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)
