Anomalies

in

Quantum Mechanics

GPSA Meeting – November 18, 2009

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Coon, Holstein, Am. J. Phys. 70 (2002) 513 – 519;

Braaten, Hammer, Annals Phys. 322 (2007) 120 – 163

Symmetry breaking – explicit, spontaneous, or anomalous

Scale-invariance anomaly and the inverse square potential

Efimov effect in three-body bound states

Outline

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Dipoles

Symmetric system

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Dipoles

Symmetric system

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Dipoles

Symmetric system

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External field bias

Explicit symmetry breaking

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Spontaneous alignment

Spontaneous symmetry breaking

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Spontaneous anti-alignment

Spontaneous symmetry breaking

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Anomalous symmetry breaking

Simple analogy

$1 $2 $3 $4 $5 $6 $7 $8

Each person receives one dollar more than he/she spends

Total cash spent equals total cash received

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Anomalous symmetry breaking

Massless fermions in one spatial dimension

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Add constant vector potential

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V (r)

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Non-relativistic Quantum Mechanics

Separation of variables:

Radial Schrödinger equation:

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V (r)

u(r)

E

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Non-interacting particle

Positive energy:

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L = 0:

L = 1:

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Invariance with respect to length scale

Scale invariance of the differential equation

Scale invariance of the solutions

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Scale invariance of the spectrum

0

p

0

p

p! p0 = p=¸ E!E0 = E=¸2

0

E

0

E

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Why not negative energy solutions?

Modified Bessel functions:

22

diverges at infinity

diverges at infinity

cusp at zero

diverges at zero

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Inverse square potential

V (r) =c

r2

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Inverse square potential

Scale invariance of the differential equation

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Positive energy:

Negative energy solutions?

still diverges at infinity

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Consider the attractive case, such that

The order of the modified Bessel function is pure imaginary

Define

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Fractal behavior near zero

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Near zero

For imaginary a

Discrete scale invariance

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E!E0 = E=¸2

E E

Bound state spectrum

0 0

¸ = ei¼®

Geometric series

¢ ¢ ¢E0¸¡4;E0¸¡2;E0;E0¸2;E0¸4; ¢ ¢ ¢

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The Efimov effect in three-body bound states

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Consider zero-range two-body interactions tuned to zero-energy two-body

resonance (infinite scattering length)

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Efimov, Sov. J. Nucl. Phys. 12 (1971) 589-595; 29 (1979) 546-553

zero range, infinite scattering length

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Cold atoms evaporatively cooled and trapped using lasers. Turn on an

external magnetic field.

Zeeman tune energy of the diatomic molecule with external magnetic field

to produce Feshbach resonance near threshold.

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In the limit of zero-range two-body interactions with zero-energy two-body

resonance

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¸ = e¼s0 ¼ 22:7

¸2 ¼ 515

Geometric series of Efimov trimers

¢ ¢ ¢E0¸¡4;E0¸¡2;E0;E0¸2;E0¸4; ¢ ¢ ¢

Confirmed in cold trap experiments using cesium atoms and also using

potassium atoms

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Zaccanti, et al., Nature Physics 5 (2009) 586 - 591

Kraemer, et al., Nature 440 (2006) 315-318

Anomalies are problems at infinity that can spoil or modify symmetries

Summary

Anomalies are most often discussed in quantum field theory, but they

also appear in simple quantum mechanics problems.

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