SamRoss
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- TL;DR
- Reading Griffiths. He derives some formulas but I'm not following.
In "Introduction to Quantum Mechanics", Griffiths derives the following formulae for counting the number of configurations for N particles.
Distinguishable particles...
$$ N!\prod_{n=1}^\infty \frac {d^{N_n}_n} {N_n !} $$
Fermions...
$$ \prod_{n=1}^\infty \frac {d_n!} {N_n!(d_n-N_n)!}$$
Bosons...
$$\prod_{n=1}^\infty \frac {(N_n+d_n-1)!} {N_n!(d_n-1)!}$$
In the above, ##N_n## stands for the number of particles in the nth state and ##d_n## stands for the degeneracy of the nth state. My confusion is not with the mathematics of combinatorics but only how it is being used here. Griffiths speaks of "picking particles" and "bins" and I'm only vaguely able to follow his argument. Furthermore, I can't search for these formulae online because they are not named in the text and I have been unsuccessful with searches for "quantum statistics", "configurations", etc. Can anyone here recognize these formulae and point me to a derivation of them? Thanks.
Distinguishable particles...
$$ N!\prod_{n=1}^\infty \frac {d^{N_n}_n} {N_n !} $$
Fermions...
$$ \prod_{n=1}^\infty \frac {d_n!} {N_n!(d_n-N_n)!}$$
Bosons...
$$\prod_{n=1}^\infty \frac {(N_n+d_n-1)!} {N_n!(d_n-1)!}$$
In the above, ##N_n## stands for the number of particles in the nth state and ##d_n## stands for the degeneracy of the nth state. My confusion is not with the mathematics of combinatorics but only how it is being used here. Griffiths speaks of "picking particles" and "bins" and I'm only vaguely able to follow his argument. Furthermore, I can't search for these formulae online because they are not named in the text and I have been unsuccessful with searches for "quantum statistics", "configurations", etc. Can anyone here recognize these formulae and point me to a derivation of them? Thanks.