- #1
zellwwf
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Hello PF :)
Consider a system of N electrons at temperature T = 0, each having a mass of m, confined to volume V. Find the number of electrons that:
a) have momentum [itex]p<p_f/2[/itex]
b) ...
The relevant equations can be derived below:
but here is a sample of them
[itex]\int F(k)D(k)d\vec{k}[/itex]
where F: How many electrons per k-state. function of magnitute of K
where D: density in k-space. (since we turned a discrete sum into an integral), function of magnitude of K too (although it's usually a constant)
I have two attempted solutions for A, and they are NOT equal, they differ by a factor of h:
Attempt 1:
note, i will be using the reduced Planck's constant, but i can't find the bar, so i will denote it by H.
[itex]\vec{p}=h \vec{k}[/itex]
[itex]\frac{\vec{p}}{h}=\vec{k}[/itex]
thus:
[itex]d\vec{p}=h d\vec{k}[/itex]
[itex]\frac{1}{h}d\vec{p}=d\vec{k}[/itex]
Now we need to calculate the density of states in K-space (too early in the morning to write it down how i got it, but i believe it's correct):
[itex]D(k) = \frac{2V}{(2 \pi)^3}[/itex]
and since:
[itex]F(k) = 1[/itex] at T = 0;
disregarding spin, as we just multiply by two if we need to 'account' for it in this question
now we can replace these results in our integral above:
[itex]\int^{\frac{p_f}{2}}_{0}F(k)D(k)d\vec{k} = \frac{V}{4h\pi^3}\int^{\frac{p_f}{2h}}_{0}d\vec{p} [/itex]
transforming to spherical coordinates reduces the integral to:
[itex]=\frac{V}{4h\pi^3}\int^{\frac{p_f}{2h}}_{0}4\pi p^2dp [/itex]
[itex]=\frac{V}{h\pi^2}\int^{\frac{p_f}{2h}}_{0}p^2dp[/itex]
[itex]=\frac{V}{24h^4\pi^2}p_f^3[/itex]
----------------------------------------------------------------------------------------------------
Now I will attempt two:
i will try to derive the integral itself for P-space, not K-Space
the only thing that i believe changes is the Density function, D(p)
since:
[itex]\vec{k}=\frac{\vec{p}}{h}=\frac{2\pi}{L}*(\vec{i}+\vec{j}+\vec{k})[/itex]
thus:
[itex]\vec{p}=\frac{2h\pi}{L}*(\vec{i}+\vec{j}+\vec{k})[/itex]
thus D(p) can be obtained:
[itex]D(p) = \frac{2L^3}{8h^3\pi^3}=\frac{V}{4h^3\pi^3}[/itex]
F(p) = 1 also, at T = 0; (again, disregarding spin)
so now we need to only evaluate the integral:
[itex]\int F(p)D(p)d\vec{p}[/itex]
[itex]=\frac{V}{4h^3\pi^3}\int^{\frac{p_f}{2}}_{0}d\vec{p}[/itex]
[itex]=\frac{V}{4h^3\pi^3}\int^{\frac{p_f}{2}}_{0}4\pi p^2dp[/itex]
[itex]=\frac{V}{3h^3\pi^2}p_f^3[/itex]
-----
Apparently they are not equal,
i know i made a mistake, or two, or three..
can someone show me :)
Homework Statement
Consider a system of N electrons at temperature T = 0, each having a mass of m, confined to volume V. Find the number of electrons that:
a) have momentum [itex]p<p_f/2[/itex]
b) ...
Homework Equations
The relevant equations can be derived below:
but here is a sample of them
[itex]\int F(k)D(k)d\vec{k}[/itex]
where F: How many electrons per k-state. function of magnitute of K
where D: density in k-space. (since we turned a discrete sum into an integral), function of magnitude of K too (although it's usually a constant)
The Attempt at a Solution
I have two attempted solutions for A, and they are NOT equal, they differ by a factor of h:
Attempt 1:
note, i will be using the reduced Planck's constant, but i can't find the bar, so i will denote it by H.
[itex]\vec{p}=h \vec{k}[/itex]
[itex]\frac{\vec{p}}{h}=\vec{k}[/itex]
thus:
[itex]d\vec{p}=h d\vec{k}[/itex]
[itex]\frac{1}{h}d\vec{p}=d\vec{k}[/itex]
Now we need to calculate the density of states in K-space (too early in the morning to write it down how i got it, but i believe it's correct):
[itex]D(k) = \frac{2V}{(2 \pi)^3}[/itex]
and since:
[itex]F(k) = 1[/itex] at T = 0;
disregarding spin, as we just multiply by two if we need to 'account' for it in this question
now we can replace these results in our integral above:
[itex]\int^{\frac{p_f}{2}}_{0}F(k)D(k)d\vec{k} = \frac{V}{4h\pi^3}\int^{\frac{p_f}{2h}}_{0}d\vec{p} [/itex]
transforming to spherical coordinates reduces the integral to:
[itex]=\frac{V}{4h\pi^3}\int^{\frac{p_f}{2h}}_{0}4\pi p^2dp [/itex]
[itex]=\frac{V}{h\pi^2}\int^{\frac{p_f}{2h}}_{0}p^2dp[/itex]
[itex]=\frac{V}{24h^4\pi^2}p_f^3[/itex]
----------------------------------------------------------------------------------------------------
Now I will attempt two:
i will try to derive the integral itself for P-space, not K-Space
the only thing that i believe changes is the Density function, D(p)
since:
[itex]\vec{k}=\frac{\vec{p}}{h}=\frac{2\pi}{L}*(\vec{i}+\vec{j}+\vec{k})[/itex]
thus:
[itex]\vec{p}=\frac{2h\pi}{L}*(\vec{i}+\vec{j}+\vec{k})[/itex]
thus D(p) can be obtained:
[itex]D(p) = \frac{2L^3}{8h^3\pi^3}=\frac{V}{4h^3\pi^3}[/itex]
F(p) = 1 also, at T = 0; (again, disregarding spin)
so now we need to only evaluate the integral:
[itex]\int F(p)D(p)d\vec{p}[/itex]
[itex]=\frac{V}{4h^3\pi^3}\int^{\frac{p_f}{2}}_{0}d\vec{p}[/itex]
[itex]=\frac{V}{4h^3\pi^3}\int^{\frac{p_f}{2}}_{0}4\pi p^2dp[/itex]
[itex]=\frac{V}{3h^3\pi^2}p_f^3[/itex]
-----
Apparently they are not equal,
i know i made a mistake, or two, or three..
can someone show me :)