2016 Quantum Physics
Text: Sears & Zemansky, University Physics
www.masteringphysics.com
Lecture notes at
https://physics.tcd.ie/assets/documents/lecture_notes
TCD JF PY1P20 2016 J.B.Pethica
Lecture 1
Summary:
Classical mechanics and waves: Particle mechanics, Basic Maths, Electromagnetism, Waves, Diffraction, Thermodynamics
Quantum phenomena:
1. The Photoelectric effect
CLASSICAL CONCEPTS
~ 1900
(i.e. things you already know)
Particle Mechanics
Newton’s Laws of motion
Force F = mass × acceleration = m a
1
2
Kinetic energy E = mv 2
Momentum p = mv
Conservation of momentum. Conservation of energy in elastic collisions
Velocity v = dx/dt
Acceleration a = dv/dt = d2x/dt2
Work done by F moving from position x1 to x2
=
∫
x1
x2
Fdx
Periodic motion
e.g. Mass on spring
ω=
S&Z Ch. 14
- Simple Harmonic Motion (SHM)
Spring constant µ Restoring force F= -µx
F = ma = m d2x/dt2 = -µx i.e. An oscillation with amplitude A and (angular) frequency ω
Solution x = A cos ωt
µ
- the ‘resonant’ frequency
m
1
2
Total energy in SHM = µ A
2
More generally….
t + φ ) Phase angle φ
x = Acos(ω
N.B. Euler notation e
iθ
= cosθ + isin θ So oscillatory motion
x = A ei(ωt + φ) Charged Particles
S&Z Ch. 21, 23
Forces on charges F = e( E + v × B) (Lorenz force)
Force is in direction of electric field E, plus at right angles to the plane of velocity v
and magnetic field B (so B does not change v or KE)
Electric potential V (‘voltage’) E = - dV/dx e.g. Potential due to point charge e =
Work done moving charge e a dist. dx through field = F dx = eE dx
x2
i.e. moving through a potential difference V =
∫ Edx
changes energy by eV x1
e.g. = change in kinetic energy for a free electron. −e
4πε r
Waves
Frequency f
Wavelength λ
Angular frequency
ω = 2πf
Plane Waves ψ = Ae
Amplitude A
(
i kx − ω t
Phase velocity v
Wavenumber
)
2π
k=
λ
ω
= fλ =
k
Intensity (energy) ∝ A2 Non-dispersive - wave velocity is constant, independent of f, λ eg. Electromagnetic waves in vacuum – speed of light c
Dispersive – wave velocity varies with f, λ e.g. water waves – Surfing (!), pond surface
dω
Group Velocity u =
dk
Diffraction Maxima for path difference = nλ
S&Z Ch. 36
n = 0,1,2,3,...... d sin θ = nλ
d
θ
2d sin θ = nλ
θ
d
Normal incidence on plane apertures Scattering from multiple planes of atoms
(Bragg)
Relativity
E = mc
And
2
m = γ m0 =
‘rest’ mass m0
S&Z Ch. 37.7, 37.8 m0
1 − v 2 c2
E = p c +m c
2
2 2
2 4
0
Thermal properties S&Z Ch. 18.4
Equipartition of energy - kBT/2 per degree of freedom (mode) e.g. 1-D oscillator - kBT
(1/2 P.E. 1/2 K.E.)
Free particle in 3-D – 3kBT/2
Oscillator in 3-D - 3kBT
QUANTUM PHENOMENA
Classical physics has problems explaining some
experiments….
The distinction between classical concepts is blurred in many
important experiments. Phenomena may not be regarded as strictly
wave-like or particle-like.
Key observations are: Photo-electric effect, Compton effect, specific heats, black-body radiation, atomic spectra, electron diffraction….
Solving these led to a revolution in thinking:
photons, wave-particle duality, uncertainty principle
& more….
A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin;
P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr;
I. Langmuir, M. Planck, M. Skłodowska-Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C.T.R. Wilson, O.W. Richardson
1927 Solvay conference
The Photoelectric Effect
Evacuated tube, 2 electrodes
E: emitter, C: collector
Light incident on E, electrons
are emitted & travel to C
Current I in external circuit
depends on V
Note: polarity of V impedes arrival of photo-electrons:
“retarding or stopping potential”
The Photoelectric Effect: what is observed
(1)  I-V dependence (for a single frequency of light)
V = Vo gives I = 0 the “stopping potential”
implies a range of electron kinetic energies from 0 to KEmax, where KEmax = eVo
(2) linear dependence of I on light intensity,
BUT Vo is unchanged by intensity
i.e. intensity of light affects number of but not
energies of electrons
(3) no time delay (“instant” emission)
(4) AND….
An important light frequency dependence…...
The frequency dependence
Vo depends linearly on f
Write Vo ∝ (f - fo)
Note: cut-off frequency (fo) below which there is no current
All these observations are incompatible with Classical Physics…
Electrons in the emitter
Electrons in metal – held in a potential ‘well’
Highest lying electrons at energy depth φ
known as “work function”
φ
Classical view: electrons accumulate energy
from incident light waves!
Therefore KE should increase with light intensity cf (1) + (2) Also, should see time lag at low intensity cf (3) Should be no minimum frequency cf (4)
To solve this PROBLEM, Einstein (1905) borrows from Planck…..
Einstein model of photoelectric effect
Light is not waves but energy “packets” (later “photons”) each photon has energy hf = ω Planck’s constant h
hf
KEmax
Photoelectron is ejected (instantly) through the complete absorption of one photon.
hf = KE + (depth in well)
Consider the highest-lying electrons
hf = KEmax + φ
KEmax = hf – φ (recall: KEmax = eVo)
eVo = hf - φ
Vo = (h/e) f - (φ/e) = (h/e)(f - f o)
φ
hf
hν
“Despite then the apparently complete success of the Einstein equation, the
physical theory of which it was designed to be the symbolic expression is found
so untenable that Einstein himself, I believe, no longer holds to it……”
(Millikan)
Summary – photoelectric effect
(using ω for frequency, = h/2π )
Observe:
1. Electrons only emitted for ω
> ω0
2. Intensity of light affects the number of electrons but NOT their energy
3. Emitted electron max. KE = (ω − ω 0 )
Conclude:
a) Photon energy E
= ω
b) Work Function φ = ω 0 is the energy required to
extract an electron from the metal.