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One form of one of the de Broglie's equations is this:

$\lambda = \frac{2\pi\hbar}{p}$

Units:

$\lambda = [m]$

$\hbar = [Js]$

$p = [\frac{kg m}{s}]$

$J=[Nm]$

How can one show with dimension analysis that the right side is equal to the left side, having unit meter?

EDIT: Sorry for having used the wrong units. But I actually had them right at some point when trying this out. But I didn't "see" the last step. If someone finds this post later and have that problem this might help them,

$[m]=\frac{[Js]}{\frac{kgm}{s}}=m\frac{Ns^2}{kgm}=m\frac{N}{kg\frac{m}{s^2}}=[m]$

This follows because N is the unit for force, and $kg\frac{m}{s^2}$ is force by Newton's laws.

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Use the identity $\textrm{N}=\textrm{kg m s}^{-2}$. –  Emilio Pisanty Sep 13 '12 at 16:20
    
Althalos, it would probably be more useful if the material you edited into the question were instead posted as an answer, since you basically answered your own question. It's perfectly fine (and encouraged, when appropriate) to answer your own questions and even accept those answers. –  David Z Sep 13 '12 at 17:53
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2 Answers

up vote 0 down vote accepted

You have the units of momentum wrong. It is $[\frac{\textrm{kg} \textrm{m}}{\textrm{s}}]$. From there it is just simple cancellation.

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You just need to use the definition of Joule (J).

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by the way, your p has incorrect units (you should only have first power for m and s) –  bobuhito Sep 13 '12 at 16:05
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