Is it possible for $Delta x$ ($sigma_x$) of any free particle wave packet to be decreasing at any time?

I don't know the answer to this yet, but here is a calculation that others might find useful in determining the answer. Let's compute the time derivative of $sigma_X$. First note that $$ frac{d}{dt}sigma_X^2 = 2sigma_Xdotsigma_X, qquad sigma_X = langle X^2rangle - langle Xrangle^2 $$ so begin{align} dotsigma_X &= frac{1}{2sigma_X}frac{d}{dt}left(langle X^2rangle - langle Xrangle^2right) &= frac{1}{2sigma_X}left(frac{d}{dt}langle X^2rangle - 2langle Xranglefrac{d}{dt}langle Xrangle right) end{align} Now use the general relation $$ frac{d}{dt}langle Orangle = frac{i}{hbar}langle[H,O]rangle $$ for an operator with no explicit time-dependence. It follows that begin{align} frac{d}{dt}langle Xrangle = frac{i}{hbar}langle[P^2/2m, X]rangle =frac{i}{2mhbar}langle -2ihbar Prangle = frac{langle Prangle}{m} end{align} so that also begin{align} frac{d}{dt}langle X^2rangle &= frac{i}{hbar} leftlangleleft[frac{P^2}{2m}, X^2right]rightrangle = frac{i}{2mhbar}(-2ihbar)langle{P,X}rangle = frac{langle {P,X}rangle}{m} end{align} where ${P, X}=PX+XP$, and therefore $$ dotsigma_X = frac{1}{2msigma_X}Big(langle {P,X}rangle - 2langle P rangle langle X rangleBig) $$ We want to know if there is a state for which at some time, $dotsigma_X<0$. Since $sigma_X>0$, the expression just derived leads us to an equivalent question; is there a state for which the following inequality holds at some time? $$ langle{P,X}rangle < 2langle Prangle langle X rangleqquad ? $$ Let me know if you find any errors in the math as I did it quickly.

1. We reformulate OP's title question (v1) as follows:

   Show that for all possible wave packets $psi(p,t)$ of a free particle, the position variance $$ {rm Var}(hat{x}) ~=~langle hat{x}^2rangle-langle hat{x}rangle^2tag{1}$$ decreases in at least some interval $[t_1,t_2]$ of time.

   As Mark Eichenlaub correctly observes in his answer and comments, a solution with decreasing ${rm Var}(hat{x})$ in $[t_1,t_2]$ can be mapped by time reversal symmetry to a solution with increasing ${rm Var}(hat{x})$ in $[-t_2,-t_1]$. Here it is used that the Hamiltonian $hat{H}=frac{hat{p}^2}{2m}$ for a free particle, and the position operator $hat{x}$ both commute with the time reversal operator $hat{T}$. See e.g. my Phys.SE answer here for further details.

   However, time reversal symmetry does logically not rule out alone the possibility that a solution has monotonically increasing ${rm Var}(hat{x})$ for all time. (It only implies that if this is the case, then there will also be a solution with monotonically decreasing ${rm Var}(hat{x})$ for all time.)

2. Explicit calculation. Consider an arbitrary wave packet $$ psi(p,t)~=~A(p)e^{-itheta(p,t)} tag{2}$$ that represents the general solution to the time-dependent Schrödinger equation in the momentum representation. Here the angle $$ theta(p,t)~=~theta_0(p) + frac{p^2 }{2m}frac{t}{hbar},qquad A(p),theta_0(p)~in~mathbb{R}, tag{3}$$ is affine in $t$. We assume that the wave packet is normalized $$ 1~=~||psi(t)||^2~=~ langle psi(t)| psi(t)rangle~=~int_{mathbb{R}} !dp~A^2 .tag{4}$$ The position operator in the momentum representation reads $$ hat{x}~=~ihbarfrac{partial }{partial p}.tag{5}$$ Therefore the position average $$langle hat{x}rangle~=~ langle psi(t)| hat{x}|psi(t)rangle ~=~hbarint_{mathbb{R}} !dp~A^2theta^{prime} tag{6}$$ is affine in $t$, and the average of the squared position $$ langle hat{x}^2rangle~=~ langle psi(t)| hat{x}^2|psi(t)rangle ~=~hbar^2 int_{mathbb{R}} !dp~(A^{prime 2} +A^2theta^{prime 2} ) tag{7} $$ is quadratic in $t$. Here primes denote differentiation wrt. $p$.

   Thus we immediately know that the position variance $$ {rm Var}(hat{x})~=~at^2+bt+ctag{8}$$ is quadratic in $t$ as well, where $a$, $b$, and $c$ are constant coefficients. It is straightforward to see from the Cauchy-Schwarz inequality $$left(int_{mathbb{R}} !dp~A^2frac{p}{m} right)^2 ~leq~ int_{mathbb{R}} !dp~left(Afrac{p}{m}right)^2 cdot int_{mathbb{R}} !dp~A^2 tag{9}$$ that the second-order coefficient $$ a~=~int_{mathbb{R}} !dp~left(Afrac{p}{m}right)^2-left(int_{mathbb{R}} !dp~A^2frac{p}{m} right)^2~geq ~0, tag{10}$$ is non-negative, cf. the normalization condition (4). In fact one may show that $a>0$ is strictly positive, and hence the ${rm Var}(hat{x})$ is a decreasing parabola for $t$ sufficiently negative, as we wanted to show.

   Sketched indirect proof of $aneq 0$: The second-order coefficient $a$ becomes zero $~Leftrightarrow~$ the Cauchy-Schwarz inequality (9) becomes an equality $~Leftrightarrow~$ $A(p) p$ is proportional to $A(p)$ $~Leftrightarrow~$ $A(p)$ is proportional to a delta function $delta(p-p_0)$. But this does not correspond to a normalizable wave packet.

3. Example. Two mutually approaching wave trains are an easy intuitive example where the position variance ${rm Var}(hat{x})$ diminishes in some time-interval $[t_1,t_2]$. But this is, in some sense, a lazy example, which sort of betrays just how universal and omnipresent this behavior is for quantum mechanics.

   For instance, as we know from the general analysis in Section 2, already the simplest possible wave packet, i.e. a single Gaussian wave packet, displays this behavior. However, that is much less intuitive, and thus that much more fascinating/mind boggling to try to understand. Let us for simplicity set $hbar=1=m$.

   A single Gaussian wave packet at $t=0$ is of the form $$ psi(p,t=0)~=~N expleft(-ipc - frac{p^2}{2}tauright) ,tag{11}$$ where $tau>0$ and $c=a+ibinmathbb{C}$ are constants. Here $N>0$ is a normalization constant. The normalization condition (4) implies that $$ N~=~left(frac{tau}{pi}right)^{frac{1}{4}}expleft(-frac{b^2}{2tau}right).tag{12}$$ Thus for arbitrary time $t$, the Gaussian wave packet reads $$ begin{align}psi(p,t)~=~&psi(p,0)expleft(-ifrac{p^2}{2}tright)cr ~=~& N expleft(-ipc - frac{p^2}{2}(tau+it)right) .end{align}tag{13}$$ In the position representation, the Gaussian wave packet becomes $$ begin{align}psi(x,t) ~=~&int_{mathbb{R}} frac{dp}{sqrt{2pi}} expleft(ipxright) psi(p,t)cr ~=~&frac{N}{sqrt{tau+it}} expleft(-frac{(x-c)^2}{2(tau+it)}right).end{align} tag{14}$$ The position probability distributions becomes $$begin{align}|psi&(x,t)|^2cr ~=~&frac{N^2}{|tau+it|} expleft(-{rm Re}frac{(x-c)^2}{tau+it}right) cr ~=~&frac{N^2}{sqrt{tau^2+t^2}} expleft(-frac{[(x-a)^2-b^2]tau-2(x-a)bt}{tau^2+t^2}right) cr ~=~&sqrt{frac{tau}{pi(tau^2+t^2)}} expleft(-frac{(x-a-frac{bt}{tau})^2tau }{tau^2+t^2} right). end{align}tag{15}$$ Therefore the position average $$ langle hat{x}rangle~=~int_{mathbb{R}} !dx~x|psi(x,t)|^2 ~=~a+frac{bt}{tau}tag{16}$$ is an affine function of $t$, while the position variance $$ {rm Var}(hat{x}) ~=~int_{mathbb{R}} !dx~(x-langle hat{x}rangle)^2|psi(x,t)|^2 ~=~frac{tau^2+t^2}{2tau}tag{17}$$ is a quadratic function of $t$.

   The time symmetric profile (17) of the position variance ${rm Var}(hat{x})$ of a single Gaussian wave packet is probably somewhat surprising to anyone who borrows his intuition from classical physics.

There is yet another solution (maybe more elementary)$^1$, with some components of the answers from Qmechanic and JoshPhysics (Currently I'm taking my first QM course and I don't quite understand the solution of Qmechanic, and this answer complement JoshPhysics's answer) the solution uses the Heisenberg Equation:

The time evolution of an operator $hat{A}$ in the Heisenberg Picture of Quantum mechanics is given by:

$$frac{dhat{A}}{dt} = frac{1}{ihbar} [hat{A},hat{H}]$$

For a free particle with $hat{H} = frac{hat{p}^2}{2m} $, the time evolution of the operators $hat{x}$ and $hat{p}$ are :

$$frac{dhat{p}(t)}{dt}=0 ::::::: frac{dhat{x}(t)}{dt}=frac{hat{p}}{m}$$

These are the equations of motion in the Heisenberg picture of a free particle, now the solutions yield the same that for a classical free particle:

$$hat{p}(t) = hat{p}(0) ::::::: hat{x}(t) =hat{x}(0)+frac{hat{p}(0)}{m}t$$

With this expressions is easy to see that (or use Ehrenfest theorem ),

$$langle hat{x}(t) rangle^2 = langle hat{x}(0) rangle^2 + frac{2t}{m} langle hat{x}(0) rangle langle hat{p}(0) rangle +frac{t^2}{m^2} langle hat{p}(0) rangle^2$$ $$langle hat{x}(t)^2 rangle = langle hat{x}(0)^2 rangle + frac{t}{m}langle {hat{x}(0),hat{p}(0)} rangle +langle hat{x}(0) rangle +frac{t^2}{m^2} langle hat{p}(0)^2 rangle $$ $$langle hat{p}(t) rangle^2 = langle hat{p}(0) rangle^2$$ $$langle hat{p}(t)^2 rangle = langle hat{p}(0)^2 rangle $$

Where we defined the anticonmutator as$ {hat{x}(0),hat{p}(0)} = hat{x}(0)hat{p}(0) +hat{p}(0) hat{x}(0)$, with this we can compute the standart deviation of the momentum and position operators of the particle, now for the position operator:

begin{align} (Delta hat{x}(t) )^2 & = langle hat{x}(t)^2 rangle - langle hat{x}(t) rangle^2 \ & = (Delta hat{x}(0) )^2 +frac{t}{m} Big( langle {hat{x}(0),hat{p}(0)} rangle- 2langle hat{x}(0) rangle langle hat{p}(0) rangle Big) + frac{t^2}{m^2} (Delta hat{p}(0) )^2 end{align}

We can see that it's an expression of the form, $at^2+bt+c$ with

$$a= (Delta hat{p}(0) )^2 ::::::: b=Big( langle {hat{x}(0),hat{p}(0)} rangle- 2langle hat{x}(0) rangle langle hat{p}(0) rangle Big) ::::::: c= (Delta hat{x}(0) )^2 $$

We can see that $ a,c geq 0 $ , however there are no restrictions for $b$, it in principle can take any value, one can imagine an initial state where $b$ is negative and very big for small $t$, the standard deviation starts decreasing, but eventually for a big enough $t$ it start increasing as one can see in the section 5 of this article Wave packet spreading: Temperature and squeezing effects with applications to quantum measurement and decoherence

Its interesting that we arrive to the same result that Joshphysics with somehow different approaches.

$^1$ Again, I post this answer to complement the already given ones, personally when I first encountered this problem I was confused with this thread, I read it but I didin't quite grasp the time reversal symetry and the other solutions where confusing or to advanced for my level. I hope this gives a light to an student searching for this in the future. A good complement for starters can be found here Heisenberg Picture: U Colorado Advanced Quantum Mechanics