Feynmans Quantum Paths (Advanced Relativity, Quantum Chromodynamics) - PowerPoint PPT Presentation

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Feynman’s Quantum Paths (Advanced ⇒ Relativity, Quantum Chromodynamics) Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Feynman: Quantum Mech ↔ Classical Mech? Generalize Classical Trajectory to QM Probability Classical Mech: single x ( t ) path = ¯ x B t b QM: waves = statistical, no path Time Dirac: Hamilton’s least-action prin t a A x b x a Position F: Look for quantum least-action principle Hamilton: space-time path variation δ calculus F: quantum particle @ B = ( x b , t b ) From all A via Green’s function ( propagator ) G � ψ ( x b , t b ) = dx a G ( x b , t b ; x a , t a ) ψ ( x a , t a ) 2 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Huygen-Feynman Quantum Wavelets Classical Becomes Quantum � B ψ ( x b , t b ) = dx a G ( b , a ) ψ ( x a , t a ) t b � i m ( x b − x a ) 2 � exp Time 2 ( t b − t a ) G ( b , a ) = � 2 π i ( t b − t a ) t a A F’s vision: ψ ↔ path x b x a Position ψ B = � all paths, A ∆ paths ∆ probabilities ∼ Huygens’s principle All paths possible! G ( b ; a ) = spherical wavelet ψ ( x b , t b ) = � wavelets Also relativity, fields 3 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Hamilton’s Principle of Least Action (Classical) Newton’s Law ≡ δ S [¯ x ( t )] = 0 "The most general motion of a physical particle moving along the classical trajectory ¯ x ( t ) from time t a to t b is along a path such that the action S [¯ x ( t )] is an extremum." δ S = S [¯ x ( t ) + δ x ( t )] − S [¯ x ( t )] = 0 (1) (Constraint) δ ( x a ) = δ ( x b ) = 0 (2) B t b [ x ( t )] = functional Time � t b S [¯ dt L [ x ( t ) , ˙ x ( t )] = x ( t )] (3) t a t a A x b x a L = Lagrangian = T [ x , ˙ x ] − V [ x ] (4) Position 4 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Connecting CM Hamilton’s Prin to QM Paths Consider Free Particle ( V = 0) � t b S [ b , a ] = dt ( T − V ) B t a t b ( x b − x a ) 2 = m x 2 ( t b − t a ) = m 2 ˙ (1) Time 2 t b − t a e iS [ b , a ] / � t a ⇒ G ( b , a ) = (2) A � 2 π i ( t b − t a ) x b x a Position � e iS [ b , a ] / � ⇒ G ( b , a ) = (3) F: QM = path integrals paths All paths ∃ , ∆ prob � ≃ 10 − 34 Js ⇒ ∼ ¯ x Mainly classical S ¯ x = extremum 5 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Relate Paths to Ground State Wave Function Hermitian ˜ H ⇒ Complete Orthonormal Set ⇒ Propagator ˜ H ψ n = E n ψ n (1) ∞ c n e − iE n t ψ n ( x ) � ψ ( x , t ) = (2) n = 0 � + ∞ dx ψ ∗ c n = n ( x , 0 ) ψ ( x , 0 ) (3) −∞ � + ∞ n ( x 0 ) ψ n ( x ) e − iE n t ψ ( x 0 , t = 0 ) � ψ ∗ → ψ ( x , t ) = dx 0 −∞ n (4) � Recall: ψ ( x b , t b ) = dx a G ( x b , t b ; x a , t a ) ψ ( x a , t a ) (5) � ψ ∗ n ( x 0 ) ψ n ( x ) e − iE n t ⇒ G ( x , t ; x 0 , 0 ) = (6) n 6 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Relate Space-Time Paths to Ψ 0 (cont) Hermitian ˜ H ⇒ Complete Orthonormal Set � n ( x 0 ) ψ n ( x ) e − iE n t ψ ∗ G ( x , t ; x 0 , t = 0 ) = (1) n Evaluate @ imaginary t (Wick rotation): � ψ ∗ n ( x 0 ) ψ n ( x ) e − E n τ G ( x , − i τ ; x 0 , t = 0 ) = (2) n Im time τ → ∞ only n = 0 For | ψ 0 | 2 : paths start & end at x 0 = x | ψ n ( x ) | 2 e − E n τ = | ψ 0 | 2 e − E 0 τ + | ψ 1 | 2 e − E 1 τ + · · · � G ( x , − i τ ; x , 0 ) = (3) n | ψ 0 ( x ) | 2 = lim τ →∞ e E 0 τ G ( x , − i τ ; x , 0 ) ⇒ (4) 7 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Break Now, Compute Later � t b t i t a t a x a 8 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Lattice Quantum Mechanics (Algorithm) Easy: Discrete Times & Positions Only! dx j x j − x j − 1 ≃ (1) dt ε � S j ≃ L j ∆ t (2) t b m ∆ x 2 ≃ − V ( x j ) ε (3) t i 2 ε Add actions for N -links t a t a G ( b , a ) ↔ � x a a − b paths Path: � links Ea path = � links Euler + Time step ε : � dx 1 · · · dx N − 1 e iS [ b , a ] G ( b , a ) = (4) 9 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Rotate t: Lagrangian ( − i τ ) = -Hamiltonian ( τ ) Wick Rotation into Imaginary Time � � dx 1 dx 2 · · · dx N − 1 e iS [ x , x 0 ] G ( x , t ; x 0 , t 0 ) = (1) N − 1 � S [ x , x 0 ] ≃ L ( x j , ˙ x j ) ε (2) j = 1 � dx � 2 x ) = T − V ( x ) = + 1 L ( x , ˙ 2 m − V ( x ) (3) dt � dx � 2 � � dx = − 1 ⇒ L x , 2 m − V ( x ) = − H (4) − id τ d τ 10 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Put Pieces Together Sum Over Paths Related to Wave Function � � τ 0 H ( τ ′ ) d τ ′ dx 1 . . . dx N − 1 e − G ( x , − i τ ; x 0 , 0 ) = (5) Individual path integral: � � H ( τ ) d τ ≃ ε E j = ε E (6) j Ground State Wave Function via Feynman: | ψ 0 ( x ) | 2 = 1 � dx 1 · · · dx N − 1 e − ε E lim (7) Z τ →∞ paths 11 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Imaginary Time Relates QM to Thermodynamics Schrödinger Equation → Heat Diffusion Equation t in QM → − i τ ∂ ( − i τ ) = −∇ 2 ∂τ = ∇ 2 ∂ψ ∂ψ i ⇒ (1) 2 m ψ 2 m ψ Boltzmann P = e − ε E weights ea Feynman path Temperature ⇔ time step: k B T = 1 ε ≡ � P = e − ε E = e −E / k B T ⇒ (2) ε ⇒ lim ε → 0 = lim T →∞ ψ 0 : long imaginary τ vs � / ∆ E Like equilibration in Ising model 12 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Summary (This is Heavy Stuff) Feynman’s Path Integral Formulation of QM QM ψ via statistical fluctuations ∼ class trajectory � Propagator( t a → t b ) G = path integral, � paths Hamilton: extremum S → path integration of H Path integral = sum trajectories on x-t lattice Paths weighted with probability e − iS / � Algorithm: ∆ path link ⇒ ∆ E (like Ising) Ψ equilibrates to ground state 13 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Break Before Algorithm Quantum Monte Carlo (QMC) Applet 14 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment A Time-Saving Trick Compute ψ ( x ) for All x ( x b ) Simultaneously � | ψ 0 ( x ) | 2 = � dx 1 · · · dx N e − ε E ( x , x 1 ,... ) t b � dx 0 · · · dx N δ ( x − x 0 ) e − ε E ( ... ) t i = t a t a Frequent x j ⇒ larger ψ ( x j ) x a EG: AB , New path + C Integrate all x sites CBD same � E i as ACB Don’t compute δ ( x ) ! Equilibrate, flip links, new E Accumulate ψ ( x ′ ) 15 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Lattice Implementation QMC.py Harmonic oscillator 1 V ( x ) = 1 2 x 2 Natural units: m = 1, 2 0.2 2 quantum classical � L: � / m ω ; t: 1 /ω ; T = 2 π Probability 0.15 1 Position 0.1 0 0.05 -1 Short T ∼ 2 T , Long t ∼ 20 T 3 0 -2 -40 -20 0 20 40 0 20 40 60 80 100 Position Time Classical: max ρ @ turning pts 4 Each x j , running sum | Ψ 0 ( x j ) | 2 5 ∆ seed; many runs > 1 long run 6 16 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Assessment and Exploration Plot classical trajectory, some actual space-time paths 1 Explore effect of smaller ∆ x , smaller ∆ t 2 � ψ 2 ( x ) , calculate: Assume ψ ( x ) = 3 � − d 2 � + ∞ � E = � ψ | H | ψ � ω ψ ∗ ( x ) dx 2 + x 2 = ψ ( x ) dx (1) � ψ | ψ � 2 � ψ | ψ � −∞ Explore effect of larger, smaller � 4 Test ψ with quantum bouncer: 5 V ( x ) = mg | x | (2) x ( t ) = x 0 + v 0 t + 1 2 gt 2 . (3) 17 / 18

QM Paths ψ 0 Lattice t → − i τ Trick Implementation Assessment Summary Feynman Path Integrals B t b Time t a A x b x a Position A different view of quantum mechanics It seems to give same answers as traditional QM Is at heart of lattice quantum chromodynamics Hard to apply beyond ground state Satisfying connection to classical mechanics 18 / 18