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The term you are looking for is "dimensional analysis"

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I suppose that the best way to argue for this is to consider the units as indicative of the vector space from which the quantities in question originate. The algebra that we have defined in physics is one such that these quantities behave under the natural rules of commutative and associate multiplication, and so when we multiply quantities $m$ and $v$ (to ...

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For dimensional consistency, firstly, I would expect the number '4' to be dimensionfull. Additionally, whether the angle is to be taken in radians or in degrees depends on where this equation came from. If I venture a guess, then I suppose at some point you differentiated a relation between $y$ and $\theta$. In that case, the relation you started of with ...

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Yes, you need to put in an extra factor of $\hbar$ to get the units to work out. This is easily seen since, if an operator $T$ generates evolution in a parameter $\tau$, the corresponding evolution operator is $e^{iT\tau}$, so $T\tau$ must be unitless. When $\tau$ is time then $T$ has to be $H/\hbar$. This is a quirk of language usage that you'll have to ...

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They are both densities, that is they have the form $$\frac{\text{base quantity}}{\text{"volume"}}$$ The base quantity has whatever units it normally has. Amperes for current, or (dimensionless) for probability. The "volume" (which is in quotes because (a) it's not necessarily 3d as in a linear charge density and (b) can exist in abstract space with ...

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Probability, as such, has no units — it is simply a dimensionless number. A probability density, however, measures probability over a unit of space (or time, or phase space, or whatever), and thus its unit is the inverse of the unit you're using to measure the space the density is distributed over. For example, if you have a probability density over ...

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All sources of light have a spread of wavelengths. There is no such thing as a light source that produces light of exactly one wavelength. Let's assume that the power emitted by your light source looks like this: I just made up this curve, but the shape of the curve doesn't matter for this discussion. The $y$ axis shows the spectral irradiance and as you ...

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Physics numerology, where physicists play around with numbers and see what comes out, with no backing theory. In other words, nobody knows why the constants have the value they do. That is why they are fundamental. If they could be derived from something more fundamental, then...

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Bob: Alice, tell me, why do the fundamental constants have the value they have? Why is the speed of light what it is? Alice: That is not a very meaningful question. Bob: What do you mean? Alice: Physics is the art of mathematically quantifying the universe we live in. So physicists map their observations to numbers. Dimensionless numbers. And as a ...

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The sample questions above referred to $c$, $G$, and $h$, all of which have units. A dimensionful constant has the value it does because of our system of units. Therefore none of the questions is meaningful. Examples: No theory can predict the value of $G$, because $G$ has to be expressed in some units. If we express it in SI units, then we're relating it ...

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