new anomalies in quantum mechanics - nc state university · 2009. 11. 18. · the efimov effect in...

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Anomalies in Quantum Mechanics GPSA Meeting November 18, 2009 1 Coon, Holstein, Am. J. Phys. 70 (2002) 513 519; Braaten, Hammer, Annals Phys. 322 (2007) 120 163

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Page 1: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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

Page 2: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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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Page 3: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Dipoles

Symmetric system

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Page 4: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Dipoles

Symmetric system

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Page 5: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Dipoles

Symmetric system

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Page 6: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

External field bias

Explicit symmetry breaking

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Page 7: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Spontaneous alignment

Spontaneous symmetry breaking

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Page 8: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Spontaneous anti-alignment

Spontaneous symmetry breaking

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Page 9: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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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Page 10: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Anomalous symmetry breaking

Massless fermions in one spatial dimension

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Page 11: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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Page 12: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Add constant vector potential

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Page 13: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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Page 14: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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Page 15: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

V (r)

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Non-relativistic Quantum Mechanics

Page 16: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Separation of variables:

Radial Schrödinger equation:

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Page 17: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

V (r)

u(r)

E

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Page 18: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Non-interacting particle

Positive energy:

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Page 19: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

L = 0:

L = 1:

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Page 20: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Invariance with respect to length scale

Scale invariance of the differential equation

Scale invariance of the solutions

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Page 21: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Scale invariance of the spectrum

0

p

0

p

p! p0 = p=¸ E!E0 = E=¸2

0

E

0

E

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Page 22: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Why not negative energy solutions?

Modified Bessel functions:

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Page 23: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

diverges at infinity

diverges at infinity

cusp at zero

diverges at zero

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Page 24: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Inverse square potential

V (r) =c

r2

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Page 25: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Inverse square potential

Scale invariance of the differential equation

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Page 26: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Positive energy:

Negative energy solutions?

still diverges at infinity

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Page 27: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Consider the attractive case, such that

The order of the modified Bessel function is pure imaginary

Define

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Page 28: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Fractal behavior near zero

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Page 29: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

Near zero

For imaginary a

Discrete scale invariance

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Page 30: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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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Page 31: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

The Efimov effect in three-body bound states

12

3

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

Page 32: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

zero range, infinite scattering length

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Page 33: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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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Page 34: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

12

3

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

resonance

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Page 35: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

¸ = 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

Page 36: New Anomalies in Quantum Mechanics - Nc State University · 2009. 11. 18. · The Efimov effect in three-body bound states 1 2 3 Consider zero-range two-body interactions tuned to

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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