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.