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SingBluSilver
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Homework Statement
I'm having a little bit of trouble getting started with this problem. Can I get a little help?
Using: (number of states in the six-dimensional region d[tex]^{3}[/tex]x d[tex]^{3}[/tex]3p) = (d[tex]^{3}[/tex]x d[tex]^{3}[/tex]p)/h[tex]^{3}[/tex]
Which provides a convenient route to the single-particle density of masses.
a) Integrate over space (of volume V) and over the direction of the momentum p to determine D(p)dp, where D(p) denotes the number of states per unit interval of momentum magnitude.
b) Adopt the non-relativistic relationship between kinetic energy and momentum, ε = p[tex]^{2}[/tex] / 2m, and determine the number of states per unit energy interval, D(ε). Do you find agreement with our previous result?
c) Consider the relativistic relationship between the total energy and momentum, ε[tex]_{rel}[/tex] = (p[tex]^{2}[/tex]c[tex]^{2}[/tex]+ m[tex]^{2}[/tex]c[tex]^{4}[/tex])[tex]^{1/2}[/tex]. determine the number of states per unit interval of total energy, D(ε[tex]_{rel}[/tex])
(The rel after epsilon is supposed to be a subscript, not sure why it went superscript)
Homework Equations
(number of states in the six-dimensional region d[tex]^{3}[/tex]x d[tex]^{3}[/tex]3p) = (d[tex]^{3}[/tex]x d[tex]^{3}[/tex]p)/h[tex]^{3}[/tex]
The Attempt at a Solution
For part a I'm not exactly sure what I'm supposed to be integrating. Do I just convert d[tex]^{3}[/tex]x to a volume and use polar coordinates to convert d[tex]^{3}[/tex]p to 4pi*p^2dp and integrate that?
I'm also not understanding what parts b and c are asking. Can anyone push me in the right direction?
Any help would be appreciated.
Thank you
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