GR8677 #88
GR8677 #88
Hello,
so here's the problem
http://grephysics.net/ans/8677/88
It's a comparison between fermi pressure, bose pressure and classical pressure at low temperatures.
I understand that, because of the pauliexclusion princeple, P(fermi) > P(bose).
But how could we reach the conclusion that P(classical) is "in between these two extremes" ?
so here's the problem
http://grephysics.net/ans/8677/88
It's a comparison between fermi pressure, bose pressure and classical pressure at low temperatures.
I understand that, because of the pauliexclusion princeple, P(fermi) > P(bose).
But how could we reach the conclusion that P(classical) is "in between these two extremes" ?

 Posts: 80
 Joined: Tue Oct 12, 2010 8:00 am
Re: GR8677 #88
$$T>T^{5/2}$$
for
$$T<1$$
for
$$T<1$$
Re: GR8677 #88
Could you please elaborate a bit on this? or, possibly, provide some reference for me to read on this specific point ?physicsworks wrote:$$T>T^{5/2}$$
for
$$T<1$$

 Posts: 80
 Joined: Tue Oct 12, 2010 8:00 am
Re: GR8677 #88
See Griffiths' QM (2nd edition), problem 1.18.
Generally, quantum effects become important at temperatures
$$T< \frac{h^2}{3 m k_B d^2}$$
wehre $$d$$ is a characteristic size of the system (interatomic spacing, for instance). Try to estimate this temperature for reasonable $$m$$ and $$d$$ and you will see that the temperature will be less than one Kelvin for most situations.
Generally, quantum effects become important at temperatures
$$T< \frac{h^2}{3 m k_B d^2}$$
wehre $$d$$ is a characteristic size of the system (interatomic spacing, for instance). Try to estimate this temperature for reasonable $$m$$ and $$d$$ and you will see that the temperature will be less than one Kelvin for most situations.
Re: GR8677 #88
What do you mean by $$T>T^{5/2}$$

 Posts: 80
 Joined: Tue Oct 12, 2010 8:00 am
Re: GR8677 #88
When $$T$$ is very small (in particular, less than 1 K), than, obviously, $$P_{classical} \propto T$$ exceeds $$P_{boson} \propto T^{5/2}$$. Of course, proportionality factors are also important, but if the temperature is extremely small (and it is, for most situations) than $$T>T^{5/2}$$ and, therefore, $$P_{classical} > P_{bosons}$$
Re: GR8677 #88
That's perfect. Thank you.
One more thing( if possible ), how could we arrive at the factor of T^5/2 ?
I don't remember seeing it in BoseEinstein distributions...
One more thing( if possible ), how could we arrive at the factor of T^5/2 ?
I don't remember seeing it in BoseEinstein distributions...
Re: GR8677 #88
Refer K Huang.ali8 wrote:That's perfect. Thank you.
One more thing( if possible ), how could we arrive at the factor of T^5/2 ?
I don't remember seeing it in BoseEinstein distributions...

 Posts: 2
 Joined: Wed Oct 26, 2011 8:07 am
Re: GR8677 #88
Hi,
thank you, exactly what I was looking for!
thank you, exactly what I was looking for!