PHY 102: Quantum Physics

Topic 3De Broglie Waves

•Wave-particle duality

•De Broglie Waves

•Particles in boxes: Energy quantisation

•Quantisation of orbital angular momentum in Bohr model

Einstein’s postulate

A beam of light can be treated as a stream of particles (PHOTONS) with zero rest mass

Each photon has energy:

hc

hfEp

where h is a constant (Planck’s constant, h ≈ 6.63 x 10-34 Js)

f, λ, c, are frequency, wavelength and velocity of light (in vacuum) respectively.

Light intensity is proportional to PHOTON FLUX (no of photons passing through unit area per second)

2222 pcmcE

consequently, particle with zero rest mass (eg photon) has momentum pgiven by:

h

c

hf

c

Ep

From Special Theory of Relativity……………

Wave-particle Duality

So, an electromagnetic wave of wavelength λ and frequency f can be thought of as a stream of particles with energy E and momentum p given by:

hfE

h

p

The de Broglie Hypothesis

In 1924, de Broglie suggested that if waves of wavelength λ were associated with particles of momentum p=h/λ, then it should also work the other way round…….

A particle of mass m, moving with velocity v has momentum p given by:

h

mvp

Kinetic Energy of particle

m

k

m

h

m

pKE

222

22

2

22

If the de Broglie hypothesis is correct, then a stream of classical particles should show evidence of wave-like characteristics……………………………………………

De Broglie wavelength of everyday objects…

Eg. Tennis ball….

Mass ~60gVelocity ~ 100mph ≈ 45 m/s

Momentum =

De Broglie wavelength =

De Broglie wavelength of a 1keV electron..

KE =

Momentum =

De Broglie wavelength =

Observation of wave-like behaviour

Just like a classical wave, effects such as diffraction and interference observed when the wave interacts with objects with dimensions of the same order as the wavelength, ie

d~So, wave-like properties not observed for everyday macroscopic objects, which have de Broglie wavelengths ~10-34 m.

What about our electrons, with λ~ 10-10 – 10-11m……??

Electron Diffraction

Electron Diffraction

sin2dn

Standing de Broglie waves

Eg electron in a “box” (infinite potential well)

V=0

V= V=

Electron “rattles” to and fro

V=0

V= V=

Standing wave formed

Wavelengths of confined states

V=0

V= V=

L

LkL

; 2

LkL

2 ;

Lk

L 3 ;

3

2

Energies of confined states

V=0

V= V=

L

In general:

L

nk

n = 1, 2, 3, 4, 5, ……….

2

22222

22 mL

n

m

kE

ie QUANTISED ENERGY LEVELS

Example calculation 1

Calculate the separation between the two lowest energy states for an electron confined in an infinite potential well of width 1nm………

Example calculation 2

Calculate the separation between the two lowest energy states for an oxygen molecule confined in a 1cm cubic box

m=2.7 x 10-26 kg

When do quantisation effects become important?

Rule of thumb: at temperatures below which kT becomes comparable with ΔE.

For our confined electron (example 1), T~10000K

For our oxygen molecule, T~10-15K !!!!!!!!

So, quantisation effects easy to observe for electron, whereas the translational motion of the gas atom in the “normal sized” box obeys classical mechanics (continuous energy distribution)

(NB kT at room temperature (300K) is about 0.025eV)

The Bohr Model (1912-13)Bohr suggested that the electrons in an atom orbit the positively-charged nucleus, in a similar way to planets orbiting the Sun

(but centripetal force provided by electrostatic attraction rather that gravitation)

Hydrogen atom: single electron orbiting positive nucleus of charge +Ze, where Z =1:

r

v

F+Ze

-e

Failure of the Classical model

The orbiting electron is an accelerating charge.

Accelerating charges emit electromagnetic waves and therefore lose energy

Classical physics predicts electron should “spiral in” to the nucleus emitting continuous spectrum of radiation as the atom “collapses”

CLASSICAL PHYSICS CAN’T GIVE US STABLE ATOMS………………..

Quantisation of angular momentum

Bohr now makes the bold assumption that the orbital angular momentum of the electron is quantised………

Since v is perpendicular to r, the orbital angular momentum is just given by L = mvr.

Bohr suggested that this is quantised, so that:

nnh

mvr 2

IMPLICATIONS???..........................................................................

Electron standing waves and the Bohr Model

Bohr’s suggestion that orbital angular momentum of electrons is quantised is equivalent to the requirement that an integer number of de Broglie wavelengths must fit into the electron orbit:

Electron standing waves and the Bohr Model

nrn 2 nne

rvm

hn 2

nnen rvmL ne

n

ne vm

L

vm

hn

2

nnh

Ln 2