Bohmian Mechanics

Paper by Jeremy BernsteinPresentation by Reece Boston

January 23, 2014

The Interpretation Problem

Schrodinger Equation:

i~∂

∂tΨ = − ~2

2m∇2Ψ + VΨ

Leads to unitary time evolution operatorU(t − t0) = exp{− i

~H(t − t0)}Dynamics so described are deterministic:

|Ψ(t)〉 = U(t − to)|Ψ(t0)〉

Measurement is random: Ψ = αΨa + βΨb

Pr(A = a) = |α|2, Pr(A = b) = |β|2

We never measure a superposition

Stern -Gerlach experiment

Stern-Gerlach splitting

Interpretations

I Copenhagen InterpretationI Everett’s Many-Worlds HypothesisI de Broglie-Bohm Pilot Wave TheoryI Statistical/Ensemble InterpretationI Transactional InterpretationI Consistent HistoriesI Stochastic MechanicsI Objective Collapse TheoriesI Quantum Mysticism/Consciousness/Solipsism

The Copenhagen Interpretation“All we touch and all we see... is all we are and all we’ll be”

I All information is in the wavefunction Ψ

I Particle has no properties until we measurethem

I Ψ can be in any arbitrary superposition,

|Ψ〉 = α|Ψ1〉+ β|Ψ2〉

I Upon measurement, Ψ collapses to aneigenstate, either Ψ1 or Ψ2

I The measurement is subjective - depends onobserver

I Schrodinger equation does not describemeasurement

I Schrodinger’s cat

Everett’s Many-Worlds

Every possible measurement is realizedWe only see one realizationDeterministic by Schrodinger EquationEntire multiverse has one wave function

|alive〉+ |dead〉

|you measure ‘alive’〉+|you measure ‘dead’〉

de Broglie-Bohm Pilot Wave Theory

I Wave/particle duality −→ wave/particle dualismI Associated with each particle is a wave function Ψ which

“pilots” it, or directs its movements and interactionsI The particle is localized and objectively has propertiesI The pilot wave is nonlocalI The pilot wave is what obeys Schrodinger EquationI The particle obeys “Bohmian Mechanics”I Hidden variables theory: Information ‘exists’ but we can’t

measure itI Nonlocal realism - satisfies Bell’s theoremI Predicts exact behavior as QM

Pilot Wave for Double Slit

EPR Paradox

Take a spin singlet state: two particles with opposite spin.

|ψ〉 =|+−〉 − | −+〉√

2

|ψ〉 =|Sx−;Sx+〉 − |Sx+; Sx−〉√

2

Send one particle to Alice, one to Bob

Bell’s Inequality

Suppose we have hidden variablesConsider spin along vectors a, b, c

Alive Bobn1 (a+, b+, c+) (a−, b−, c−)n2 (a+, b+, c−) (a−, b−, c+)n3 (a+, b−, c+) (a−, b+, c−)n4 (a−, b+, c+) (a+, b−, c−)n5 (a+, b−, c−) (a−, b+, c+)n6 (a−, b+, c−) (a+, b−, c+)n7 (a−, b−, c+) (a+, b+, c−)n8 (a−, b−, c−) (a+, b+, c+)

Pr(a+, b+) = n3+n5 ≤ (n3+n7)+(n5+n2) = Pr(a+, c+)+Pr(c+, b+)

But in QM Pr(a = +, b = +) = 12 sin2 θab

2Not always possible: No real local theory

Bohmian Mechanics

Write ψ = R exp{ i~S}, for modulus R and phase S/~Insert in to Schrodinger Equation, separate in to real and imaginary

∂R

∂t= − 1

2m

[R∇2S + 2∇R · ∇S

]∂S

∂t=

~2

2m

∇2R

R− 1

2m(∇S)2 − V

If we use Pr = ψ∗ψ = R2 in first, then we get the usual continuityequation, written as

∂ Pr

∂t+∇ · (Pr∇S/m)

If j = Pr∇S/m, suggests that∇S/m is a velocity.

Bohmian Mechanics

Let’s look at second equationClassical Hamilton-Jacobi:

∂S

∂t+

1

2m(∇S)2 + V = 0

We have∂S

∂t+

1

2m(∇)2 + V − ~2

2m

∇2R

R= 0

Suggests identifying the last term as “Quantum Potential”

U = − ~2

2m(∇2R/R)

Both potentials - classical and quantum - will influence particlemotion

Bohmian Mechanics

Suppose the particle has objective position Q(t)Then identify Q(t) = ∇S/mWe write “Newton’s Equation”: mQ(t) = −∇(V (Q) + U(Q))From Q Pr = j = 1/mIm(ψ∗∇ψ), we get

Q =1

m

∂

∂QIm [logψ(Q, t)]

Solve Schrodinger Equation for ψUse to solve Bohm equation for QMore particles, more equations, but only one ψSince ψ(Q1, . . . ,QN) and Qk = f (ψ), nonlocality

Explaining Measurement

Let Ψ(x , y) = ψ(x)φ(y)Let A be some observable of system;

I non-degenerateI discrete spectrumI complete eigenspace

Let ψ = ψα, an eigenstate of ALet them interact ψα(x)φ0(y) −→ ψα(x)φα(y) = Ψα(x , y)Let ψ =

∑α cαψα −→ Ψ =

∑α cαΨα

We measure A = α with Pr(α) = |cα|2

Explaining Collapse

Conditional wave function, ψ(x) =∫

Ψ(x , y)dyIf we measure A = β, then φ(y) = 0 unless y corresponds to β

ψ(x) =

∫A=β

∑α

cαψα(x)φα(y)dy =

∫A=β

cβψβ(x)φβ(y)dy

=

[∫A=β

cβφβ(y)dy

]ψβ(x) = Nψβ(x)

The conditional wave function is transformed from ψ to ψβThere is NO COLLAPSE

Simple ExamplesFree Particle:

ψ = e ikx ⇒ Imlogψ = kx ⇒ x =k

m

Harmonic Oscillator Ground State:

ψ = e−mωx2/2~e−iωt/2 ⇒ Im logψ = −ωt/2⇒ x = 0

Harmonic Oscillator Coherent State:

ψα = e−mω(x−α)2/2~e−iωt/2

= exp

{− mω

2~ (x − α cos 2ωt)2

+i mω~ (−xα sinωt − 12α

2 sin 2ωt − ωt/2)

}=⇒ x = −ω

~α sinωt

Spin

Spin

We never actually measure spin, we measure position.For magnetic systems, p→ p− ASpin becomes a property of wave function, now has two components

Ψ→[

Ψ+

Ψ−

]Spin is not a property of the particle.Particle will take one or the other of the trajectories through aStern-Gerlach device.

Conclusions?

New ‘ontology’ that treats particles as real and wave functions asnonlocal.Added formalism: quantum potential and particle trajectoryQuantum Potential:

U = − ~2

4m

[∇2 Pr

Pr− (∇Pr)2

2 Pr

]Particle Trajectory:

Q(t) =1

m

∂

∂QIm logψ(Q)

Useful in computer modeling - calculate wavefunction once, modifyalong trajectories

Conclusions?

Questions?