[PDF]Lecture Notes GeomMech Part 2 - Imperial College London
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Imperial College London
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by DD Holm - 2011 - Cited by 91 - Related articles
Sep 24, 2011 - WSPC/Book Trim Size for 9in by 6in. Contents. Preface xv. 1 Galileo. 1. 1.1 Principle of Galilean ... 2.3.2 Infinitesimal transformations of a Lie group. 40 ..... inside a uniformly moving ship would be unable to determine by mea-.[PDF]Geometric Mechanics, Part II: Rotating, Translating and ...
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Imperial College London
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by DD Holm - Cited by 92 - Related articles
1 Galileo. 1. 1.1 Principle of Galilean relativity . . . . . . . . . . . . . . 1. 1.2 Galilean transformations . . . . . . . . . . . . . . . . . ... 2.3.2 Infinitesimal transformations of a Lie group . . 31 ..... ship are unable to determine by measurements made inside it whether.
Covariance of the Schr¨odinger
equation under low velocity boosts.
A. B. van Oosten, Theor. Chem.& Mat. Sci. Centre,
University of Groningen, Nijenborgh 4, Groningen 9747
AG, The Netherlands
It is well-known that Schr¨odinger wave functions are not covariant
under Galilean boosts. To obtain the correct result the boost
transformation, t0 = t and ~x0 = ~x − ~vt, must be followed by the
phase shift ±Á = 1
2mv2t +m~v · ~r. A generally accepted approach
is to absorb the phase shift into the Galilean boost, construct the
Schr¨odinger group and claim Galilean invariance of the Schr¨odinger
wave function. Here I address the physical meaning of the phase
shift. It is not a coordinate transformation since it depends on
the mass of the Schr¨odinger particle. Consequently, one needs as
many Schr¨odinger groups as there are distinct masses. The phase
shift does not follow from Lorentz boost per se in the low velocity
limit. Covariance of the non-relativistic quantum mechanical
kinetic energy and momentum under pure coordinate transformations
can be satisfied only by the boost t0 = t
³
1 + 1
2
v2
c2 + ~v·~r
c2
´
and ~x0 = ~x − ~vt. Thus proper time and relativity of simultaneity
are seen to be the roots of non-relativistic quantum mechanical
inertia.
Keywords
equation under low velocity boosts.
A. B. van Oosten, Theor. Chem.& Mat. Sci. Centre,
University of Groningen, Nijenborgh 4, Groningen 9747
AG, The Netherlands
It is well-known that Schr¨odinger wave functions are not covariant
under Galilean boosts. To obtain the correct result the boost
transformation, t0 = t and ~x0 = ~x − ~vt, must be followed by the
phase shift ±Á = 1
2mv2t +m~v · ~r. A generally accepted approach
is to absorb the phase shift into the Galilean boost, construct the
Schr¨odinger group and claim Galilean invariance of the Schr¨odinger
wave function. Here I address the physical meaning of the phase
shift. It is not a coordinate transformation since it depends on
the mass of the Schr¨odinger particle. Consequently, one needs as
many Schr¨odinger groups as there are distinct masses. The phase
shift does not follow from Lorentz boost per se in the low velocity
limit. Covariance of the non-relativistic quantum mechanical
kinetic energy and momentum under pure coordinate transformations
can be satisfied only by the boost t0 = t
³
1 + 1
2
v2
c2 + ~v·~r
c2
´
and ~x0 = ~x − ~vt. Thus proper time and relativity of simultaneity
are seen to be the roots of non-relativistic quantum mechanical
inertia.
Keywords
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