Physical Chemistry 20130503 week 5 Friday May 3, 2013 page 1
The DeBroglie Hypothesis (1923)
-French physicist
-said matter should behave like waves
E=hν and E=mc2 so hν=mc2
c
=ν
λ
so
hc
=mc 2
λ
and
h
=mc
λ
λ=
h
mc
Matter can’t go the speed of light, so he modified that last formula for classical objects:
λ=
h
mv
DeBroglie' s equation
λ=
h
p
λ in m
h in Js
v in m⁄s
m in kg
p is linear momentum
λ is a wave property but m is a particle property so DeBroglie’s equation shows wave/particle duality.
Wave/particle duality was the topic of DeBroglie’s thesis. Einstein read it and said it was interesting, so
the thesis was approved.
Electrons are standing waves.
There are two kinds of waves:
1. travelling waves: position of crests and troughs are moving
2. standing waves: position of crests and troughs are stationary
2πr=nλ
r is radius
r=
h
mvr=n ( )
2π
nλ
2π
n=1,2,3…
angular momentum quantized
For macroscopic objects, λ is extremely small and unobservable.
me = 9.1*10-31kg = mass of electron
v = 1.0*106m/s = speed of electron in 1st orbit
λ = 7*10-10m=7Å detectable
macroscopic object:
m=1.0g
v=1.0cm/s
λ=7*10-27cm=7*10-19Å
For macroscopic objects, quantum effects are unobservable.
See uncertainty handout.
Electron diffraction shows electrons have wavelike properties. Neutron diffraction requires a nuclear
reactor.
vx is velocity along x axis.
px is momentum along x axis.
A͞C⟘ D͞P
A͞ P͞ =C͞ P͞
CD has to be .5λ for destructive interference (minimum intensity)
Condition for 1st diffraction minimum = D͞P - A͞ P͞ = .5λ = C͞ D͞
d>>w so α=0
⦟ACD=90°
w is slit width
⦟PAC=⦟P=90°
⦟PDE=⦟DAC=θ
1
D͞C͞ 2 λ λ
sinθ=
=
=
A͞ D͞ 1 w w
2
90°=⦟ADC+θ
λ is wavelength
θ=90°-⦟ADC
w is slit width
That’s for the first diffraction minimum, which happens at both points P and Q.
px=psinθ p is momentum
+θ or –θ
Δpx=psinθ-(-psinθ)=2psinθ
sin is odd function so sin(-θ)=-sin(θ)
cos is even function so cos(-θ)=cos(θ)
∆px =2p
∆px =2p
Δx=w
λ
w
use DeBroglie
h 2h
=
wp w
λ=
h
p
p is total momentum
w is very narrow (width of slit)
∆px =
2h
∆x
∆x∆px =2h
∆px ∆x≥
h
4π
different answers depending on how it ' s derived
h is Plank ' s constant (very small)
uncertainty in momentum times uncertainty in position is at least Plank’s constant
This is not due to measurement error or anything like that. It’s fundamental to the particle.
ΔxΔpx≥h
ΔyΔpy≥h
ΔzΔpz≥h
Bohr fascinated with orbits
Planets have definite orbits. An electron in a definite orbit would mean Δr=0 so Δp=∞ and Δv=∞.
r is radius of orbit, p is momentum of electron, v is velocity of electron
angular momentum:
mvr=n
1s shell has no orbiter angular momentum: l(lower case l)=0
h
2π