new anomalies in quantum mechanics - nc state university · 2009. 11. 18. · the efimov effect in...
TRANSCRIPT
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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12
3
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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