=
E
h
One would also like to have
E=
1 2
p2
mv particle =
2
2m
hω = E =
hω =
1 2
p 2 ( hk )
mv particle =
=
2
2m
2m
2
h2 2
k
2m
12
What are we looking for ? A wave so that:
This wave should be solution of a wave equation
hω =
h2 2
k
hω =
2m
ω2 = V2k 2
∂ 2u
∂ 2u
= V2 2
∂t 2
∂x
Ψ ∝n
2
α
h2 2
k
2m
αω = β k 2
∂u
∂2u
=β 2
∂t
∂x
Nota bene: this is only possible
with a complex wave ei(kx-ωt).
It does not work with sine or cosine
functions
The intensity of the
wave is proportional to
the density of particles
How to choose α and β ?
Insert ei(kx-ωt) into α
∂u
∂ 2u
=β 2
∂t
∂x
Heisenberg’s
uncertainty
principle
α ( −iω ) ei( kx −ωt ) = β ( ik )( ik ) ei( kx −ωt ) = − β k 2ei ( kx −ωt )
α ( −i ) = h
h2
β =−
2m
−
∂u
h2 ∂2u
= ih
2
2m ∂x
∂t
hω =
−
h2 2
k
2m
h 2 ∂ 2Ψ
∂Ψ
= ih
2m ∂x 2
∂t
As ei(kx-ωt) is a complex wave we
cannot simply take the square, but…
2
Ψ ∝n
Free particle wave function
Ψ ( x, t ) = Aei (kx −ωt )
Many
ei (kx −ωt )
are necessary to localize
a wave function
∆x
is a solution of Schrödinger’s equation
h 2 ∂ 2Ψ
∂Ψ
−
= ih
2
2m ∂x
∂t
∆k∆x ≥ 1
Ψ ( x, t ) = Aei (kx −ωt ) Ae − i (kx −ωt ) = A2
2
2
Ψ = constant
This is not realistic for a
particle
∆k
13
Heisenberg’s Uncertainty Principle
Heisenberg’s Uncertainty Principle
∆k∆x ≥ 1
∆ (h k )∆ x ≥ h
h
2
h
∆E ∆t ≥
2
∆p x ∆x ≥
∆p x ∆x ≥ h
3 times quantum physics
∆p∆x ≥ h
Electron
tunneling
Spin of the
electron
Heisenberg’s
uncertainty
principle
Many
ei (kx −ωt )
are necessary to localize
a wave function
The velocity of a particle is given by the group
velocity
Vg =
dω
dk
Vg =
dω
dk
hω =
h2 2
k
2m
hω =
h2 2
k
2m
d ω hk
=
=
dk
m
p mv
=
=
= v particle
m
m
Vg =
Atoms cannot
be vanishingly
small
d ω hk
=
=
dk
m
p mv
=
=
= v particle
m
m
Vg =
14
The group velocity is completely different
from the phase velocity
V phase =
hω =
ω
ω
k
dω
dk
hω =
h2 2
k
2m
2
h 2
k
2m
d ω hk
=
=
dk
m
p mv
=
=
= v particle
m
m
hk
=
k 2m
p
m v v particle
=
=
=
2m 2m
2
V phase =
Vg =
Vg =
=
Guessing
Schrödinger’s
equation for a
particle in a
potential
Potential energy
Total energy
F
Etotal=Ekinetic + U
x
= ½mv2 +½Kx2
v
x
x
F
U= ½Kx2
Force F=-Kx
U= ½Kx2
Work= ÛFdx
x
= - Potential energy U
x
How to introduce potential energy ?
−
Insert ei(kx-ωt) into
h ∂Ψ
∂Ψ
= ih
2m ∂x 2
∂t
2
2
gives
Stationary states
We look for solutions of the time-dependent
Schrödinger equation
h2 2
k = hω
2m
Kinetic
energy
Kinetic
energy
+
−
=
Total
energy
in the form
h 2 ∂ 2Ψ
∂Ψ
+ U ( x)Ψ = ih
∂t
2m ∂x 2
Ψ ( x, t ) = Φ ( x ) e
2
h 2
k + U ( x ) = hω
2m
−
h 2 ∂ 2Ψ
∂Ψ
+ U ( x)Ψ = ih
2m ∂x 2
∂t
−
−i
Et
h
and find
h 2 ∂ 2Φ
+ U ( x)Φ = EΦ
2m ∂x 2
This is the time-independent Schrödinger equation for
stationary states (standing waves)
15