Quantum Ideas. Syllabus:

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1 Quantum Ideas Syllabus: The success of classical physics, measurements in classical physics. The nature of light, the ultraviolet catastrophe, the photoelectric effect and the quantisation of radiation. Atomic spectral lines and the discrete energy levels of electrons in atoms, the Frank-Hertz experiment and the Bohr model of an atom. Magnetic dipoles in homogeneous and inhomogeneous magnetic fields and the Stern- Gerlach experiment showing the quantisation of the magnetic moment. The Uncertainty principle by considering a microscope and the momentum of photons, zero point energy, stability and size of atoms. Measurements in quantum physics, the impossibility of measuring two orthogonal components of magnetic moments. The EPR paradox, entanglement, hidden variables, non-locality and Aspect's experiment, quantum cryptography and the BB84 protocol. Schrödinger's cat and the many-world interpretation of quantum mechanics. Interferometry with atoms and large molecules. Amplitudes, phases and wavefunctions. Interference of atomic beams, discussion of two-slit interference, Bragg diffraction of atoms, quantum eraser experiments. A glimpse of quantum engineering and quantum computing. Schrödinger's equation and boundary conditions. Solution for a particle in an infinite potential well, to obtain discrete energy levels and wavefunctions.

2 Failure of Classical Physics Photoelectric Effect Blackbody Radiation Uncertainty Principle Interference of massive particles De Brogie Wavelength Spectral lines Schrödinger s Equation Quantisation of radiation Planck s Hypothesis Wave-Particle Duality Quantised energy levels Quantum Physics Structure of Matter Atom model (Bohr) Molecules, Solid state, etc. Modern Applications Quantum Cryptography Quantum Computing Paradoxa in early Gedanken-experiments Entanglement Superposition Probabilities Role of the observer Break-down of the Wavefunction Many-world interpretation

3 Classical Physics: classical mechanics (Newton; F = m a) electricity and magnetism (Coulomb, Faraday, Maxwell) electromagnetic waves (rf... light... x-ray... gamma) thermodynamics (energy conservation, equilibration, statistical mechanics) accurate measurement of all observables (position x(t)and momentum p(t) ) Quantum Physics: probabilistic - not deterministic (Einstein: Good does not play dice ) probability wave function ψ(x,t) to describe a particle superposition and entanglement non-local behaviour ( Spooky interaction at a distance that bothered Einstein) uncertainty principle: Δx Δp ħ/2 and ΔE Δt ħ/2

illumination with a mercury lamp, filtering a single spectral line. Cathode metal with a binding energy (or work function) of Ebind = 2.02 ev yellow, 578nm, 5.19E+14 Hz, Ekin = 0.

7 illumination with a mercury lamp, filtering a single spectral line. Cathode metal with a binding energy (or work function) of Ebind = 2.02 ev yellow, 578nm, 5.19E+14 Hz, Ekin = 0.13 ev green, 546nm, 5.50E+14 Hz, Ekin = 0.27 ev blue, 436nm, 6.88E+14 Hz, Ekin = 0.81 ev violet, 405nm, 7.41E+14 Hz, Ekin = 1.02 ev Planck s constant is obtained from the slope of the kinetic energy, Ekin(ν)

9 Hints for quantisation: a) threshold (minimum frequency required): resonance phenomenon quantised medium or light b) linear in the intensity (for ν=const). electron number proportional to photon number c) photo current insensitive to ν (provided hν > Ebind) no change of the electron current if photon flux constant albeit the intensity is increasing: Iphoto ν d) no delay direct evidence! it lasts seconds until a single atom accumulates enough energy, so the radiation cannot be continuous

13 Blackbody = Cavity? - Multiple reflections -> absorption of incident light - Thermal equilibrium -> Walls <-> Cavity modes - Spectral energy density ρ(ν)dν R(ν)dν (Radiance through whole) - Boundary conditions: Nodes on the walls - Standing waves along x,y,z Consider a 2D problem and decompose λ into λx = λ / cos(α) λy = λ / sin(α) with nx λx = 2L etc... ==> nx = (2L/ λ) cos(α) and ny = (2L/ λ) sin(α) square and add these conditions (generalise into 3D): (2L/ λ) 2 = nx 2 + ny 2 + nz 2

14 Number of Modes in the cavity with frequencies smaller than ν: - sphere of radius R = (nx 2 + ny 2 + nz 2 ) = 2L/ λ = 2L/c ν - mode number N(ν) = 4/3 π R 3 2/8 =... - same for N(ν+dν) =... Mode number in the interval ν...ν+dν ΔN = N(ν+dν) - N(ν) = 8π ν 2 L 3 / c 3 dν Spectral density per unit volume ρ(ν)dν = ΔΝ/L 3 = 8π ν 2 / c 3 dν

$1/t*1.* "{& ZD Lutb- *,id7 # L \"= )'/-x \7 = Vi*x f-) r, L Lou- L,.$

15 1/t*1.* "{& ZD Lutb- *,id7 # L \"= )'/-x \7 = Vi*x f-) r, L Lou- L,.n-L h n-, "l ll fzl '- tl,*tt h" \, - ZLI ', ), -- zli h,o h)r' 7L.,\ ZL J. ql ;LJ -' hr' * 'rt = *" -D 6 Lo f*,*c'j l" 3?, R

$Sil*'L i. ll h - slpaa, svj;u, IZ, //,-Lu o(,--l = l,/o/,*,l ll 70., hor= = \r 81,,' L ll (fry..)) = ( ts' 7'trLt.r #,l --,,Ln 6 4 I 7'.$

16 Sil*'L i. ll h - slpaa, svj;u, IZ, //,-Lu o(,--l = l,/o/,*,l ll 70., hor= = \r 81,,' L ll (fry..)) = ( ts' 7'trLt.r #,l --,,Ln 6 4 I 7'.,{u u*( 3o3 Z g'l.,-:,j'-' (ot:l:.t ryn{r.^ , Jrt r?,t 6N 0t) do =,({,t,ju1-//u) StiJ',t= y #'(y')'-'=8o4u 1 = 8try" t, a.'dy v 3Y'Jt' + "(*L fu u--:l e'sb*,- I

18 Blackbody Radiation Average energy <E>per frequency mode, using the Boltzmann distribution P(E) ne nhν kt E = with n=0 E n P(E) hν = P(E) n=0 n=0 e nhν kt n=0 ( ) = 1 B Bexp hν kt = hν A B and A Aexp( hν ) kt = Bexp( hν ) kt A B = 1 exp hν kt ( ) 1 and E = hν ( ) 1 exp hν kt

19 Planck s law Spectral energy density (energy per unit volume in the frequency range ν...ν+dν): ρ(ν)dν = 8πν 2 c 3 hν ( ) 1 = 8πhν exp hν kt 3 1 c 3 ( ) 1 exp hν kt total energy density: ρ = ρ(ν)dν = 8π 5 k 4 15(hc) 3 T 4 = σt 4

$Ao^*'l a,- ol a q- Pt L- ft t^j/ c = hg E;-,h,-! =,,o' = h?^ -t Pc ]o' \ II l- ll(*" ( s tttt = t*t C De--Lnjt.$

22 Ao^*'l a,- ol a q- Pt L- ft t^j/ c = hg E;-,h,-! =,,o' = h?^ -t Pc ]o' \ II l- ll(*" ( s tttt = t*t C De--Lnjt.,"*(*rlL \ = % '7* " I ii.,") l,o,ouin ",u\"::f_?], +k tul/...* Lo[u,*, ej L k",j, 14'"'r'*iltLt=?x -t glno u).t/.' i (G,t) = uflz (w - aili v;ll a.?rt)

$?r s&t o {_r"/ trll* :'-.6'ao' ^. L"l-- o rt "kn tj,l- tu" +-J"{ to.(-7; f =r^ P, '9^l CO',otte*{i* z c)"?o s;" q?.*6 Ar 'I (al 'er - )'s;"o (t L-%*o J*, J*r \t.$

24 ?r s&t o {_r"/ trll* :'-.6'ao' ^. L"l-- o rt "kn tj,l- tu" +-J"{ to.(-7; f =r^ P, '9^l CO',otte*{i* z c)"?o s;" q?.*6 Ar 'I (al 'er - )'s;"o (t L-%*o J*, J*r \t.7, - 2fi,*'a / Y n*t -D d,.rk.- gh -, - "/.J** C.'->un-hn*"A!Jpu:lL [n}ux{ut- fair,'tq,.t.k,a- [=hf,=b= b 'lt\t?^'=":h.-zutlc' ' < t I e ^: ^1 + 1i1t' 'a Ln.* (I{- \,)2,o" L (i-*)- rt^*a s Zh(t.*re)

$-C(l"^: J \ / t = Ao L l-a-e) a-/f0/t> 'I t.$

25 c9 r'^ I rr Vqu!/(e.-C(l"^: J \ / t = Ao L l-a-e) a-/f0/t> 'I t...r14 v"rolo-lq,/ Ao- _,.' t 7". _-,ov -t2 = Z.?3 - t0'- nelq

28 Hanbury-Brown and Twiss: Intensity correlation measurements - dead time of the detectors - beam splitter - pair of photon counters - cross correlation Single-photon emitters: - single atoms or ions - crystal defects (Quantum dots)

QUANTUM MECHANICS Chapter 12

QUANTUM MECHANICS Chapter 12 Colours which appear through the Prism are to be derived from the Light of the white one Sir Issac Newton, 1704 Electromagnetic Radiation (prelude) FIG Electromagnetic Radiation