Discover and Define Physical Quantities and Units

Find canonical units for derivatives of physical quantity variables and create new quantities and unit

Consider the quantum harmonic oscillator equation:

In[1]:=

equation=

[

ℏ

′′

[

]



ℰ

[

];

Use

QuantityVariableCanonicalUnit

to examine the canonical units and unit dimensions of the terms in the equation:

In[2]:=

GridPrependThrough[{Identity,QuantityVariableCanonicalUnit,QuantityVariableDimensions}[#]]&/@

[

ℏ

′′

[

]

ℰ

[

],{"Term","Canonical Units","Unit Dimensions"},FrameAll

Out[2]=

Term	Canonical Units	Unit Dimensions
1 2 m 2 x 2 ω ψ [ x ]	2 Hertz Kilograms 2 Meters	{{LengthUnit,2},{MassUnit,1},{TimeUnit,-2}}
- 1 2 2 ℏ ′′ ψ [ x ] m	2 ReducedPlanckConstant Kilograms 2 Meters	{{LengthUnit,2},{MassUnit,1},{TimeUnit,-2}}
ℰ ψ [ x ]	Joules	{{LengthUnit,2},{MassUnit,1},{TimeUnit,-2}}

Confirm the canonical units are compatible:

In[3]:=

CompatibleUnitQ

"Hertz"

"Kilograms"

"Meters"

,"Joules",CompatibleUnitQ

"ReducedPlanckConstant"

"Kilograms"

"Meters"

,"Joules"

Out[3]=

{True,True}

Examine the canonical unit of a multidimensional function:

In[4]:=

QuantityVariableCanonicalUnit

(2,1)

[

]

Out[4]=

3/2

Meters

Seconds

Define physical quantities in terms of combinations of existing physical quantities, independent physical quantities, or even a list of unit dimensions:

In[5]:=

x=QuantityVariable["x","ElectricPotential"/"Altitude"];q=QuantityVariable["Q",IndependentPhysicalQuantity["Widgets"]/"Height"];s=QuantityVariable["s",{{"ElectricCurrentUnit",1},{"LengthUnit",2},{IndependentUnitDimension["Widgets"],-1},{"MassUnit",-1},{"TimeUnit",3}}];

Derive canonical units based on these physical quantities:

In[6]:=

QuantityVariableCanonicalUnit/@{x,q,s}

Out[6]=



Volts

Meters

IndependentUnit[Widgets]

Meters

Amperes

Meters

Seconds

GramsIndependentUnit[Widgets]



Determine their unit dimensions:

In[7]:=

Grid[{#,QuantityVariableDimensions[#]}&/@{x,q,s},FrameAll]

Out[7]=

x	{{ElectricCurrentUnit,-1},{LengthUnit,1},{MassUnit,1},{TimeUnit,-3}}
Q	{{LengthUnit,-1},{IndependentUnitDimension[Widgets],1}}
s	{{ElectricCurrentUnit,1},{LengthUnit,2},{MassUnit,-1},{TimeUnit,3},{IndependentUnitDimension[Widgets],-1}}

Find a combination of the variables that match the dimensions of a specific physical quantity:

In[8]:=

DimensionalCombinations[{x,q,s},"Distance",GeneratedParametersNone]

Out[8]=





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Contributed by: Wolfram Staff

Discover and Define Physical Quantities and Units

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