Problem about momentum uncertainty

[sktsasus]

Homework Statement

According to the Big Bang model of cosmology, the universe has been expanding since some initial time (call it t = 0) when the temperature was infinite. At early times, the temperature T scales as t^1/2 . The current temperature is about 3K. Consider the part of space which corresponds to the currently visible universe. It is about 10^10Mpc. Consider a proton confined to this part of space. Assuming that the proton is an elementary particle, then at what temperature does the momentum uncertainty become so large that we cannot tell if there is one photon in the volume or more than one?

Homework Equations

delta x*delta p being greater than or equal to h/4pi

The Attempt at a Solution

I could not quite figure out how to move forwards after the initial formula. There does not seem to be any values for x, p or h I can insert to gain some values.

Any help?

 

[anathema]

Hello,

Thank you for your question. To solve this problem, we need to use the Heisenberg uncertainty principle, which states that the uncertainty in position (delta x) and momentum (delta p) of a particle cannot be simultaneously known with complete precision. This principle is described by the formula you mentioned: delta x * delta p >= h/4pi, where h is Planck's constant.

In this case, we are trying to determine the temperature at which the momentum uncertainty of a proton becomes so large that we cannot distinguish between one photon or more than one in the given volume. To do this, we need to set the momentum uncertainty equal to the minimum uncertainty allowed by the Heisenberg uncertainty principle, which is h/4pi.

So, our equation would be: delta x * delta p = h/4pi

Now, we know that delta x is the size of the volume in which the proton is confined, which is given as 10^10Mpc. We also know that delta p is related to the temperature by the formula delta p = kT, where k is the Boltzmann constant. Therefore, our equation becomes:

10^10Mpc * kT = h/4pi

We are trying to solve for the temperature (T), so we can rearrange the equation to get:

T = h/4pi * 1/(10^10Mpc * k)

Now, we just need to plug in the values for h, pi, and k to get the temperature at which the momentum uncertainty becomes large enough. I will leave that part for you to calculate. I hope this helps! Let me know if you have any further questions.