2017 Quantum Physics

Text: Sears & Zemansky, University Physics www.masteringphysics.com

Lecture notes at

www.tcd.ie/Physics/study/current/undergraduate/lecture-notes/py1p20

TCD JF PY1P20 2017 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

Momentum p = mv Kinetic energy

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 =

�

E =12mv 2

Fdx

x2

x1∫

Periodic motion - Simple Harmonic Motion (SHM) S&Z Ch. 14

e.g. Mass on spring Spring constant µ Restoring force F= -µx

F = ma = m d2x/dt2 = -µx Solution x = A cos ωt

i.e. An oscillation with amplitude A and (angular) frequency ω

- the ‘resonant’ frequency

Total energy in SHM =

More generally…. Phase angle φ

N.B. Euler notation So oscillatory motion x = A ei(ωt + φ)

x = Acos(ωt + φ)

ω =

µm

eiθ = cosθ + isinθ

12µA2

Charged Particles S&Z Ch. 21, 23

Forces on charges (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

i.e. moving through a potential difference changes energy by eV

e.g. = change in kinetic energy for a free electron.

�

F = e E + v × B( )

=

−e4πεr

V = Edx

x1

x2

∫

Waves

Frequency f Wavelength λ Phase velocity v

Angular frequency Wavenumber

Plane Waves

Amplitude A 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

Group Velocity

�

= fλ =ωk

�

ω = 2πf

�

k =2πλ

�

u =dωdk

ψ = Aei kx−ω t( )

Diffraction S&Z Ch. 36

Maxima for path difference = nλ n = 0,1,2,3,......

Normal incidence on plane apertures Scattering from multiple planes of atoms (Bragg)

d

θ d

2d sinθ = nλ

�

d sinθ = nλ

θ

Relativity S&Z Ch. 37.7, 37.8

‘rest’ mass m0

And

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

E2 = p2c2 + m0

2c4

E = mc2

m = γ m0 =

m0

1− v2 c2

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”

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…..

φ

Einstein model of photoelectric effect Light is not waves but energy “packets” (later “photons”)

each photon has energy hf = ℏω Planck’s constant h 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

KEmax

hνhf

hf

“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, ℏ = h/2π )

Observe: 1. Electrons only emitted for 2. Intensity of light affects the number of electrons but NOT their energy 3. Emitted electron max. KE

Conclude:

a) Photon energy

b) Work Function is the energy required to extract an electron from the metal.

�

ω > ω0

�

= ω −ω0( )

�

E = ω

�

φ = ω0