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== Air Max 2014 1 ==
{{quantum mechanics | cTopic = Background }}
The '''old quantum theory''' is a collection of results from the years 1900–1925 which predate modern [[quantum mechanics]]. The theory was never complete or self-consistent, but was a collection of [[heuristic]] prescriptions which are now understood to be the first quantum corrections to [[classical mechanics]].<ref>{{cite book|last = ter Haar|first =D.|title =The Old Quantum Theory|publisher=Pergamon Press|year=1967|pages = 206|isbn = 0-08-012101-2}}</ref> The [[Bohr model]] was the focus of study, and [[Arnold Sommerfeld]]<ref>{{cite book|first=Arnold |last= Sommerfeld|title=Atombau und Spektrallinien'| publisher=Friedrich Vieweg und Sohn|location=Braunschweig|year=1919|isbn=3-87144-484-7}}</ref> made a crucial contribution by quantizing the z-component of the [[angular momentum]], which in the old quantum era was called ''space quantization'' (Richtungsquantelung). This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of [[quantum degeneracy]]. The theory would have correctly explained the [[Zeeman effect]], except for the issue of electron [[Spin (physics)|spin]].


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The main tool was '''Bohr–Sommerfeld quantization''', a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model of the atom; the system can only be in one of these states and not in any states in between. The theory did not extend to chaotic motions.
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== Basic principles ==
The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys [[classical mechanics]] except that not every motion is allowed, only those motions which obey the ''old quantum condition'':


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where the <math>p_i</math> are the momenta of the system and the <math>q_i</math> are the corresponding coordinates. The quantum numbers <math>n_i</math> are ''integers'' and the integral is taken over one period of the motion at constant energy (as described by the [[hamiltonian mechanics|Hamiltonian]]). The integral is an area in phase space, which is a quantity called the action and is quantized in units of [[Planck constant|Planck's constant]]. For this reason, Planck's constant was often called the ''quantum of action''.


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In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates <math>q_i</math> in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.
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The motivation for the old quantum condition was the [[correspondence principle]], complemented by the physical observation that the quantities which are quantized must be [[adiabatic invariant]]s. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.


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=== Harmonic oscillator ===
The simplest system in the old quantum theory is the [[harmonic oscillator]], whose [[Hamiltonian (quantum mechanics)|Hamiltonian]] is:


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The level sets of ''H'' are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule:


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a result which was known well before, and used to formulate the old quantum condition. This result differs by <math>\frac{1}{2}\hbar \omega</math> from the results found with the help of [[quantum mechanics]]. This constant is neglected in the derivation of the ''old quantum theory'', and its value can not be determined using it.


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The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a [[Boltzmann weight]]:
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:<math>
U = {\sum_n \hbar\omega n e^{-\beta n\hbar\omega} \over \sum_n e^{-\beta n \hbar\omega}} = {\hbar \omega e^{-\beta\hbar\omega} \over 1 - e^{-\beta\hbar\omega}},\;\;\;{\rm where}\;\;\beta = \frac{1}{kT},
</math>


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''kT'' is [[Boltzmann constant]] times the [[Thermodynamic temperature|absolute temperature]], which is the temperature as measured in more natural units of energy. The  quantity <math>\beta</math> is more fundamental in thermodynamics than the temperature, because it is the [[thermodynamic potential]] associated to the energy.
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From this expression, it is easy to see that for large values of <math>\beta</math>, for very low temperatures, the average energy U in the Harmonic oscillator approaches zero very quickly, exponentially fast. The reason is that kT is the typical energy of random motion at temperature T, and when this is smaller than <math>\scriptstyle \hbar\omega</math>, there is not enough energy to give the oscillator even one quantum of energy. So the oscillator stays in its ground state, storing next to no energy at all.


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This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the [[specific heat]], so the specific heat is exponentially small at low temperatures, going to zero like
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::<math> \exp(-\hbar\omega/kT) </math>


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At small values of <math>\beta</math>, at high temperatures, the average energy U is equal to <math>1/\beta = kT</math>. This reproduces the [[equipartition theorem]] of classical thermodynamics: every harmonic oscillator at temperature ''T'' has energy ''kT'' on average. This means that the specific heat of an oscillator is constant in classical mechanics and equal to&nbsp;''k''. For a collection of atoms connected by springs, a reasonable model of a solid, the total specific heat is equal to the total number of oscillators times&nbsp;''k''. There are overall three oscillators for each atom, corresponding to the three possible directions of independent oscillations in three dimensions. So the specific heat of a classical solid is always 3k per atom, or in chemistry units, 3R per [[Mole (unit)|mole]] of atoms.
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Monatomic solids at room temperatures have approximately the same specific heat of 3k per atom, but at low temperatures they don't. The specific heat is smaller at colder temperatures, and it goes to zero at absolute zero. This is true for all material systems, and this observation is called the [[third law of thermodynamics]]. Classical mechanics cannot explain the third law, because in classical mechanics the specific heat is independent of the temperature.


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This contradiction between classical mechanics and the specific heat of cold materials was noted by [[James Clerk Maxwell]] in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. Einstein resolved this problem in 1906 by proposing that atomic motion is quantized. This was the first application of quantum theory to mechanical systems. A short while later, [[Peter Debye]] gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see [[Einstein solid]] and [[Debye model]]).
相关的主题文章:
 
<ul>
=== One-dimensional potential: U=0 ===
 
One-dimensional problems are easy to solve. At any energy ''E'', the value of the momentum ''p'' is found from the conservation equation:
  <li>[http://chenyang90.gotoip2.com/forum.php?mod=viewthread&tid=85689&fromuid=1926 http://chenyang90.gotoip2.com/forum.php?mod=viewthread&tid=85689&fromuid=1926]</li>
 
 
:<math>
  <li>[http://bbs.920yl.net/forum.php?mod=viewthread&tid=2003038&fromuid=141157 http://bbs.920yl.net/forum.php?mod=viewthread&tid=2003038&fromuid=141157]</li>
\sqrt{2m(E - V(q))} = p
 
</math>
  <li>[http://profile.lcbc-leb.com/activity/p/104666/ http://profile.lcbc-leb.com/activity/p/104666/]</li>
 
 
which is integrated over all values of ''q'' between the classical ''turning points'', the places where the momentum vanishes. The integral is easiest for a ''particle in a box'' of length ''L'', where the quantum condition is:
  <li>[http://www.histoirepassion.eu/spip.php?article1078/ http://www.histoirepassion.eu/spip.php?article1078/]</li>
 
 
:<math>
</ul>
2\int_0^L p \, dq = nh
</math>
 
which gives the allowed momenta:
 
:<math>
p= {nh \over 2L}
</math>
 
and the energy levels
 
:<math>
E_n= {p^2 \over 2m} = {n^2 h^2 \over 8mL^2}
</math>
 
=== One-dimensional potential: U=Fx ===
 
Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force ''F'' binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.
 
:<math>
2 \int_0^{\frac{E}{F}} \sqrt{2m(E - Fx)}\ dx= n h
</math>
 
so that the quantum condition is
 
:<math>
{4\over 3} \sqrt{2m}{ E^{3/2}\over F } = n h
</math>
 
which determines the energy levels,
 
:<math>
E_n = \left({3nhF\over 4\sqrt{2m}} \right)^{2/3}
</math>
 
In the specific case F=mg, the particle is confined by the gravitational potential of the earth and the "wall" here is the surface of the earth.
 
=== One-dimensional potential: U=kx^2 ===
 
=== Rotator ===
Another simple system is the rotator. A rotator consists of a mass ''M'' at the end of a massless rigid rod of length ''R'' and in two dimensions has the Lagrangian:
:<math>
L = {MR^2 \over 2} \dot\theta^2
</math>
 
which determines that the angular momentum ''J'' conjugate to <math>\theta</math>, the [[polar coordinates|polar angle]], <math>\scriptstyle  J = MR^2 \dot\theta</math>. The old quantum condition requires that ''J'' multiplied by the period of <math>\theta</math> is an integer multiple of Planck's constant:
 
:<math>
2\pi J = n h
\,</math>
 
the angular momentum to be an integer multiple of <math>\scriptstyle \hbar</math>. In the [[Bohr model]], this restriction imposed on circular orbits was enough to determine the energy levels.
 
In three dimensions, a rigid rotator can be described by two angles — <math>\scriptstyle \theta</math> and <math>\scriptstyle \phi</math>, where <math>\scriptstyle \theta</math> is the inclination relative to an arbitrarily chosen ''z''-axis while <math>\scriptstyle \phi</math> is the rotator angle in the projection to the ''x''–''y'' plane. The kinetic energy is again the only contribution to the Lagrangian:
 
:<math>
L = {MR^2\over 2} \dot\theta^2 + {MR^2\over 2} (\sin(\theta)\dot\phi)^2
\,</math>
 
And the conjugate momenta are <math>\scriptstyle p_\theta = \dot\theta</math> and <math>\scriptstyle p_\phi=\sin(\theta)^2 \dot\phi</math>. The equation of motion for <math>\scriptstyle \phi</math> is trivial: <math>\scriptstyle p_\phi</math> is a constant:
 
:<math>
p_\phi = l_\phi
\,</math>
 
which is the ''z''-component of the angular momentum. The quantum condition demands that the integral of the constant <math>\scriptstyle l_\phi</math> as <math>\scriptstyle \phi</math> varies from 0 to <math>2\pi</math> is an integer multiple of ''h'':
 
:<math>
l_\phi = m \hbar
\,</math>
 
And ''m'' is called the [[magnetic quantum number]], because the ''z'' component of the angular momentum is the magnetic moment of the rotator along the ''z'' direction in the case where the particle at the end of the rotator is charged.
 
Since the three-dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the ''z''-component of the angular momentum to be the integers ''l'',''m''. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox: how can the orientation of the angular momentum relative to the arbitrarily chosen ''z''-axis be quantized? This seems to pick out a direction in space.
 
This phenomenon, the quantization of angular momentum about an axis, was given the name ''space quantization'', because it seemed incompatible with rotational invariance. In modern quantum mechanics, the angular momentum is quantized the same way, but the discrete states of definite angular momentum in any one orientation are [[quantum superposition]]s of the states in other orientations, so that the process of quantization does not pick out a preferred axis. For this reason, the name "space quantization" fell out of favor, and the same phenomenon is now called the quantization of angular momentum.
 
=== Hydrogen atom ===
The angular part of the Hydrogen atom is just the rotator, and gives the quantum numbers ''l'' and ''m''. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved.
 
For a fixed value of the total angular momentum ''L'', the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):
 
:<math>
H= { p^2 \over 2 } + {l^2 \over 2 r^2 } - {1\over r}.
</math>
 
Fixing the energy to be (a negative) constant and solving for the radial momentum ''p'', the quantum condition integral is:
 
:<math>
2 \int \sqrt{2E - {l^2\over r^2} + { 2\over r}}\ dr=  k h
</math>
 
which is elementary, and gives a new quantum number ''k'' which determines the energy in combination with ''l''. The energy is:
 
:<math>
E= -{1 \over 2 (k + l)^2}
</math>
 
and it only depends on the sum of ''k'' and ''l'', which is the ''principal quantum number'' ''n''. Since ''k'' is positive, the allowed values of ''l'' for any given ''n'' are no bigger than ''n''. The energies reproduce those in the [[Bohr model]], except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values.
 
The semiclassical hydrogen atom is called the [[Arnold Sommerfeld|Sommerfeld]] model, and its orbits are ellipses of various sizes at discrete inclinations. The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the [[Stern–Gerlach experiment]].
'''Bohr–Sommerfeld theory''' is a part of the development of [[quantum mechanics]] and describes the possibility of atomic [[energy levels]] being split by a [[magnetic field]].
 
=== Relativistic orbit ===
[[Arnold Sommerfeld]] derived the relativistic solution of atomic energy levels.<ref>{{cite book | author = [[Arnold Sommerfeld]] | title = Atombau und Spektrallinien | publisher = Braunschweig | year = 1924 | isbn = 3-87144-484-7}}</ref> We will start this derivation with the relativistic equation for energy in the [[electric potential]]
 
:<math>W={m_\mathrm{0} c^2} \left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1 \right) - k \frac{Z e^2}{r}</math>
 
After substitution <math>u = \frac{1}{r}</math> we get
 
:<math>\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 1 + \frac{W}{m_\mathrm{0} c^2} + k \frac{Z e^2}{m_\mathrm{0} c^2} u</math>
 
For momentum <math>p_\mathrm{r} = m \dot{r}</math>, <math>p_\mathrm{\varphi} = m r^2 \dot{\varphi}</math> and their ratio <math>\frac{p_\mathrm{r}}{p_\mathrm{\varphi}} = - \frac{du}{d \varphi}</math> the equation of motion is (see [[Binet equation]])
 
:<math>\frac{d^2 u}{d \varphi ^2} = - \left( 1 - k^2 \frac{Z^2 e^4}{c^2 p_\mathrm{\varphi}^2} \right) u + \frac{m_\mathrm{0} kZe^2}{p_\mathrm{\varphi}^2} \left( 1+\frac{W}{m_\mathrm{0} c^2} \right) = - \omega_\mathrm{0}^2 u + K </math>
 
with solution
 
:<math> u = \frac{1}{r} = K + A \cos \omega_\mathrm{0} \varphi </math>
 
The angular shift of [[periapsis]] per revolution is given by
 
:<math> \varphi_\mathrm{s} = 2 \pi \left(\frac{1}{\omega_\mathrm{0}} - 1\right) \approx 4 \pi^3 k^2 \frac{Z^2 e^4}{c^2 n_\mathrm{\varphi}^2 h^2} </math>
 
With the quantum conditions
 
:<math> \oint p_\mathrm{\varphi} \, d \varphi = 2 \pi p_\mathrm{\varphi} = n_\mathrm{\varphi} h</math>
 
and
 
:<math> \oint p_\mathrm{r} \, d r = p_\mathrm{\varphi} \oint \left( \frac{1}{r} \frac{dr}{d \varphi} \right)^2 \, d \varphi = n_\mathrm{r} h</math>
 
we will obtain energies
 
:<math> \frac{W}{m_\mathrm{0} c^2} = \left( 1 + \frac{\alpha ^2 Z^2}{(n_\mathrm{r} + \sqrt{n_\mathrm{\varphi} ^2 - \alpha ^2 Z^2} )^2} \right) ^{-1/2} - 1</math>
 
where <math>\alpha</math> is the [[fine-structure constant]]. This solution is same as the solution of the [[Dirac equation]].<ref>{{cite journal|author=Ya I Granovski |url=http://www.iop.org/EJ/article/1063-7869/47/5/L06/PHU_47_5_L06.pdf |title=Sommerfeld formula and Dirac's theory |journal=Physics-Uspekhi |volume=47 |issue=5 |pages=523&ndash;524 |year=2004|doi=10.1070/PU2004v047n05ABEH001885|bibcode = 2004PhyU...47..523G }}</ref>
 
== De Broglie waves ==
In 1905, Einstein noted that the entropy of the quantized electromagnetic field oscillators in a box is, for short wavelength, equal to the entropy of a gas of point particles in the same box. The number of point particles is equal to the number of quanta. Einstein concluded that the quanta could be treated as if they were localizable objects
(see<ref name=einsta>{{cite journal | last = Einstein | first = Albert | authorlink = Albert Einstein | title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt | journal = Annalen der Physik | volume = 17 | pages = 132–148 | year = 1905 | url = http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf | accessdate = 2008-02-18|bibcode = 1905AnP...322..132E |doi = 10.1002/andp.19053220607 | issue = 6 }}</ref> page 139/140), particles of light, and named them [[photon]]s.  
 
Einstein's theoretical argument was based on [[thermodynamics]], on counting the number of states, and so was not completely convincing. Nevertheless, he concluded that light had attributes of [[wave particle duality|both waves and particles]], more precisely that an electromagnetic standing wave with frequency <math>\omega</math> with the quantized energy:
 
:<math>
E = n\hbar\omega
\,</math>
 
should be thought of as consisting of n photons each with an energy <math>\scriptstyle \hbar\omega</math>. Einstein could not describe how the photons were related to the wave.
 
The photons have momentum as well as energy, and the momentum had to be <math>\scriptstyle \hbar k</math> where <math>k</math> is the wavenumber of the electromagnetic wave. This is required by relativity, because the momentum and energy form a [[four-vector]], as do the frequency and wave-number.
 
In 1924, as a PhD candidate, [[Louis de Broglie]] proposed a new interpretation of the quantum condition. He suggested that all matter, electrons as well as photons, are described by waves obeying the relations.
 
:<math>
p = \hbar k
</math>
 
or, expressed in terms of wavelength <math>\lambda</math> instead,
 
:<math>
p = {h \over \lambda}
</math>
 
He then noted that the quantum condition:
 
:<math>
\int p \, dx = \hbar \int k \, dx = 2\pi\hbar n
</math>
 
counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of <math>2\pi</math>. Expressed in wavelengths, the number of wavelengths along a classical orbit must be an integer. This is the condition for constructive interference, and it explained the reason for quantized orbits—the matter waves make [[standing waves]] only at discrete frequencies, at discrete energies.
 
For example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls. The condition becomes:
 
:<math>
n\lambda = 2L
\,</math>
 
so that the quantized momenta are:
 
:<math>
p = \frac{nh}{2L}</math>
 
reproducing the old quantum energy levels.
 
This development was given a more mathematical form by Einstein, who noted that the phase function for the waves: <math>\theta(J,x)</math> in a mechanical system should be identified with the solution to the [[Hamilton–Jacobi equation]], an equation which even [[William Rowan Hamilton|Hamilton]] considered to be the short-wavelength limit of wave mechanics.
 
These ideas led to the development of the [[Schrödinger equation]].
 
== Kramers transition matrix ==
The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, [[Hendrik Kramers]] was able to find heuristics for describing how emission and absorption should be calculated.
 
Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency:
 
:<math>
X_n(t) = \sum_{k=-\infty}^{\infty} e^{ik\omega t} X_{n;k}
</math>
 
The index ''n'' describes the quantum numbers of the orbit, it would be ''n''–''l''–''m'' in the Sommerfeld model. The frequency <math>\omega</math> is the angular frequency of the orbit <math>\scriptstyle 2\pi/T_n</math> while ''k'' is an index for the Fourier mode. Bohr had suggested that the ''k''-th harmonic of the classical motion correspond to the transition from level ''n'' to level ''n''−''k''.
 
Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to  <math>|X_k|^2</math>, as it would be in classical mechanics. The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels.
 
This idea led to the development of [[matrix mechanics]].
 
== Limitations of the old quantum theory ==
The old quantum theory had some limitations:<ref>{{cite book |first=G.S. |last=Chaddha |title=Quantum Mechanics |publisher=New Age international |location=New Dehli |year=2006 |isbn=81-224-1465-6 |pages=8–9 |url=http://books.google.com/books?id=Bzj2JcPeAHAC&pg=PA9}}</ref>
* The old quantum theory provides no means to calculate the intensities of the spectral lines.
* It fails to explain the anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected).
* It cannot quantize chaotic systems, i.e. dynamical systems in which trajectories are neither closed nor periodic and whose analytical form does not exist. This presents a problem for systems as simple as a 2-electron atom which is classically chaotic analogously to the famous gravitational [[three-body problem]].
However it can be used to describe atoms with more than one electron (e.g. Helium) and the Zeeman effect.<ref>{{cite journal|last=E.A. Solov’ev|title=Classical approach in atomic physics|journal= European Physical Journal D|year=2011|volume=65|pages=331–351|doi=10.1140/epjd/e2011-20261-6|arxiv = 1003.4387 |bibcode = 2011EPJD...65..331S|first1=E. A.|issue=3 }}</ref>
It was later understood that the old quantum theory is in fact the [[WKB approximation|semi-classical approximation]] (also called quasi-classical) to the Schrödinger equation<ref>{{cite book
|author=[[Lev Landau|L.D. Landau]], [[Evgeny Lifshitz|E.M. Lifshitz]]
|year=1977
|title=Quantum Mechanics: Non-Relativistic Theory
|edition=3rd |volume=Vol. 3
|publisher=[[Pergamon Press]]
|isbn=978-0-08-020940-1
}}</ref> whose limitations are still under investigation.
 
== History ==
The old quantum theory was sparked by the work of [[Max Planck]] on the emission and absorption of light, and began in earnest after the work of [[Albert Einstein]] on the specific heats of solids. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.
 
In 1913, [[Niels Bohr]] identified the [[correspondence principle]] and used it to formulate a [[Bohr Model|model]] of the [[hydrogen atom]] which explained the [[Atomic emission spectrum|line spectrum]]. In the next few years [[Arnold Sommerfeld]] extended the quantum rule to arbitrary integrable systems making use of the principle of [[adiabatic invariant|adiabatic invariance]] of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.
 
Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. [[Max Planck]] introduced the [[zero point energy]] and [[Arnold Sommerfeld]] semiclassically quantized the relativistic hydrogen atom. [[Hendrik Kramers]] explained the [[Stark effect]]. [[Satyendra Nath Bose|Bose]] and Einstein gave the correct quantum statistics for photons.
 
Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with [[Werner Heisenberg]] to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating [[matrix mechanics]].
 
In 1924, [[Louis de Broglie]] introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 [[Erwin Schrödinger]] found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrödinger's wave mechanics developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called [[Transformation theory (quantum mechanics)|transformation theory]].
 
==References==
{{reflist}}
 
==Further reading==
* {{Cite book| editor-first=J. |editor-last= Thewlis|title=Encyclopaedic Dictionary of Physics| year=1962}}
* {{cite journal
| last = Pais
| first = Abraham
| title = Max Born's Statistical Interpretation of Quantum Mechanics
| journal = Science
| volume =218
| issue =4578
| pages =1193–8
| publisher =
| location =
| url =http://cds.cern.ch/record/141137/files/cer-000052203.pdf
| jstor =
| doi =10.1126/science.218.4578.1193
| year = 1982
| pmid = 17802457 |bibcode = 1982Sci...218.1193P }} Address to annual meeting of the Optical Society of America October 21, 1982 (Tucson AZ). Retrieved 2013-09-8 
 
 
{{DEFAULTSORT:Old Quantum Theory}}
[[Category:Quantum mechanics]]
[[Category:History of physics| ]]
 
{{Link GA|uk}}
[[de:Quantenphysik#Frühe Quantentheorien]]

Revision as of 15:17, 13 April 2013

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The main tool was Bohr–Sommerfeld quantization, a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model of the atom; the system can only be in one of these states and not in any states in between. The theory did not extend to chaotic motions.

Basic principles

The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition:

H(p,q)=Epidqi=nih

where the pi are the momenta of the system and the qi are the corresponding coordinates. The quantum numbers ni are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). The integral is an area in phase space, which is a quantity called the action and is quantized in units of Planck's constant. For this reason, Planck's constant was often called the quantum of action.

In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates qi in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.

The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.

Examples

Harmonic oscillator

The simplest system in the old quantum theory is the harmonic oscillator, whose Hamiltonian is:

H=p22m+mω2q22.

The level sets of H are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule:

E=nω,

a result which was known well before, and used to formulate the old quantum condition. This result differs by 12ω from the results found with the help of quantum mechanics. This constant is neglected in the derivation of the old quantum theory, and its value can not be determined using it.

The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a Boltzmann weight:

U=nωneβnωneβnω=ωeβω1eβω,whereβ=1kT,

kT is Boltzmann constant times the absolute temperature, which is the temperature as measured in more natural units of energy. The quantity β is more fundamental in thermodynamics than the temperature, because it is the thermodynamic potential associated to the energy.

From this expression, it is easy to see that for large values of β, for very low temperatures, the average energy U in the Harmonic oscillator approaches zero very quickly, exponentially fast. The reason is that kT is the typical energy of random motion at temperature T, and when this is smaller than ω, there is not enough energy to give the oscillator even one quantum of energy. So the oscillator stays in its ground state, storing next to no energy at all.

This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the specific heat, so the specific heat is exponentially small at low temperatures, going to zero like

exp(ω/kT)

At small values of β, at high temperatures, the average energy U is equal to 1/β=kT. This reproduces the equipartition theorem of classical thermodynamics: every harmonic oscillator at temperature T has energy kT on average. This means that the specific heat of an oscillator is constant in classical mechanics and equal to k. For a collection of atoms connected by springs, a reasonable model of a solid, the total specific heat is equal to the total number of oscillators times k. There are overall three oscillators for each atom, corresponding to the three possible directions of independent oscillations in three dimensions. So the specific heat of a classical solid is always 3k per atom, or in chemistry units, 3R per mole of atoms.

Monatomic solids at room temperatures have approximately the same specific heat of 3k per atom, but at low temperatures they don't. The specific heat is smaller at colder temperatures, and it goes to zero at absolute zero. This is true for all material systems, and this observation is called the third law of thermodynamics. Classical mechanics cannot explain the third law, because in classical mechanics the specific heat is independent of the temperature.

This contradiction between classical mechanics and the specific heat of cold materials was noted by James Clerk Maxwell in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. Einstein resolved this problem in 1906 by proposing that atomic motion is quantized. This was the first application of quantum theory to mechanical systems. A short while later, Peter Debye gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see Einstein solid and Debye model).

One-dimensional potential: U=0

One-dimensional problems are easy to solve. At any energy E, the value of the momentum p is found from the conservation equation:

2m(EV(q))=p

which is integrated over all values of q between the classical turning points, the places where the momentum vanishes. The integral is easiest for a particle in a box of length L, where the quantum condition is:

20Lpdq=nh

which gives the allowed momenta:

p=nh2L

and the energy levels

En=p22m=n2h28mL2

One-dimensional potential: U=Fx

Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force F binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.

20EF2m(EFx)dx=nh

so that the quantum condition is

432mE3/2F=nh

which determines the energy levels,

En=(3nhF42m)2/3

In the specific case F=mg, the particle is confined by the gravitational potential of the earth and the "wall" here is the surface of the earth.

One-dimensional potential: U=kx^2

Rotator

Another simple system is the rotator. A rotator consists of a mass M at the end of a massless rigid rod of length R and in two dimensions has the Lagrangian:

L=MR22θ˙2

which determines that the angular momentum J conjugate to θ, the polar angle, J=MR2θ˙. The old quantum condition requires that J multiplied by the period of θ is an integer multiple of Planck's constant:

2πJ=nh

the angular momentum to be an integer multiple of . In the Bohr model, this restriction imposed on circular orbits was enough to determine the energy levels.

In three dimensions, a rigid rotator can be described by two angles — θ and ϕ, where θ is the inclination relative to an arbitrarily chosen z-axis while ϕ is the rotator angle in the projection to the xy plane. The kinetic energy is again the only contribution to the Lagrangian:

L=MR22θ˙2+MR22(sin(θ)ϕ˙)2

And the conjugate momenta are pθ=θ˙ and pϕ=sin(θ)2ϕ˙. The equation of motion for ϕ is trivial: pϕ is a constant:

pϕ=lϕ

which is the z-component of the angular momentum. The quantum condition demands that the integral of the constant lϕ as ϕ varies from 0 to 2π is an integer multiple of h:

lϕ=m

And m is called the magnetic quantum number, because the z component of the angular momentum is the magnetic moment of the rotator along the z direction in the case where the particle at the end of the rotator is charged.

Since the three-dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the z-component of the angular momentum to be the integers l,m. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox: how can the orientation of the angular momentum relative to the arbitrarily chosen z-axis be quantized? This seems to pick out a direction in space.

This phenomenon, the quantization of angular momentum about an axis, was given the name space quantization, because it seemed incompatible with rotational invariance. In modern quantum mechanics, the angular momentum is quantized the same way, but the discrete states of definite angular momentum in any one orientation are quantum superpositions of the states in other orientations, so that the process of quantization does not pick out a preferred axis. For this reason, the name "space quantization" fell out of favor, and the same phenomenon is now called the quantization of angular momentum.

Hydrogen atom

The angular part of the Hydrogen atom is just the rotator, and gives the quantum numbers l and m. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved.

For a fixed value of the total angular momentum L, the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):

H=p22+l22r21r.

Fixing the energy to be (a negative) constant and solving for the radial momentum p, the quantum condition integral is:

22El2r2+2rdr=kh

which is elementary, and gives a new quantum number k which determines the energy in combination with l. The energy is:

E=12(k+l)2

and it only depends on the sum of k and l, which is the principal quantum number n. Since k is positive, the allowed values of l for any given n are no bigger than n. The energies reproduce those in the Bohr model, except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values.

The semiclassical hydrogen atom is called the Sommerfeld model, and its orbits are ellipses of various sizes at discrete inclinations. The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the Stern–Gerlach experiment. Bohr–Sommerfeld theory is a part of the development of quantum mechanics and describes the possibility of atomic energy levels being split by a magnetic field.

Relativistic orbit

Arnold Sommerfeld derived the relativistic solution of atomic energy levels.[3] We will start this derivation with the relativistic equation for energy in the electric potential

W=m0c2(11v2c21)kZe2r

After substitution u=1r we get

11v2c2=1+Wm0c2+kZe2m0c2u

For momentum pr=mr˙, pφ=mr2φ˙ and their ratio prpφ=dudφ the equation of motion is (see Binet equation)

d2udφ2=(1k2Z2e4c2pφ2)u+m0kZe2pφ2(1+Wm0c2)=ω02u+K

with solution

u=1r=K+Acosω0φ

The angular shift of periapsis per revolution is given by

φs=2π(1ω01)4π3k2Z2e4c2nφ2h2

With the quantum conditions

pφdφ=2πpφ=nφh

and

prdr=pφ(1rdrdφ)2dφ=nrh

we will obtain energies

Wm0c2=(1+α2Z2(nr+nφ2α2Z2)2)1/21

where α is the fine-structure constant. This solution is same as the solution of the Dirac equation.[4]

De Broglie waves

In 1905, Einstein noted that the entropy of the quantized electromagnetic field oscillators in a box is, for short wavelength, equal to the entropy of a gas of point particles in the same box. The number of point particles is equal to the number of quanta. Einstein concluded that the quanta could be treated as if they were localizable objects (see[5] page 139/140), particles of light, and named them photons.

Einstein's theoretical argument was based on thermodynamics, on counting the number of states, and so was not completely convincing. Nevertheless, he concluded that light had attributes of both waves and particles, more precisely that an electromagnetic standing wave with frequency ω with the quantized energy:

E=nω

should be thought of as consisting of n photons each with an energy ω. Einstein could not describe how the photons were related to the wave.

The photons have momentum as well as energy, and the momentum had to be k where k is the wavenumber of the electromagnetic wave. This is required by relativity, because the momentum and energy form a four-vector, as do the frequency and wave-number.

In 1924, as a PhD candidate, Louis de Broglie proposed a new interpretation of the quantum condition. He suggested that all matter, electrons as well as photons, are described by waves obeying the relations.

p=k

or, expressed in terms of wavelength λ instead,

p=hλ

He then noted that the quantum condition:

pdx=kdx=2πn

counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of 2π. Expressed in wavelengths, the number of wavelengths along a classical orbit must be an integer. This is the condition for constructive interference, and it explained the reason for quantized orbits—the matter waves make standing waves only at discrete frequencies, at discrete energies.

For example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls. The condition becomes:

nλ=2L

so that the quantized momenta are:

p=nh2L

reproducing the old quantum energy levels.

This development was given a more mathematical form by Einstein, who noted that the phase function for the waves: θ(J,x) in a mechanical system should be identified with the solution to the Hamilton–Jacobi equation, an equation which even Hamilton considered to be the short-wavelength limit of wave mechanics.

These ideas led to the development of the Schrödinger equation.

Kramers transition matrix

The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, Hendrik Kramers was able to find heuristics for describing how emission and absorption should be calculated.

Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency:

Xn(t)=k=eikωtXn;k

The index n describes the quantum numbers of the orbit, it would be nlm in the Sommerfeld model. The frequency ω is the angular frequency of the orbit 2π/Tn while k is an index for the Fourier mode. Bohr had suggested that the k-th harmonic of the classical motion correspond to the transition from level n to level nk.

Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to |Xk|2, as it would be in classical mechanics. The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels.

This idea led to the development of matrix mechanics.

Limitations of the old quantum theory

The old quantum theory had some limitations:[6]

  • The old quantum theory provides no means to calculate the intensities of the spectral lines.
  • It fails to explain the anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected).
  • It cannot quantize chaotic systems, i.e. dynamical systems in which trajectories are neither closed nor periodic and whose analytical form does not exist. This presents a problem for systems as simple as a 2-electron atom which is classically chaotic analogously to the famous gravitational three-body problem.

However it can be used to describe atoms with more than one electron (e.g. Helium) and the Zeeman effect.[7] It was later understood that the old quantum theory is in fact the semi-classical approximation (also called quasi-classical) to the Schrödinger equation[8] whose limitations are still under investigation.

History

The old quantum theory was sparked by the work of Max Planck on the emission and absorption of light, and began in earnest after the work of Albert Einstein on the specific heats of solids. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.

In 1913, Niels Bohr identified the correspondence principle and used it to formulate a model of the hydrogen atom which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.

Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. Max Planck introduced the zero point energy and Arnold Sommerfeld semiclassically quantized the relativistic hydrogen atom. Hendrik Kramers explained the Stark effect. Bose and Einstein gave the correct quantum statistics for photons.

Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with Werner Heisenberg to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating matrix mechanics.

In 1924, Louis de Broglie introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrödinger's wave mechanics developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory.

References

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Further reading

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Address to annual meeting of the Optical Society of America October 21, 1982 (Tucson AZ). Retrieved 2013-09-8

Template:Link GA de:Quantenphysik#Frühe Quantentheorien

  1. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  3. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  5. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  6. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  8. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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