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- Homework Statement
- Converting between wavenumber and wavelength

- Homework Equations
- None

By dimensional analysis, we have that the wavenumber: $$k = \frac{\text{radians}}{\text{distance}}$$

And the wavelength:

$$\lambda = \frac{\text{distance}}{1 \text{wave}}$$

Then:

$$\lambda k = \frac{\text{radians}}{\text{distance}}\frac{\text{distance}}{1 \text{wave}} = \frac{\text{radians}}{1 \text{wave}}$$

Now:

$$2\pi \text{radians} = 1 \text{wave} $$

$$\frac{\text{radians}}{1 \text{wave}} = \frac{1}{2\pi} $$

So

$$\lambda k = \frac{1}{2\pi} $$

But this contradicts the "well known"

$$\lambda k = 2\pi $$

So where did I go wrong?

And the wavelength:

$$\lambda = \frac{\text{distance}}{1 \text{wave}}$$

Then:

$$\lambda k = \frac{\text{radians}}{\text{distance}}\frac{\text{distance}}{1 \text{wave}} = \frac{\text{radians}}{1 \text{wave}}$$

Now:

$$2\pi \text{radians} = 1 \text{wave} $$

$$\frac{\text{radians}}{1 \text{wave}} = \frac{1}{2\pi} $$

So

$$\lambda k = \frac{1}{2\pi} $$

But this contradicts the "well known"

$$\lambda k = 2\pi $$

So where did I go wrong?