Spring Flashcards
In 1D, how is wavenumber related to wavelength?
For a particle in a box, how is energy level spacing related to L (length of sides of box)?
Energy level spacing decreases as L increases (levels get closer together)
Roughly, how is density of states found in 1D?
By finding the number of k states between k and k+Δk.
- by dividing width from k to k+Δk by the gap between each state
The taking this number of states and dividing it by the gap Δk
How is partition function found using density of k states?
For 2D and 3D what does this represent?
Number of intersections of equally spaced lines which correspond to an allowed energy level
What is E(k)? What varies between dimensions?
How is the number of states between k and k+Δk related to the number of states between E and E+ΔE?
What is the formula for D(E)?
For typical E, which dimensions have D(E) independent of E?
2D only
D(E) for 3D depends on sqrt E
D(E) for 1D is proportional to 1/sqrt E
Is D(E) always the same for each dimension?
No, D(E) depends on E’s dependence on k so varies e.g. for an electron in a graphene sheet (E=ℏsk)
What is the equation for Z without E explicitly subbed in?
How can Z be evaluated using D(E)?
Subbing in D(E) and evaluating over dE
In words, what is energy density of states?
The number of states with energy between E and E+ΔE, divided by ΔE
Do both D(E) and D(k) change for materials?
No, D(k) always the same for each dimension (1D, 2D, 3D) but D(E) can change for non-parabolic relations between E and k
What does the Boltzmann probability distribution tell us? What is the equation?
The probability that any one of the particles occupies a particular quantum state of energy E
How can the Boltzmann probability distribution be used to find average number of particles in each state? What are we assuming here?
Where Na is amount of particles in whole box.
We assume na(E) «1 so we don’t worry about several particles on one level
From the average number of particles in a state, na, how can average number of particles with energy between E and E+dE be found?
Z is?
The partition function
na(E) is ?
Number of particles in a particular energy state
What is λD? Why do we introduce it?
Thermal deBroglie Wavelength corresponding to a particle whose total KE is kBT.
Introduced to simplify Z
How does energy distribution relate to volume of gas?
It’s independent.
What is α(E)? (in words and as a product of other values)
Number of particles per unit energy between E and E+dE.
= naD(E)
dN=α(E)dE
How does α(E) (particles per unit energy) depend on E?
Product of an exponential term and a sqrt dependence so gives rise to a peak in dependence.
On a graph of α(E) against E, roughly what value does the peak take?
3/2KBT