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I know that the orbitals of nitrogen overlap better with the 1s orbitals of hydrogen, but I don't know how to use this information to say why ammonia is stronger than $\ce{PH3}$.

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From the Wikipedia article for phosphine:

The low dipole moment and almost orthogonal bond angles lead to the conclusion that in PH3 the P-H bonds are almost entirely pσ(P) – sσ(H) and phosphorus 3s orbital contributes little to the bonding between phosphorus and hydrogen in this molecule. For this reason, the lone pair on phosphorus may be regarded as predominantly formed by the 3s orbital of phosphorus. ... This electronic structure leads to a lack of nucleophilicity and an ability to form only weak hydrogen bonds.

And, one might assume, likely also explains the lowered basicity of phosphine relative to ammonia.

Ammonia is classically $sp^3$-hybridized, which entails that the lone pair resides in an an $sp^3$-orbital. Based on empirical evidence (some of which is mentioned above), phosphine is less $sp^3$-hybridized, with the lone pair residing in an orbital that is more like an $s$-orbital than an $sp^3$-orbital. Thus phosphine's orbital has more s-character, which means the electron pair will be held closer to the nucleus and be less inclined to bond. This is an argument rationalizing phosphine's lower basicity than ammonia.

Countering this argument, one might argue that phosphine's lone pair orbital is a $3s$, which is inherently bigger than the $2s$-orbital ammonia uses for hybridization. So "after"* hybridization, ammonia's $sp^3$-orbital is certainly bigger than its original $2s$ orbital ($2s$ has more s-character than $sp^3$), however, it may not actually be bigger than phosphine's $3s$. Thus, perhaps ammonia's hybridized $sp^3$-orbital is not so different in size than phosphine's $3s$-orbital, I suppose we would need a quantum computation to know for sure.

At the end of the day, we are only trying to rationalize an observed phenomenon after the fact. In spite all of our human ideas of orbitals, hybridization, etc, nature does as it pleases. Perhaps our models can offer a correct explanation and someone can interpret it better than I did (which shouldn't be too difficult), but also for some phenomena we must realize that our models don't fit perfectly and there are exceptions to every trend.

*Quotes used because ammonia is inherently hybridized, the act of nitrogen's $\{s, p_x, p_y, p_z\}$ atomic orbitals hybridizing when the molecule forms is a human idea that helps explain empirical observation, but may not necessarily take place physically. Indeed, the very $\{s, p_x, p_y, p_z\}$ orbitals themselves are only exact solutions for the hydrogen atom (as treated by quantum mechanics), and are only approximations of the orbitals of all other elements. But let's not get bogged down in all that.

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