Wave Collapse Doesnt Matter Chris Stucchio, Courant Institute Joint - PowerPoint PPT Presentation

Wave Collapse Doesn’t Matter Chris Stucchio, Courant Institute Joint work with Avy Soffer, Juerg Frohlich and Michael Sigal.

Quantum Mechanics - consensus

Wavefunction State of the universe ψ ( x 1 , . . . , x N , t ) = position of i ’th particle x i = time t

Probability Theory Probability distribution of particle configurations | ψ ( x 1 , . . . , x N , t ) | 2 dx 1 . . . dx N

Evolution i ∂ t ψ = H ψ Schrodinger Equation

Physical Facts • Suppose we allow the wavefunction evolves to a “split” state √ √ ψ ( x, T ) = λφ ( x − L ) + 1 − λφ ( x + L ) • Meaning of this wavefunction: particle near x = L with probability λ particle near x = − L with probability 1 − λ Repeated “measurements” of particle position will yield same result

Physical Facts • In the absence of measurement, interference effects are observed. • Split, recombine than measure: P ( x ) = | φ 1 ( x ) | 2 + | φ 2 ( x ) | 2 + 2 ℜ φ 1 ( x ) φ 2 ( x ) interference • Split, measure, recombine, then measure again: term P ( x ) = | φ 1 ( x ) | 2 + | φ 2 ( x ) | 2

Copenhagen Interpretation and Wave Collapse “Textbook Quantum Mechanics”

How to predict outcome of experiments • “Prepare” initial wavefunction. √ √ ψ ( x, 0) = λφ ( x − L ) + 1 − λφ ( x + L ) • Allow it to evolve under Schrodinger equation. • “Measure” the position of the particle.

How to predict outcome of experiments √ √ λφ ( x − L ) + 1 − λφ ( x + L ) probability = λ probability = 1 − λ measurement φ ( x − L ) φ ( x + L )

Problems with this interpretation • Why do certain states of the universe constitute a measurement? • Why are measurements special? • Are there experiments which are not measurements?

Are all wavefunctions possible? (Not normalized)

Decoherence

Dynamics in configuration space Configuration space is really big. Many particles moving a small distance adds up. √ | (1 , 1 , . . . , 1) − (0 , 0 , . . . , 0) | = 3 N

Measurements are not special • Between measurements, the system remains on a low-dimensional submanifold of configuration space. • Measurements are interactions in which many degrees of freedom become relevant. √ O ( N ) • After measurement, different states are a distance apart. This implies no interference, since: 2 ℜ ¯ ψ 1 ( x ) ψ 2 ( x ) ≈ 0

An unmeasured interaction

An unmeasured interaction

The measurement process

The measurement Interaction Switched On process

A realistic example

The measurement process • Want to measure the position of a quantum particle. • Measurement apparatus is a many-body quantum fluid (BEC), which interacts with particle. • Use conventional methods to measure position of the splash.

The measurement process • Want to measure the position of a quantum particle. • Measurement apparatus is a many-body quantum fluid (BEC), which interacts with particle. • Use conventional methods to measure position of the splash. The particle is here.

Many Body Schrodinger equation   p ( x − y j ) + 1  − ∆ x 2 M + − ∆ y � �  ψ ( x, � V N V N i ∂ t ψ ( x, � y, t ) = 2 m + s ( y i − y j ) y, t ) 2 j i � = j N � ψ 0 ( x, y ) = φ ( x ) χ ( y j ) j =1 = Position of particle to be measured x = Position of j -th fluid particle y j V N P ( x − y j ) = Interaction between particle and fluid V N S ( y i − y j ) = Internal fluid interaction

Hydrodynamic Formulation ∂ t ρ ( x, � y, t ) + ∇ · [ ρ ( x, � y, t ) v ( x, � y, t )] = 0 − � ∂ t � v ( x, � y, t ) + � v ( x, � y, t ) · ∇ � v ( x, � y, t ) = ∇ V ( x, � y ) y, t ) | 2 ρ ( x, � y, t ) = | ψ ( x, � � ∇ = ( M − 1 ∇ x , m − 1 ∇ � y 1 , . . . , m − 1 ∇ � y N ), particle and the mass of the light particle. N � � y j ) + 1 V N V N V ( x, � y ) = p ( x − � s ( � y j ) y i − � 2 j =1 i � = j � � ρ ( x, � y, t ) ρ ( x, � y, t ) + ∆ x + ∆ y � � ρ ( x, � y, t ) ρ ( x, � y, t ) M m

Multiconfiguration Reduction • Many-Body Schrodinger equation is hard. Solution: derive reduced equation. ρ ( x, y 1 , t ) , v x ( x, y 1 , t ) and v y ( x, y 1 , t ) • Reduced variables N � ρ ( x, � y, t ) = ρ ( x, � y j , t ) j =1   N � v ( x, , t ) = v x ( x, y j , t ) , v y ( x, y 1 , t ) , . . . , v y ( x, y N , t ) �   j =1

Derivation of reduced equation � � � N � ∂ t ρ ( x, � y j , t ) + ρ ( x, � y j , t ) v x ( x, � y l , t ) ∇ x · ρ ( x, � y j , t ) � l =1 � � N + 1 � N ρ ( x, � y j , t ) ∇ x · ρ ( x, � y j , t ) � v x ( x, � y l , t ) l =1 � + ∇ y j · [ ρ ( x, � y j , t ) � v y ( x, � y j , t )] = 0 The velocity equation can be rewritten as follows:

Derivation of reduced equation � � � N � ∂ t ρ ( x, � y j , t ) + ρ ( x, � y j , t ) v x ( x, � y l , t ) ∇ x · ρ ( x, � y j , t ) � l =1 � � N + 1 � N ρ ( x, � y j , t ) ∇ x · ρ ( x, � y j , t ) � v x ( x, � y l , t ) l =1 � + ∇ y j · [ ρ ( x, � y j , t ) � v y ( x, � y j , t )] = 0 We will reduce this The velocity equation can be rewritten as follows:

Derivation of reduced equation • Equation for velocities: � N N � � � � v x ( x, � y j , t ) + v x ( x, � y k , t ) v x ( x, � y j , t ) ∂ t � � · ∇ x � j =1 k =1 � = −∇ x + � v y ( x, � y j , t ) · ∇ y j � v x ( x, � y j , t ) M V ( x, � y ) � N � � v y ( x, � y j , t ) + v x ( x, � y k , t ) v y ( x, � y j , t ) ∂ t � � · ∇ x � k =1 y j , t ) = −∇ � y j + � v y ( x, � y j , t ) · ∇ y � v y ( x, � m V ( x, � y ) (5b)

Derivation of reduced equation • Equation for velocities: � N N � � � � v x ( x, � y j , t ) + v x ( x, � y k , t ) v x ( x, � y j , t ) ∂ t � � · ∇ x � j =1 k =1 � = −∇ x + � v y ( x, � y j , t ) · ∇ y j � v x ( x, � y j , t ) M V ( x, � y ) � N � � v y ( x, � y j , t ) + v x ( x, � y k , t ) v y ( x, � y j , t ) ∂ t � � · ∇ x � k =1 y j , t ) = −∇ � y j + � v y ( x, � y j , t ) · ∇ y � v y ( x, � m V ( x, � y ) (5b)

Mean field for velocity • We need not consider a general point in configuration space, only typical ones. • Probability distribution of fluid particle: ρ ( x, y j , t ) dy j � ρ ( x, y j , t ) dy j • Central limit theorem: N � v x ( x, y, t ) ρ ( x, y, t ) dy � v x ( x, y j , t ) → ( N − 1) � ρ ( x, y, t ) dy j =2

Mean Field for Potential N � � y j ) + 1 V N V N V ( x, � y ) = p ( x − � s ( � y j ) y i − � 2 j =1 i � = j � � ρ ( x, � y, t ) ρ ( x, � y, t ) + ∆ x + ∆ y � � ρ ( x, � y, t ) ρ ( x, � y, t ) M m • Similar tricks can be used for the potential: V N � s ( y 1 − y ) ρ ( x, y, t ) dy � V N s ( y 1 − y j ) → ( N − 1) � ρ ( x, y, t ) dy j � =1

Probably Approximately Correct • How accurate is this? • Mcdiarmid’s inequality. For a vector of i.i.d. variables, if sup | f ( x 1 , . . . , x i − 1 , x i , x i +1 , x N ) − f ( x 1 , . . . , x i − 1 , x i , ˆ x i +1 , x N ) | < C x, ˆ x i • then: − 2 ǫ 2 � � P ( | f ( � x ) − E [ f ( � x )] | > ǫ ) ≤ 2 exp NC 2

Probably Approximately Correct • Implication:   N � V N p ( x − y ) ρ ( x, y ) dy � V N  | p ( x − y j ) − ( N − 1) | ≥ N ǫ P  � ρ ( x, y ) dy j =1 2 ǫ 2 N � � ≤ 2 exp − || V N p ( x ) || L ∞ • The probability distribution is w.r.t. conditional distribution of fluid particle: ρ ( x, y ) P ( y ) = � ρ ( x, y ) dy

Y-indepence of equations • Equations depend (to order O(N)) not on but on it’s expected v x ( x, y, t ) value: �� v x ( x, y ′ , t ) ρ ( x, y ′ , t ) dy ′ � � ∂ t ρ ( x, y, t ) + ( N − 1)( ∇ x · ρ ( x, y, t )) d � y l � ρ ( x, y ′ , t ) dy ′ �� v x ( x, y ′ , t ) ρ ( x, y ′ , t ) dy ′ � � + ( N − 1) ρ ( x, y, t ) ∇ x · d � y l � ρ ( x, y ′ , t ) dy ′ + ∇ x · [ ρ ( x, y, t ) � v ( x, y, t )] + ∇ y [ ρ ( x, y, t ) � v y ( x, y, t )] = 0 (7)

Y-indepence of equations • Equations depend (to order O(N)) not on but on it’s expected v x ( x, y, t ) value: N � v x ( x, y ′ , t ) ρ ( x, y ′ , t ) dy ′ �� � � � v x ( x, � y j , t ) + ( N − 1) v x ( x, � y j , t ) ∂ t � · ∇ x � � ρ ( x, y ′ , t ) dy ′ j =1 � + � v x ( x, � y j , t ) · ∇ x � v x ( x, � y j , t ) + � v y ( x, � y j , t ) · ∇ y � v x ( x, � y j , t ) N � � −∇ x − ∇ x � M V N = p ( x − � y j ) M V q j =1