Some more considerations are necessary: If \omega has the dimension of an inverse length and \tau has dimension of inverse time (Hertz), then the sum \omega + \frac{i}{\tau} has incompatible units. With other words, if the chemical potenial \mu has unit Gibbs then the sum \omega + \frac{i}{\tau} needs dimension
In[121]:= UnitConvert["Gibbs"/"ReducedPlanckConstant"]
Out[121]= Quantity[9.482522*10^33, 1/("Moles" "Seconds")]
with other words Hertz/Moles. Let’s say you fix that, you can in principle go ahead as follows
Clear[e, k, \[HBar], c, \[Omega], T, \[Tau], \[Sigma]]
e = Quantity["ElementaryCharge"];
k = Quantity["BoltzmannConstant"];
\[HBar] = Quantity["ReducedPlanckConstant"];
c = Quantity["SpeedOfLight"];
(* \[Omega]=2 \[Pi]/\[Lambda];
\[Lambda] = Quantity[445,"Nanometers"]; *)
\[Tau] = Quantity[2178, "Hertz"];
T = Quantity[300, "Kelvin"];
\[Sigma][\[Mu]_, \[Omega]_] :=
I (e^2 k T)/(\[Pi] \[HBar] (\[Omega] + I/\[Tau])) (\[Mu]/(k T) +
2 Log[Exp[-\[Mu]/(k T)] + 1]) +
I e^2/(4 \[Pi] \[HBar]) Log[(2 Abs[\[Mu]] - \[HBar] (\[Omega] +
I/\[Tau]))/(2 Abs[\[Mu]] + \[HBar] (\[Omega] + I/\[Tau]))]
calling it like
\[Sigma][Quantity[1.4, "Gibbs"], 2 \[Pi]/Quantity[445, "Nanometers"]]
to get an error message
Quantity::compat: 1/(Nanometers) and 1/(Hertz) are incompatible units >>
of course.
Thank you very much!
I’m sorry for my mistake that omega has the dimension of inverse time Hertz beause of omega=c*k[SpeedOfLight multiply VectorNumber]
right now ,I want define [Mu] as a variable with dimension of ElectronVolt.
I can’t get it in my computer’s mathmatica as following code : [Mu]=QuantityVariable[[Mu],“ElectronVolt”]
how to rectity my falut to get my purpose to plot [Sigma] vs diffrent [Mu] ranging from 0 eV to 0.5 eV.
farther more.
I want plot sigma with variable [Mu] ranging from 0eV(ElectronVolts) to 0.5eV;[Lamda] ranging from 1400nm to 1600nm ,which changes the values of [Omega]. and parameters [tau] ranging from 0.1ps[picoseconds] to 0.5ps.
Would you please help me to rectify my fault .Thanks!
Well
In[33]:= Clear[e, k, \[HBar], c, \[Omega], T, \[Tau], \[Sigma]]
e = Quantity["ElementaryCharge"];
k = Quantity["BoltzmannConstant"];
\[HBar] = Quantity["ReducedPlanckConstant"];
c = Quantity["SpeedOfLight"];
(* \[Omega]=2 \[Pi] c/\[Lambda];
\[Lambda] = Quantity[445,"Nanometers"];
\[Tau] =Quantity[0.2,"Picoseconds"]; *)
T = Quantity[300, "Kelvin"];
\[Sigma][\[Mu]_, \[Omega]_, \[Tau]_] :=
I (e^2 k T)/(\[Pi] \[HBar]^2 (\[Omega] + I/\[Tau])) (\[Mu]/(k T) +
2 Log[Exp[-\[Mu]/(k T)] + 1]) +
I e^2/(4 \[Pi] \[HBar]) Log[(2 Abs[\[Mu]] - \[HBar] (\[Omega] +
I/\[Tau]))/(2 Abs[\[Mu]] + \[HBar] (\[Omega] + I/\[Tau]))]
In[40]:= \[Sigma][Quantity[0.2, "ElectronVolts"],
2 \[Pi] c/Quantity[1400, "Nanometers"], Quantity[0.2, "Picoseconds"]]
Out[40]= Quantity[(1.90904 - 0.0426355 I), (
"BoltzmannConstant" (
"ElementaryCharge")^2 "Kelvins" \
"Picoseconds")/("ReducedPlanckConstant")^2]
In[41]:= UnitConvert[%]
Out[41]= Quantity[(0.0000608367 - 1.3587*10^-6 I), (("Amperes")^2 (
"Seconds")^3)/("Kilograms" ("Meters")^2)]
In[46]:= QuantityMagnitude[\[Sigma][Quantity[0.2, "ElectronVolts"],
2 \[Pi] c/Quantity[1400, "Nanometers"],
Quantity[0.2, "Picoseconds"]]]
Out[46]= 1.90904 - 0.0426355 I
and it is plottable like
Plot3D[ReIm[QuantityMagnitude[\[Sigma][Quantity[0.4, "Electronvolts"],
2 \[Pi] c/Quantity[x, "Nanometers"],
Quantity[y, "Picoseconds"]]]], {x, 1400, 1600}, {y, 0.1, 0.5}, PlotPoints -> 10, MaxRecursion -> 0]
The real part is above. If one unuses the options PlotPoints -> 10, MaxIterations -> 0 the plot takes a very long time. Using Quantity in numerics and graphics is definitely an overkill borderlining abuse, the usual method is to compute dimensionless expressions by remembering the actual physical dimension only in plot labels and axes labels.
Thank you! that’s very nice of you !
