TITLE: Avoiding extraneous solutions
QUESTION [1 upvotes]: When solving quadratic equations like $\sqrt{x+1} + \sqrt{x-1} = \sqrt{2x + 1}$  we are told to solve naively, for example we would get $x \in \{\frac{-\sqrt{5}}{2},\frac{\sqrt{5}}{2}\}$, even though the first solution doesn't work, and then try all the solutions and eliminate the extraneous ones. This is not a very elegant algorithm! How would one use the fact that $\sqrt{x}^2= |x|$ to avoid having to check answers?

REPLY [2 votes]: If you ensure that
$$
\begin{cases}
x+1\ge0\\
x-1\ge0\\
2x+1\ge0
\end{cases}
$$
then you can square both sides, because they are guaranteed to exist and, when $a,b\ge0$, $a=b$ if and only if $a^2=b^2$.
The conditions above are equivalent to $x\ge1$.
Squaring we get
$$
x+1+2\sqrt{x^2-1}+x-1=2x+1
$$
that simplifies to
$$
2\sqrt{x^2-1}=1
$$
and you can square again, because both sides are non negative. This gives
$$
4x^2=5.
$$
Since you know that $x\ge1$, the only solution is
$$
x=\frac{\sqrt{5}}{2}.
$$


TITLE: Moments of the position operator and wavepacket spreading
QUESTION [4 upvotes]: I've noticed that when papers in mathematical physics concern themselves with the rate at which a wavepacket spreads, they almost always try to bound the moments of the position operator (the operator whose eigenvalue gives the position of the particle).
What precisely is the connection between those moments and wavepacket spreading?

REPLY [4 votes]: Well, the absolute value squared of a wave packet $\Psi_t(x)$ has the interpretation of a time-dependent probability distribution $P_t(x)=|\Psi_t(x)|^2$ for the stochastic variable $x$ (position on a line, or in two- or three-dimensional space). As for any probability distribution, it makes sense to try to characterize it by its moments $\mu_n(t)=\int x^n P_t(x) dx$. These are by definition moments of the position operator $\hat{x}$, $\mu_n(t)=\langle\Psi_t|\hat{x}^n|\Psi_t\rangle$.
One then speaks of wavepacket spreading if the variance $\mu_2(t)-\mu_1^2(t)$ increases with time, and so bounding the second moment of the position operator is a sensible thing to do.