User:John David Wright/sandbox

see #Wavenumber comment to user:Hankwang

Quantity in the International Vocabulary of Basic and General Terms in Metrology

A quanitity is often expressed using a reference unit. The technical term "quantity" appears extensively in an authoritative brochure entitled The International System of Units distributed by the Bureau International des Poids et Mesures. The introduction of this brochure states that, "The terms quantity and unit are defined in the International Vocabulary of Basic and General Terms in Metrology, the VIM." The International Vocabulary of Basic and General Terms in Metrology contains authoritative terms compiled by the International Organization for Standardization and offers the following definition for the term "quantity".

quantity

property of a phenomenon, body, or substance, to which a magnitude can be assigned

NOTES

1 The concept ‘quantity’ can be subdivided in two levels, general concept and individual concept.

2 Symbols for quantities are given in the International Standard ISO 31:1992, Quantities and units.

3 In laboratory medicine, where lists of individual quantities are presented, each designation is conventionally given in the exhaustive and unambiguous IUPAC/IFCC format “System Component; kind-of-quantity”, where 'System' is the object under consideration, mostly having a component of special interest, and 'kind-of-quantity' (or 'kind-of-property' if nominal properties are included) is a label for general concepts such as ‘length’, ‘diameter’, and ‘amount-of-substance concentration’. An example of an individual concept under ‘quantity’ could be 'Plasma (Blood) Sodium ion; amount-of-substance concentration equal to 143 mmol/l in a given person at a given time’.

4 A vector or a tensor can also be a quantity if all its components are quantities.

Unit in The International System of Units brochure

The unit is simply a particular example of the quantity concerned which is used as a reference, and the number is the ratio of the value of the quantity to the unit. For a particular quantity, many different units may be used.

See also Quantities, Units and Symbols in Physical Chemistry

Wavenumber comment to user:Hankwang

In his revision of "...which confuses the role of a unit and its dimension" to "...which confuses the role of a dimension with that of the name of a quantity" user:Hankwang seems to regard the word "dimension" as a synonym of "unit". User:Mbeychok holds the same belief as implied by his article on Units conversion by factor-label.

Consider the example in the physical quantity article. An analogous example for wavenumber would read as follows.

If a certain value of wavenumber is written as

${\tilde {\nu }}=\{{\tilde {\nu }}\}[{\tilde {\nu }}]=300\,\mathrm {cm} ^{-1}$

then

${\tilde {\nu }}$ represents the physical quantity

300 is the numerical value $\{{\tilde {\nu }}\}$

$\mathrm {cm} ^{-1}$ represents the unit $[{\tilde {\nu }}]$, i.e., the reciprocal centimeter.

I can also make the statement that the quantity 300 cm^-1 has dimensions of wavenumber. Using this example, let us consider of the wavenumber article and the following versions of the setence in question.

1. In colloquial usage, the unit cm^-1 is sometimes pronounced as "wavenumber", which confuses the role of a unit with that of a quantity.

2. In colloquial usage, the unit cm^-1 is sometimes pronounced as "wavenumber", which confuses the role of a unit with that of its physical dimension.

3. In colloquial usage, the unit cm^-1 is sometimes pronounced as "wavenumber", which confuses the role of a dimension with that of the name of a quantity.

Concerning sentence 1, a unit itself is a quantity. Concerning sentence 2, I was using dimension in the same sense that the first table in the Centimeter gram second system of units article uses the term dimension. Wavenumber is a dimension just like frequency is a dimension. A search of the internet for "has dimensions of wave number" or "has dimensions of wavenumber" or "has dimensions of frequency" produces examples from peer-reviewed scientific journals. However, the term dimension could also imply "inverse length" instead of "wavenumber". Concerning sentence 3, the word "dimension" is not a synonym of "unit". Therefore the meaning of all three setences is not exactly clear.

Thus I make the following suggestions:

a. In colloquial usage, the cgs unit of wavenumber, cm^-1, is sometimes called a "wavenumber", which confuses the name of a unit with that of a quantity.

b. In colloquial usage, the cgs unit of wavenumber, cm^-1, is sometimes called a "wavenumber", which confuses the name for a unit with the name for a kind-of-quantity. (This sentence conveys the possibility of confusion with the m^-1)

c. In colloquial usage, the cgs unit of wavenumber, cm^-1, is sometimes called a "wavenumber", which confuses the name of a unit with that of a general concept.

1. In colloquial usage, the cgs unit of wavenumber, cm^-1, is sometimes called a "wavenumber", which confuses the name of a unit with the name of a general quantity.

2. In colloquial usage, the symbol cm^-1 is sometimes called a "wavenumber", which confuses the name of a symbol with the name of a more general quantity.

3. In colloquial usage, the unit cm^-1 is sometimes called a "wavenumber", which confuses the unit with the kind-of-quantity it represents.

4. In colloquial usage, the unit cm^-1 is sometimes called a "wavenumber". Thus the unit is confused with the kind of physical quantity it represents.

5. In colloquial usage, the symbol for the unit of wavenumber, cm^-1, is sometimes called a "wavenumber". Thus the symbol is confusingly given the same name as the kind of physical quantity it represents.

Introduction

A wave vector is a vector that specifies the wavenumber and direction of propagation for a wave. The magnitude of the wave vector indicates the wavenumber. The direction vector of the wave vector indicates the direction of wave propagation.

For example consider a plane wave. A common representation of the oscillation at time (t) and a single point in space (z) along the direction of propagation is:

$\psi \left(t,z\right)=A\cos \left(\varphi +kz+\omega t\right),$

where A is the amplitude, φ is the starting phase of the wave, k is the angular wavenumber, and ω is the angular frequency. We can easily extend the formula by substituting the dot product of the wave vector k and the position vector r for the scalar product of the wavenumber k and the variable z as follows:

$\psi \left(t,{\mathbf {r} }\right)=A\cos \left(\varphi +{\mathbf {k} }\cdot {\mathbf {r} }+\omega t\right).$

In three dimensions the the wave vector (${\mathbf {k} }$) is formally written:

${\mathbf {k} }=[k_{x},k_{y},k_{z}].$

or

${\mathbf {k} }=k_{x}{\hat {x}}+k_{y}{\hat {y}}+k_{z}{\hat {z}}.$

or other ways depending on one's choice of vector notation. Here $k_{x}$, $k_{y}$, and $k_{z}$ are the wave vector's individual components of each cartesian coordinate $x$, $y$, and $z$. The wave vector has a magnitude, i.e., angular wavenumber ($k$) of:

$k={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}.$

and direction vector (${\hat {k}}$) of:

${\hat {k}}={\mathbf {k} }/{k}.$

The direction of propagation is also uniquely determined by the three angles A, B, and C between the wave vector and the coordinate axes x, y, and z, respectively. These angles are given by the following relationships:

$A=\arccos \left({\frac {k_{x}}{k}}\right),B=\arccos \left({\frac {k_{y}}{k}}\right),C=\arccos \left({\frac {k_{z}}{k}}\right).$

In special relativity

A wave packet of nearly monochromatic light can be characterized by the wave vector

$k^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)\,$

which, when written out explicitly in its covariant and contravariant forms is

$k^{\mu }=\left({\frac {\omega }{c}},k^{1},k^{2},k^{3}\right)\,$ and

$k_{\mu }=\left({\frac {\omega }{c}},-k_{1},-k_{2},-k_{3}\right).\,$

The magnitude of this wave vector is then

$k^{2}=k^{\mu }k_{\mu }=k^{0}k_{0}-k^{1}k_{1}-k^{2}k_{2}-k^{3}k_{3}\,$

$={\frac {\omega ^{2}}{c^{2}}}-{\vec {k}}^{2}=0.\,$

That last step where it equals zero, is a result of the fact that, for light,

$k={\frac {\omega }{c}}.\,$

Lorentz transform

Taking the Lorentz transform of the wave vector is one way to derive the Relativistic Doppler effect. The Lorentz matrix is defined as

$\Lambda ={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}.$

In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the lorentz transform as follows. Note that the source is in a frame S^s and earth is in the observing frame, S^obs. Applying the lorentz transformation to the wave vector

$k_{s}^{\mu }=\Lambda _{\nu }^{\mu }k_{\mathrm {obs} }^{\nu }\,$

and choosing just to look at the $\mu =0$ component results in

$k_{s}^{0}=\Lambda _{0}^{0}k_{\mathrm {obs} }^{0}+\Lambda _{1}^{1}k_{\mathrm {obs} }^{1}+\Lambda _{2}^{2}k_{\mathrm {obs} }^{2}+\Lambda _{3}^{3}k_{\mathrm {obs} }^{3}\,$

${\frac {\omega _{s}}{c}}\,$	$=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma k_{\mathrm {obs} }^{1}\,$
	$\quad =\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma {\frac {\omega _{\mathrm {obs} }}{c}}\cos \theta .\,$

So

${\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta \cos \theta )}}\,$

Source moving away

As an example, to apply this to a situation where the source is moving strait away from the observer ($\theta =\pi$), this becomes:

${\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1+\beta )}}={\frac {\sqrt {1-\beta ^{2}}}{1+\beta }}={\frac {\sqrt {(1+\beta )(1-\beta )}}{1+\beta }}={\frac {\sqrt {1-\beta }}{\sqrt {1+\beta }}}\,$

Source moving towards

To apply this to a situation where the source is moving strait towards the observer ($\theta =0$), this becomes:

${\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {\sqrt {1+\beta }}{\sqrt {1-\beta }}}\,$

See Also

[1]

References

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4.

Category:Wave mechanics