- #1
JK423
Gold Member
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Say you have a free particle, non relativistic, and you want to calculate the density of states (number of states with energy E-E+dE).
In doing that, textbooks apply periodic boundary conditions (PBC) in a box of length L, and they get L to infinity, and in this way the states become countable.
I have some questions and i'd really like your help. Thank you in advance!
1) Can someone explain to me what is the physical meaning of choosing PBC for the particle's wavefunction?
2)Why to choose boundary conditions at all for the free particle, since there are no boundaries..?
3) If PBC is just a mathematical trick that just gives the correct result, is there another more natural way of obtaining the density of states?
4) If we apply PBC, then the wavevector is quantized: k=2πn/L, where n=integer. When we take continuous limit L-->Infinity we get Δk-->dk because L is at the denominator and we can consider k to be a continuous variable. The problem is that when we make the integration, k's limits go from -Infinity to +Infinity and at these limits the approximation of treating k as a continuous variable is false, since 'n' is comparable in magnitude with L. Which means that at the limits (infinity) for large n, the integral should be replaced by sum.
So, how do we justify this? How can we prove that the error of keeping the integral all the way to infinity is negligible?
One idea that comes to my mind is the fact that the density of states is usually integrated with someother function f(k) which falls of rapidly when k-->Infinity, so the contribution of very large k is negligible. You think that this is correct?
In doing that, textbooks apply periodic boundary conditions (PBC) in a box of length L, and they get L to infinity, and in this way the states become countable.
I have some questions and i'd really like your help. Thank you in advance!
1) Can someone explain to me what is the physical meaning of choosing PBC for the particle's wavefunction?
2)Why to choose boundary conditions at all for the free particle, since there are no boundaries..?
3) If PBC is just a mathematical trick that just gives the correct result, is there another more natural way of obtaining the density of states?
4) If we apply PBC, then the wavevector is quantized: k=2πn/L, where n=integer. When we take continuous limit L-->Infinity we get Δk-->dk because L is at the denominator and we can consider k to be a continuous variable. The problem is that when we make the integration, k's limits go from -Infinity to +Infinity and at these limits the approximation of treating k as a continuous variable is false, since 'n' is comparable in magnitude with L. Which means that at the limits (infinity) for large n, the integral should be replaced by sum.
So, how do we justify this? How can we prove that the error of keeping the integral all the way to infinity is negligible?
One idea that comes to my mind is the fact that the density of states is usually integrated with someother function f(k) which falls of rapidly when k-->Infinity, so the contribution of very large k is negligible. You think that this is correct?