Capstone

Michael A. Baker

Youngstown State University

21 July 2016

Introduction to Quantum Mechanics

States in Hilbert Space

[x , p] = i~

Superposition of Basis States

|φ〉 = a1 |1〉+ ...+ an |n〉

Schrodinger Equation, analog to Newton

i~∂t |φ〉 = H |φ〉

Introduction to Quantum Mechanics

States in Hilbert Space

[x , p] = i~

Superposition of Basis States

|φ〉 = a1 |1〉+ ...+ an |n〉

Schrodinger Equation, analog to Newton

i~∂t |φ〉 = H |φ〉

Introduction to Quantum Mechanics

States in Hilbert Space

[x , p] = i~

Superposition of Basis States

|φ〉 = a1 |1〉+ ...+ an |n〉

Schrodinger Equation, analog to Newton

i~∂t |φ〉 = H |φ〉

Harmonic Oscillator

Consider the following Hamiltonian...

H =p2

2m+

1

2~ω2x2

Position Basis Solution

Weyl Algebraic Solution via Ladder Operators

a =

√mω

2~x + i

√1

2mω~p

a† =

√mω

2~x − i

√1

2mω~p

En = (n +1

2~ω

Multi-particle Systems

Treat Multple Particle as Direct Product of Single ParticleSystems

Interchange of Identical Particles descibed by same state (upto scaling)

Pick up phase depending on type of particle: Bosons (+1)/Fermions (−1)

Negative Sign leads to Pauli Exclusion Principle

Multi-particle Systems

Treat Multple Particle as Direct Product of Single ParticleSystems

Interchange of Identical Particles descibed by same state (upto scaling)

Pick up phase depending on type of particle: Bosons (+1)/Fermions (−1)

Negative Sign leads to Pauli Exclusion Principle

Multi-particle Systems

Treat Multple Particle as Direct Product of Single ParticleSystems

Interchange of Identical Particles descibed by same state (upto scaling)

Pick up phase depending on type of particle: Bosons (+1)/Fermions (−1)

Negative Sign leads to Pauli Exclusion Principle

Multi-particle Systems

Treat Multple Particle as Direct Product of Single ParticleSystems

Interchange of Identical Particles descibed by same state (upto scaling)

Pick up phase depending on type of particle: Bosons (+1)/Fermions (−1)

Negative Sign leads to Pauli Exclusion Principle

Lieb-Liniger Model

1-D Interacting Boson Model with Periodic B.C.

H = −N∑

j=1

∂2

2j

+ 2c∑

1<=i<j<=N

δ(xi − xj )

ψ(x1, ..., xN) =∑

P

a(P)exp(iN∑

j=1

kpj xj )

Lieb-Liniger Model

1-D Interacting Boson Model with Periodic B.C.

H = −N∑

j=1

∂2

2j

+ 2c∑

1<=i<j<=N

δ(xi − xj )

ψ(x1, ..., xN) =∑

P

a(P)exp(iN∑

j=1

kpj xj )

Lieb-Liniger Model

1-D Interacting Boson Model with Periodic B.C.

H = −N∑

j=1

∂2

2j

+ 2c∑

1<=i<j<=N

δ(xi − xj )

ψ(x1, ..., xN) =∑

P

a(P)exp(iN∑

j=1

kpj xj )

Experimental

D. Weiss Lab at Penn State

Rb-87 in Light Trap (Harmonic)

Fits 1-D interacting bose model, acts like fermi gas at largecoupling

Experimental

D. Weiss Lab at Penn State

Rb-87 in Light Trap (Harmonic)

Fits 1-D interacting bose model, acts like fermi gas at largecoupling

Experimental

D. Weiss Lab at Penn State

Rb-87 in Light Trap (Harmonic)

Fits 1-D interacting bose model, acts like fermi gas at largecoupling

Crescimanno

Spectral Equivalence of Bosons and Fermions in 1-DHarmonic Potential

Flow Monotonically Increasing

What does flow look like for intermediate coupling?

Crescimanno

Spectral Equivalence of Bosons and Fermions in 1-DHarmonic Potential

Flow Monotonically Increasing

What does flow look like for intermediate coupling?

Crescimanno

Spectral Equivalence of Bosons and Fermions in 1-DHarmonic Potential

Flow Monotonically Increasing

What does flow look like for intermediate coupling?

Zuo-Gao Approach

Model the system of bosons using Quartic Interaction

L = tr(∂tM∂tM†) + tr(MM†) + g4tr(MM†MM†)

Taking the Large-N Limit,

N2ε(g4) = NεF−∫

dx

3π(2εF − x2 − 2g4x

4)θ(2εF − x2 − 2g4x4))

N =

∫dx

π(2εF − x2 − 2g4x

4)12 θ(2εF − x2 − 2g4x

4)

Zuo-Gao Approach

Model the system of bosons using Quartic Interaction

L = tr(∂tM∂tM†) + tr(MM†) + g4tr(MM†MM†)

Taking the Large-N Limit,

N2ε(g4) = NεF−∫

dx

3π(2εF − x2 − 2g4x

4)θ(2εF − x2 − 2g4x4))

N =

∫dx

π(2εF − x2 − 2g4x

4)12 θ(2εF − x2 − 2g4x

4)

Zuo-Gao Approach

Deal with Collective Field for Convenience in a general system.

φ(x) =

∫dk

2πe ikx tr(e−ikM) =

∑δ(xi − xj )

Effectively a ”density”, which can describe our system. As φis a function of x , we can Fourier transform.

Consider the general Hamiltonian

H =1

2

∑p2i +

1

2

∑v(xi , xj ) +

∑V (xi )

Zuo-Gao Approach

Deal with Collective Field for Convenience in a general system.

φ(x) =

∫dk

2πe ikx tr(e−ikM) =

∑δ(xi − xj )

Effectively a ”density”, which can describe our system. As φis a function of x , we can Fourier transform.

Consider the general Hamiltonian

H =1

2

∑p2i +

1

2

∑v(xi , xj ) +

∑V (xi )

Zuo-Gao Approach

Deal with Collective Field for Convenience in a general system.

φ(x) =

∫dk

2πe ikx tr(e−ikM) =

∑δ(xi − xj )

Effectively a ”density”, which can describe our system. As φis a function of x , we can Fourier transform.

Consider the general Hamiltonian

H =1

2

∑p2i +

1

2

∑v(xi , xj ) +

∑V (xi )

Zuo-Gao Approach

Consider the general Hamiltonian

H =1

2

∑p2i +

1

2

∑v(xi , xj ) +

∑V (xi )

For the L-L Harmonic Potential, we have V (x) = 12ω

2x2 andv(x , y) = gδ(x − y), so

H =

Zuo-Gao Approach

Consider the general Hamiltonian

H =1

2

∑p2i +

1

2

∑v(xi , xj ) +

∑V (xi )

For the L-L Harmonic Potential, we have V (x) = 12ω

2x2 andv(x , y) = gδ(x − y), so

H =

Coupling and Chemical Potential

Taking the derivative w.r.t. ρ(u) leads to

π2

2ρ2(u) + αρ(u) = E − 1

2ω2u2

Simple application of Quadratic Formula yields

ρ(x) =−a +

√a2 + 2π2E − x2

π2

However, from QM, we have a normalization condition onρ(u), leading to

π

2= −

α√ω

√Eαω√

2π+( α2

ω

2π2+

Eαω

)asin

(√Eα

α2

2π2ω+ Eα

ω

)Numerically Solve for Energy

Coupling and Chemical Potential

Taking the derivative w.r.t. ρ(u) leads to

π2

2ρ2(u) + αρ(u) = E − 1

2ω2u2

Simple application of Quadratic Formula yields

ρ(x) =−a +

√a2 + 2π2E − x2

π2

However, from QM, we have a normalization condition onρ(u), leading to

π

2= −

α√ω

√Eαω√

2π+( α2

ω

2π2+

Eαω

)asin

(√Eα

α2

2π2ω+ Eα

ω

)Numerically Solve for Energy

Coupling and Chemical Potential

Taking the derivative w.r.t. ρ(u) leads to

π2

2ρ2(u) + αρ(u) = E − 1

2ω2u2

Simple application of Quadratic Formula yields

ρ(x) =−a +

√a2 + 2π2E − x2

π2

However, from QM, we have a normalization condition onρ(u), leading to

π

2= −

α√ω

√Eαω√

2π+( α2

ω

2π2+

Eαω

)asin

(√Eα

α2

2π2ω+ Eα

ω

)Numerically Solve for Energy

Chemical Potential

Chemical Potential

Veff = N2

∫du(π2

6ρ(u)3 − (Eα −

1

2ω2u2)ρ(u) +

α

2ρ(u)2

)Solved Analytically

Chemical Potential versus Coupling

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 100 200 300 400 500 600 700 800 900 1000

Pote

nti

al

Coupling

Potential Versus Coupling

Small Coupling Limit

???

Chemical Potential versus Coupling (Small Coupling)

-1.74

-1.72

-1.7

-1.68

-1.66

-1.64

-1.62

-1.6

-1.58

-1.56

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Pote

nti

al

Coupling

Potential Versus Coupling at Small Alpha

Large Coupling Limit

???

Chemical Potential versus Coupling (Large Coupling)

LargeAlphaRange.pdf

Conclusion

Yep

Bibliography

Burden, R., Faires, D., and Burden, A. , Numerical Analysis,Cengage, Boston, MA, 2011.

Strauss, W. , Partial Differential Equations: An Introduction,Wiley, Danvers, MA, 2008.