Abstract
We rederive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.
| Original language | English |
|---|---|
| Article number | 030102 |
| Journal | Physical Review A - Atomic, Molecular, and Optical Physics |
| Volume | 86 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - Sep 24 2012 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Atomic and Molecular Physics, and Optics
Cite this
Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator. / Mandilara, A.; Cerf, N. J.
In: Physical Review A - Atomic, Molecular, and Optical Physics, Vol. 86, No. 3, 030102, 24.09.2012.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator
AU - Mandilara, A.
AU - Cerf, N. J.
PY - 2012/9/24
Y1 - 2012/9/24
N2 - We rederive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.
AB - We rederive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.
UR - http://www.scopus.com/inward/record.url?scp=84866686189&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84866686189&partnerID=8YFLogxK
U2 - 10.1103/PhysRevA.86.030102
DO - 10.1103/PhysRevA.86.030102
M3 - Article
VL - 86
JO - Physical Review A
JF - Physical Review A
SN - 1050-2947
IS - 3
M1 - 030102
ER -