GR8677 #88

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ali8
Posts: 100
Joined: Sat Aug 29, 2009 8:20 am

GR8677 #88

Post by ali8 » Sat Jul 02, 2011 8:02 am

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 pauli-exclusion princeple, P(fermi) > P(bose).

But how could we reach the conclusion that P(classical) is "in between these two extremes" ?

physicsworks
Posts: 80
Joined: Tue Oct 12, 2010 8:00 am

Re: GR8677 #88

Post by physicsworks » Sat Jul 02, 2011 8:52 am

$$T>T^{5/2}$$
for
$$T<1$$

ali8
Posts: 100
Joined: Sat Aug 29, 2009 8:20 am

Re: GR8677 #88

Post by ali8 » Sat Jul 02, 2011 8:56 am

physicsworks wrote:$$T>T^{5/2}$$
for
$$T<1$$
Could you please elaborate a bit on this? or, possibly, provide some reference for me to read on this specific point ?

physicsworks
Posts: 80
Joined: Tue Oct 12, 2010 8:00 am

Re: GR8677 #88

Post by physicsworks » Sat Jul 02, 2011 9:20 am

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.

ali8
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Re: GR8677 #88

Post by ali8 » Sat Jul 02, 2011 10:00 am

What do you mean by $$T>T^{5/2}$$

physicsworks
Posts: 80
Joined: Tue Oct 12, 2010 8:00 am

Re: GR8677 #88

Post by physicsworks » Sat Jul 02, 2011 10:53 am

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, $$P_{classical} > P_{bosons}$$

ali8
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Re: GR8677 #88

Post by ali8 » Sat Jul 02, 2011 12:11 pm

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 Bose-Einstein distributions...

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sphy
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Re: GR8677 #88

Post by sphy » Sun Jul 03, 2011 1:41 am

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 Bose-Einstein distributions...
Refer K Huang.

sharlene006
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Joined: Wed Oct 26, 2011 8:07 am

Re: GR8677 #88

Post by sharlene006 » Wed Oct 26, 2011 8:32 am

Hi,
thank you, exactly what I was looking for!



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