Question on the uncertainty principle

Question

The problem statement:

Measurement detects a position of a proton with accuracy of $\pm10pm$. How much is the position uncertainty $1s$ later? Assume the speed of a proton $v\ll c$.

What i understand:

I know that in general it holds that:

\begin{align} \Delta x \Delta p \geq \frac{\hbar}{2} \end{align}

This means i can calculate momentum uncertainty for the first measurement:

\begin{align} \Delta x_1\Delta p_1 &\geq \frac{\hbar}{2}\\ \Delta p_1 &\geq \frac{\hbar}{2\Delta x_1}\\ \Delta p_1 &\geq \frac{1.055\times10^{-34}Js}{2\cdot 10\times10^{-12}m \rlap{~\longleftarrow \substack{\text{should I insert 20pm}\\\text{instead of 10pm?}}}}\\ \Delta p_1 &\geq 5.275\times10^{-24} \frac{kg m}{s}\\ \Delta p_1 &\geq 9.845 keV/c \end{align}

Question:

Using the position uncertainty $\Delta x_1$ I calculated the momentum uncertainty in first measurement $\Delta p_1$.

How do I calculate the position uncertainty $\Delta x_2$ after $1s$? I am not sure what happens $1s$ later but is a momentum uncertainty conserved so that it holds $\Delta p_1 = \Delta p_2$? I know that the problem expects me to use the clasic relation $p=m_ev$ somehow but how?

Answer 1

You are on the right track in calculating the uncertainty in momentum using the uncertainty principle.

The new position will be

$$x_2 = x_1 + \frac{p}{m} t$$

There is a well known technique of error propagation which works like

$\delta(f(x_1, x_2, x_i, ...) = \sqrt{\Sigma\left(\frac{\partial f }{\partial x_i} \delta x_i\right)^2 } $,

where $\delta x_i$ means the uncertainty in $x_i$, which is an independent coordinate (including momenta and times) of the motion. You sum over every measurement that has an uncertainty.

This comes from the Taylor series.

Applying this, you will get $$\delta x_2 = \sqrt{\delta x_1^2 + \left(\frac{t}{m} \delta p\right)^2}$$

EDIT -

I thought about this a little more and I think that the addition in quadrature is not so appropriate here. Usually you use this for measurement uncertainties, where you look for one-sigma intervals, but for quantum mechanics, where you look for complete uncertainty, it might be more correct to add the components directly.

$$\delta x_2 = \delta x_1 + \left(\frac{t}{m} \delta p\right)$$

Answer 2

Let's suppose you did the measurement in the rest frame of the proton. You've worked out that $\Delta p$ is $5.275 \times 10^{-24}$ kg.m/s. This means the probability distribution of the momentum will be some curve (usually taken to be a gaussian) with a half width of $5.275 \times 10^{-24}$. To convert this to a velocity probability distribution just divide by the proton mass, $1.673 \times 10^{-27}$ kg, and you'll find the velocity probability distribution half width is $3,150$ m/s. Therefore after 1 second the probability distribution of the proton position will have a half width of $3,150$ m.