I assume you are referring to interconverting between #lambda#, #nu#, and #tildenu#, or something like that? I say this because #lambda# is typically wavelength in #"nm"#, #nu# is typically frequency in #"s"^-1#, and #tildenu# means energy in wavenumbers (#"cm"^-1#). If you want #"nm"^(-1)#, just reciprocate #"nm"#.
There are four possibilities for conversions that I could cover:
- #lambda->nu#
- #nu->tildenu#
- #nu->lambda#
- #tildenu->nu#
However, recognize that if you can do 1 and 2, you have done 3 and 4 backwards, and if you can do 1 and 2 consecutively, you can go straight from #lambda# to #tildenu# (same with 3 and 4 but #tildenu# to #lambda#). So, I will only show 1 and 2.
Suppose we have #lambda = 600nm# for yellow light and we want its frequency in #s^-1#.
What we want is to convert from a unit of length to a unit of reciprocal time, which requires something that has #"length"/"time"# units... The speed of light works great here, and it's about #3xx10^8 "m/s"#. Therefore:
- Reciprocate the wavelength
- Convert to #"m"#
- Multiply by the speed of light
#overbrace((1/(600 cancel("nm"))))^(lambda)xx((10^9 cancel("nm"))/(1 cancel("m")))xx(3xx10^8 cancel("m")/"s") = underbrace(color(blue)(5.bar55xx10^(-3) "s"^-1))_(nu)#
This is fairly straightforward. We have #1/"s"# and want #1/"cm"#. Suppose we have a frequency of #6xx10^(-3) "s"^-1#.
- Divide by the speed of light
- Convert to #"cm"#
#overbrace((6xx10^(-3) 1/cancel("s")))^(nu)xx(cancel("s")/(3xx10^8 cancel("m")))xx((1 cancel("m"))/(100 "cm")) = underbrace(color(blue)(2xx10^(-13) "cm"^(-1)))_(tildenu)#
