Quantum limit

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_limit

 

A quantum limit in physics is a limit on measurement accuracy at quantum scales. Depending on the context, the limit may be absolute (such as the Heisenberg limit), or it may only apply when the experiment is conducted with naturally occurring quantum states (e.g. the standard quantum limit in interferometry) and can be circumvented with advanced state preparation and measurement schemes.

The usage of the term standard quantum limit or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical observable of a system under study that does not commute with itself at different times leads to such limits. In short, it is the Heisenberg uncertainty principle that is the cause .

A schematic description of how physical measurement process is described in quantum mechanics

 

A more detailed explanation would be that any measurement in quantum mechanics involves at least two parties, an Object and a Meter. The former is the system whose observable, say ${\displaystyle {\hat {x}}}$, we want to measure. The latter is the system we couple to the Object in order to infer the value of ${\displaystyle {\hat {x}}}$ of the Object by recording some chosen observable, ${\displaystyle {\hat {\mathcal {O}}}}$, of this system, e.g. the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as indirect measurements (see pp. 38–42 of ). So any measurement is a result of interaction and that acts in both ways. Therefore, the Meter acts on the Object during each measurement, usually via the quantity, ${\displaystyle {\hat {\mathcal {F}}}}$, conjugate to the readout observable ${\displaystyle {\hat {\mathcal {O}}}}$, thus perturbing the value of measured observable ${\displaystyle {\hat {x}}}$ and modifying the results of subsequent measurements. This is known as back action (quantum) of the Meter on the system under measurement.

At the same time, quantum mechanics prescribes that readout observable of the Meter should have an inherent uncertainty, ${\displaystyle \delta {\hat {\mathcal {O}}}}$, additive to and independent of the value of the measured quantity ${\displaystyle {\hat {x}}}$. This one is known as measurement imprecision or measurement noise. Because of the Heisenberg uncertainty principle, this imprecision cannot be arbitrary and is linked to the back-action perturbation by the uncertainty relation:

${\displaystyle \Delta {\mathcal {O}}\Delta {\mathcal {F}}\geqslant \hbar /2\,,}$

where ${\displaystyle \Delta a={\sqrt {\langle {\hat {a}}^{2}\rangle -\langle {\hat {a}}\rangle ^{2}}}}$ is a standard deviation of observable ${\displaystyle a}$ and ${\displaystyle \langle {\hat {a}}\rangle }$ stands for expectation value of ${\displaystyle a}$ in whatever quantum state the system is. The equality is reached if the system is in a minimum uncertainty state. The consequence for our case is that the more precise is our measurement, i.e the smaller is ${\displaystyle \Delta {\mathcal {\delta O}}}$, the larger will be perturbation the Meter exerts on the measured observable ${\displaystyle {\hat {x}}}$. Therefore, the readout of the meter will, in general, consist of three terms:

${\displaystyle {\hat {\mathcal {O}}}={\hat {x}}_{\mathrm {free} }+\delta {\hat {\mathcal {O}}}+\delta {\hat {x}}_{BA}[{\hat {\mathcal {F}}}]\,,}$

where ${\displaystyle {\hat {x}}_{\mathrm {free} }}$ is a value of ${\displaystyle {\hat {x}}}$ that the Object would have, were it not coupled to the Meter, and ${\displaystyle \delta {\hat {x_{BA}}}[{\hat {\mathcal {F}}}]}$ is the perturbation to the value of ${\displaystyle {\hat {x}}}$ caused by back action force, ${\displaystyle {\hat {\mathcal {F}}}}$. The uncertainty of the latter is proportional to ${\displaystyle \Delta {\mathcal {F}}\propto \Delta {\mathcal {O}}^{-1}}$. Thus, there is a minimal value, or the limit to the precision one can get in such a measurement, provided that ${\displaystyle \delta {\hat {\mathcal {O}}}}$ and ${\displaystyle {\hat {\mathcal {F}}}}$ are uncorrelated.

 

The terms "quantum limit" and "standard quantum limit" are sometimes used interchangeably. Usually, "quantum limit" is a general term which refers to any restriction on measurement due to quantum effects, while the "standard quantum limit" in any given context refers to a quantum limit which is ubiquitous in that context. 

Examples

Displacement measurement

Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement. In the scheme shown in Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body ${\displaystyle M}$. The position ${\displaystyle x}$ of ${\displaystyle M}$ is probed periodically with time interval ${\displaystyle \vartheta }$. We assume mass ${\displaystyle M}$ large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process. 

Simplified scheme of optical measurement of mechanical object position

 

Then each ${\displaystyle j}$-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position ${\displaystyle x(t_{j})}$ at the moment of reflection:

${\displaystyle {\hat {\phi }}_{j}^{\mathrm {refl} }={\hat {\phi }}_{j}-2k_{p}{\hat {x}}(t_{j})\,,}$

(1)

where ${\displaystyle k_{p}=\omega _{p}/c}$, ${\displaystyle \omega _{p}}$ is the light frequency, ${\displaystyle j=\dots ,-1,0,1,\dots }$ is the pulse number and ${\displaystyle {\hat {\phi }}_{j}}$ is the initial (random) phase of the ${\displaystyle j}$-th pulse. We assume that the mean value of all these phases is equal to zero, ${\displaystyle \langle {\hat {\phi }}_{j}\rangle =0}$, and their root mean square (RMS) uncertainty ${\displaystyle \langle ({\hat {\phi ^{2}}}\rangle -\langle {\hat {\phi }}\rangle ^{2})^{1/2}}$ is equal to ${\displaystyle \Delta \phi }$.

The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector can be done using, e.g. homodyne or heterodyne detection scheme (see Section 2.3 in  and references therein), or other such read-out techniques.

In this example, light pulse phase ${\displaystyle {\hat {\phi }}_{j}}$ serves as the readout observable ${\displaystyle {\mathcal {O}}}$ of the Meter. Then we suppose that the phase ${\displaystyle {\hat {\phi }}_{j}^{\mathrm {refl} }}$ measurement error introduced by the detector is much smaller than the initial uncertainty of the phases ${\displaystyle \Delta \phi }$. In this case, the initial uncertainty will be the only source of the position measurement error:

${\displaystyle \Delta x_{\mathrm {meas} }={\frac {\Delta \phi }{2k_{p}}}\,.}$

(2)

For convenience, we renormalise Eq. (1) as the equivalent test-mass displacement:

${\displaystyle {\tilde {x}}_{j}\equiv -{\frac {{\hat {\phi }}_{j}^{\mathrm {refl} }}{2k_{p}}}={\hat {x}}(t_{j})+{\hat {x}}_{\mathrm {fl} }(t_{j})\,,}$

(3)

where

${\displaystyle {\hat {x}}_{\mathrm {fl} }(t_{j})=-{\frac {{\hat {\phi }}_{j}}{2k_{p}}}}$

are the independent random values with the RMS uncertainties given by Eq. (2). 

Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to

${\displaystyle {\hat {p}}_{j}^{\mathrm {after} }-{\hat {p}}_{j}^{\mathrm {before} }={\hat {p}}_{j}^{\mathrm {b.a.} }={\frac {2}{c}}{\hat {\mathcal {W}}}_{j}\,,}$

(4)

where ${\displaystyle {\hat {p}}_{j}^{\mathrm {before} }}$ and ${\displaystyle {\hat {p}}_{j}^{\mathrm {after} }}$ are the test-mass momentum values just before and just after the light pulse reflection, and ${\displaystyle {\mathcal {W}}_{j}}$ is the energy of the ${\displaystyle j}$-th pulse, that plays the role of back action observable ${\displaystyle {\hat {\mathcal {F}}}}$ of the Meter. The major part of this perturbation is contributed by classical radiation pressure:

${\displaystyle \langle {\hat {p}}_{j}^{\mathrm {b.a.} }\rangle ={\frac {2}{c}}{\mathcal {W}}\,,}$

with ${\displaystyle {\mathcal {W}}}$ the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator. The random part, which cannot be compensated, is proportional to the deviation of the pulse energy:

${\displaystyle {\hat {p}}^{\mathrm {b.a.} }(t_{j})={\hat {p}}_{j}^{\mathrm {b.a.} }-\langle {\hat {p}}_{j}^{\mathrm {b.a.} }\rangle ={\frac {2}{c}}{\bigl (}{\hat {\mathcal {W}}}_{j}-{\mathcal {W}}{\bigr )}\,,}$

and its RMS uncertainly is equal to

${\displaystyle \Delta p_{\mathrm {b.a.} }={\frac {2\Delta {\mathcal {W}}}{c}}\,,}$

(5)

with ${\displaystyle \Delta {\mathcal {W}}}$ the RMS uncertainty of the pulse energy. 

Assuming the mirror is free (which is a fair approximation if time interval between pulses is much shorter than the period of suspended mirror oscillations, ${\displaystyle \vartheta \ll T}$), one can estimate an additional displacement caused by the back action of the ${\displaystyle j}$-th pulse that will contribute to the uncertainty of the subsequent measurement by the ${\displaystyle j+1}$ pulse time ${\displaystyle \vartheta }$ later:

${\displaystyle {\hat {x}}_{\mathrm {b.a.} }(t_{j})={\frac {{\hat {p}}^{\mathrm {b.a.} }(t_{j})\vartheta }{M}}\,.}$

Its uncertainty will be simply

${\displaystyle \Delta x_{\mathrm {b.a.} }(t_{j})={\frac {\Delta {p}_{\mathrm {b.a.} }(t_{j})\vartheta }{M}}\,.}$

If we now want to estimate how much has the mirror moved between the ${\displaystyle j}$ and ${\displaystyle j+1}$ pulses, i.e. its displacement ${\displaystyle \delta {\tilde {x}}_{j+1,j}={\tilde {x}}(t_{j+1})-{\tilde {x}}(t_{j})}$, we will have to deal with three additional uncertainties that limit precision of our estimate:

${\displaystyle \Delta {\tilde {x}}_{j+1,j}={\Bigl [}(\Delta x_{\rm {meas}}(t_{j+1}))^{2}+(\Delta x_{\rm {meas}}(t_{j}))^{2}+(\Delta x_{\rm {b.a.}}(t_{j}))^{2}{\Bigr ]}^{1/2}\,,}$

where we assumed all contributions to our measurement uncertainty statistically independent and thus got sum uncertainty by summation of standard deviations. If we further assume that all light pulses are similar and have the same phase uncertainty, thence ${\displaystyle \Delta x_{\rm {meas}}(t_{j+1})=\Delta x_{\rm {meas}}(t_{j})\equiv \Delta x_{\rm {meas}}=\Delta \phi /(2k_{p})}$ . 

Now, what is the minimum this sum and what is the minimum error one can get in this simple estimate? The answer ensues from quantum mechanics, if we recall that energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:

${\displaystyle \Delta {\mathcal {W}}\Delta \phi \geq {\frac {\hbar \omega _{p}}{2}}\,.}$

Therefore, it follows from Eqs. (2 and 5) that the position measurement error ${\displaystyle \Delta x_{\mathrm {meas} }}$ and the momentum perturbation ${\displaystyle \Delta p_{\mathrm {b.a.} }}$ due to back action also satisfy the uncertainty relation:

${\displaystyle \Delta x_{\mathrm {meas} }\Delta p_{\mathrm {b.a.} }\geq {\frac {\hbar }{2}}\,.}$

Taking this relation into account, the minimal uncertainty, ${\displaystyle \Delta x_{\mathrm {meas} }}$, the light pulse should have in order not to perturb the mirror too much, should be equal to ${\displaystyle \Delta x_{\mathrm {b.a.} }}$ yielding for both ${\displaystyle \Delta x_{\mathrm {min} }={\sqrt {\frac {\hbar \vartheta }{2M}}}}$. Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:

${\displaystyle \Delta {\tilde {x}}_{j+1,j}\geqslant {\Bigl [}2(\Delta x_{\rm {meas}})^{2}+{\Bigl (}{\frac {\hbar \vartheta }{2M\Delta x_{\rm {meas}}}}{\Bigr )}^{2}{\Bigr ]}^{1/2}\geqslant {\sqrt {\frac {3\hbar \vartheta }{2M}}}\,,}$

This is the Standard Quantum Limit for such a 2-pulse procedure. In principle, if we limit our measurement to two pulses only and do not care about perturbing mirror position afterwards, the second pulse measurement uncertainty, ${\displaystyle \Delta x_{\rm {meas}}(t_{j+1})}$, can, in theory, be reduced to 0 (it will yield, of course, ${\displaystyle \Delta p_{\rm {b.a.}}(t_{j+1})\to \infty }$) and the limit of displacement measurement error will reduce to:

${\displaystyle \Delta {\tilde {x}}_{SQL}={\sqrt {\frac {\hbar \vartheta }{M}}}\,,}$

which is known as the Standard Quantum Limit for the measurement of free mass displacement. 

This example represents a simple particular case of a linear measurement. This class of measurement schemes can be fully described by two linear equations of the form~(3) and (4), provided that both the measurement uncertainty and the object back-action perturbation (${\displaystyle {\hat {x}}_{\mathrm {fl} }(t_{j})}$ and ${\displaystyle {\hat {p}}^{\mathrm {b.a.} }(t_{j})}$ in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case). 

Usage in quantum optics

In the context of interferometry or other optical measurements, the standard quantum limit usually refers to the minimum level of quantum noise which is obtainable without squeezed states.

There is additionally a quantum limit for phase noise, reachable only by a laser at high noise frequencies.

In spectroscopy, the shortest wavelength in an X-ray spectrum is called the quantum limit.

Misleading relation to the classical limit

Note that due to an overloading of the word "limit", the classical limit is not the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the Armstrong limit). In "classical limit", "limit" is used in the sense of a limiting process. (Note that there is no simple rigorous mathematical limit which fully recovers classical mechanics from quantum mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)

Correspondence principle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Correspondence_principle

In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.

The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom.

The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter.
Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondence principle.

Quantum mechanics

The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classical mechanics and classical electrodynamics. If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics. Bohr's correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large. A. Sommerfeld (1921) referred to the principle as "Bohrs Zauberstab" (Bohr's magic wand).

The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". "Restricted QCC" refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered.

The post-1925 new quantum theory came in two different formulations. In matrix mechanics, the correspondence principle was built in and was used to construct the theory. In the Schrödinger approach classical behavior is not clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws.

The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The principles of quantum mechanics are broad: states of a physical system form a complex vector space and physical observables are identified with Hermitian operators that act on this Hilbert space. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.

Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because the statistical interpretation only gives the probabilities of different classical outcomes, Bohr has argued that quantum physics does not reduce to classical mechanics similarly as classical mechanics emerges as an approximation of special relativity at small velocities. He argued that classical physics exists independently of quantum theory and cannot be derived from it. His position is that it is inappropriate to understand the experiences of observers using purely quantum mechanical notions such as wavefunctions because the different states of experience of an observer are defined classically, and do not have a quantum mechanical analog. The relative state interpretation of quantum mechanics is an attempt to understand the experience of observers using only quantum mechanical notions. Niels Bohr was an early opponent of such interpretations. 

Many of these conceptual problems, however, resolve in the phase-space formulation of quantum mechanics, where the same variables with the same interpretation are utilized to describe both quantum and classical mechanics. 

Other scientific theories

The term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theory to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit". 

For example,

• Einstein's special relativity satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the speed of light (example below);
• General relativity reduces to Newtonian gravity in the limit of weak gravitational fields;
• Laplace's theory of celestial mechanics reduces to Kepler's when interplanetary interactions are ignored;
• Statistical mechanics reproduces thermodynamics when the number of particles is large;
• In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that the inherited factors are protein coding genes.

In order for there to be a correspondence, the earlier theory has to have a domain of validity—it must work under some conditions. Not all theories have a domain of validity. For example, there is no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically dominant for 18 centuries, does not have any domain of validity (on the other hand, it can sensibly be said that the falling of objects through the air ("natural motion") constitutes a domain of validity for a part of Aristotle's mechanics).

Examples

Bohr model

If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted are integer multiples of 1/T.

In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of 1/T, so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/T must have nearby energy levels which differ in energy by h/T, and they should be equally spaced near that level,

${\displaystyle \Delta E_{n}={h \over T(E_{n})}~.}$

Bohr worried whether the energy spacing 1/T should be best calculated with the period of the energy state ${\displaystyle E_{n}}$, or ${\displaystyle E_{n+1}}$, or some average—in hindsight, this model is only the leading semiclassical approximation. 

Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r^3/2. The energy scales as 1/r, so the level spacing formula amounts to

${\displaystyle \Delta E\propto {1 \over r^{3 \over 2}}\propto E^{3 \over 2}.}$

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. 

The angular momentum L of the circular orbit scales as √r. The energy in terms of the angular momentum is then

${\displaystyle E\propto {1 \over r}\propto {1 \over L^{2}}.}$

Assuming, with Bohr, that quantized values of L are equally spaced, the spacing between neighboring energies is

${\displaystyle \Delta E\propto {1 \over (L+\hbar )^{2}}-{1 \over L^{2}}\approx -{2\hbar \over L^{3}}\propto -E^{3 \over 2}.}$

This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be ħ, so the angular momentum should be an integer multiple of ħ,

${\displaystyle L={nh \over 2\pi }=n\hbar ~.}$

This is how Bohr arrived at his model. Since only the level spacing is determined heuristically by the correspondence principle, one could always add a small fixed offset to the quantum number— L could just as well have been (n+.338) ħ. 

Bohr used his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that he was able to get so much from what is only the leading order approximation. A less heuristic treatment accounts for needed offsets in the ground state L², cf. Wigner–Weyl transform.

One-dimensional potential

Bohr's correspondence condition can be solved for the level energies in a general one-dimensional potential. Define a quantity J(E) which is a function only of the energy, and has the property that

${\displaystyle {dJ \over dE}=T}$

This is the analog of the angular momentum in the case of the circular orbits. The orbits selected by the correspondence principle are the ones that obey J = nh for n integer, since

${\displaystyle \Delta E=E_{n+1}-E_{n}={dE \over dJ}(J_{n+1}-J_{n})={1 \over T}\,\Delta J}$

This quantity J is canonically conjugate to a variable θ which, by the Hamilton equations of motion changes with time as the gradient of energy with J. Since this is equal to the inverse period at all times, the variable θ increases steadily from 0 to 1 over one period.

The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J,θ coordinates is that of a half-cylinder, capped off at J = 0, which is the motionless orbit at the lowest value of the energy. These coordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-p space. 

The area enclosed by an orbit is invariant under canonical transformations, so it is the same in x-p space as in J-θ. But in the J-θ coordinates, this area is the area of a cylinder of unit circumference between 0 and J, or just J. So J is equal to the area enclosed by the orbit in x-p coordinates too,

${\displaystyle J=\int _{0}^{T}p{dx \over dt}\,dt~.}$

The quantization rule is that the action variable J is an integer multiple of h. 

Multiperiodic motion: Bohr–Sommerfeld quantization

Bohr's correspondence principle provided a way to find the semiclassical quantization rule for a one degree of freedom system. It was an argument for the old quantum condition mostly independent from the one developed by Wien and Einstein, which focused on adiabatic invariance. But both pointed to the same quantity, the action. 

Bohr was reluctant to generalize the rule to systems with many degrees of freedom. This step was taken by Sommerfeld, who proposed the general quantization rule for an integrable system,

${\displaystyle J_{k}=hn_{k}.\,}$

Each action variable is a separate integer, a separate quantum number.

This condition reproduces the circular orbit condition for two dimensional motion: let r,θ be polar coordinates for a central potential. Then θ is already an angle variable, and the canonical momentum conjugate is L, the angular momentum. So the quantum condition for L reproduces Bohr's rule:

${\displaystyle \int _{0}^{2\pi }Ld\theta =2\pi L=nh.}$

This allowed Sommerfeld to generalize Bohr's theory of circular orbits to elliptical orbits, showing that the energy levels are the same. He also found some general properties of quantum angular momentum which seemed paradoxical at the time. One of these results was that the z-component of the angular momentum, the classical inclination of an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradict rotational invariance. This was called space quantization for a while, but this term fell out of favor with the new quantum mechanics since no quantization of space is involved. 

In modern quantum mechanics, the principle of superposition makes it clear that rotational invariance is not lost. It is possible to rotate objects with discrete orientations to produce superpositions of other discrete orientations, and this resolves the intuitive paradoxes of the Sommerfeld model. 

The quantum harmonic oscillator

Here is a demonstration of how large quantum numbers can give rise to classical (continuous) behavior. 

Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values,

${\displaystyle E=(n+1/2)\hbar \omega ,\ n=0,1,2,3,\dots ~,}$

where ω is the angular frequency of the oscillator.

However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify that our idea of macroscopic systems fall within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A, is

${\displaystyle E={\frac {m\omega ^{2}A^{2}}{2}}.}$

Thus, the quantum number has the value

${\displaystyle n={\frac {E}{\hbar \cdot \omega }}-{\frac {1}{2}}={\frac {m\omega A^{2}}{2\hbar }}-{\frac {1}{2}}}$

If we apply typical "human-scale" values m = 1kg, ω = 1 rad/s, and A = 1 m, then n ≈ 4.74×10³³. This is a very large number, so the system is indeed in the correspondence limit. 

It is simple to see why we perceive a continuum of energy in this limit. With ω = 1 rad/s, the difference between each energy level is ħω ≈ 1.05 × 10⁻³⁴J, well below what we normally resolve for macroscopic systems. One then describes this system through an emergent classical limit. 

Relativistic kinetic energy

Here we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classical expression, for speeds that are much slower than the speed of light, v ≪ c. 

Einstein's mass-energy equation

${\displaystyle E={\frac {m_{0}}{\sqrt {1-v^{2}/c^{2}}}}c^{2}~,}$

where the velocity, v is the velocity of the body relative to the observer, ${\displaystyle m_{0}\ }$ is the rest mass (the observed mass of the body at zero velocity relative to the observer), and c is the speed of light. 

When the velocity v vanishes, the energy expressed above is not zero, and represents the rest energy,

${\displaystyle E_{0}=m_{0}c^{2}.\ }$

When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, by definition, the kinetic energy,

${\displaystyle T=E-E_{0}={\frac {m_{0}c^{2}}{\sqrt {1-v^{2}/c^{2}}}}\ -\ m_{0}c^{2}~.}$

Using the approximation

${\displaystyle (1+x)^{n}\approx 1+nx\ }$

for ${\displaystyle |x|\ll 1\ }$

we get, when speeds are much slower than that of light, or v ≪ c,

${\displaystyle T\ }$	${\displaystyle =m_{0}c^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)\ }$
	${\displaystyle =m_{0}c^{2}\left(\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}-1\right)\ }$
	${\displaystyle \approx m_{0}c^{2}\left((1-(-{\begin{matrix}{\frac {1}{2}}\end{matrix}})v^{2}/c^{2})-1\right)\ }$
	${\displaystyle =m_{0}c^{2}\left({\begin{matrix}{\frac {1}{2}}\end{matrix}}v^{2}/c^{2}\right)\ }$
	${\displaystyle ={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{0}v^{2}~,}$

which is the Newtonian expression for kinetic energy.