text: sears & zemansky, university physics www ... quantum physics text: sears & zemansky,...
TRANSCRIPT
2018 Quantum Physics
Text: Sears & Zemansky, University Physicswww.masteringphysics.com
Lecture notes at
www.tcd.ie/Physics/study/current/undergraduate/lecture-notes/py1p20
TCD JF PY1P20 2018 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 MechanicsNewton’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
∫
WavesFrequency 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
θd
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 electrodesE: emitter, C: collector
Light incident on E, electrons are emitted & travel to C
Current I in external circuitdepends 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 currentAll 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 hPhotoelectron is ejected (instantly) through the complete absorption of one photon.
hf = KE + (depth in well)
Consider the highest-lying electronshf = KEmax + φKEmax = 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 energy3. 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