## GR8677 #88

ali8
Posts: 103
Joined: Sat Aug 29, 2009 8:20 am

### 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 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

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

ali8
Posts: 103
Joined: Sat Aug 29, 2009 8:20 am

### Re: GR8677 #88

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

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
Posts: 103
Joined: Sat Aug 29, 2009 8:20 am

### Re: GR8677 #88

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

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

ali8
Posts: 103
Joined: Sat Aug 29, 2009 8:20 am

### 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 Bose-Einstein distributions...

sphy
Posts: 209
Joined: Sun Jan 30, 2011 7:23 am

### Re: GR8677 #88

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
Posts: 2
Joined: Wed Oct 26, 2011 8:07 am

### Re: GR8677 #88

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