Solutions to discretisation exercises #993
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mfherbst
reviewed
Aug 10, 2024
docs/src/guide/discretisation.jl
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| @@ -63,7 +63,7 @@ end; | |||
| # !!! tip "Exercise 2" | |||
| # Show that | |||
| # ```math | |||
| # \langle e_G, e_{G'}\rangle = ∫_0^{2π} e_G(x) e_{G'}(x) d x = δ_{G, G'} | |||
| # \langle e_G, e_{G'}\rangle = ∫_0^{2π} e_G(x)\ast e_{G'}(x) d x = δ_{G, G'} | |||
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Thank you, I will fix that.
docs/src/guide/discretisation.jl
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| @@ -118,7 +118,8 @@ plot(real(ψ); label="") | |||
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| # Again this should match with the result above. | |||
| # | |||
| # **Exercise 4:** Look at the Fourier coefficients of $\psi$ ($\psi$_fourier) | |||
| # !!! "Exercise 4" | |||
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I think the syntax here and the next line is not correct (some keyword missing after !!! & indention in next lines
mfherbst
reviewed
Aug 14, 2024
docs/src/guide/discretisation.jl
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| @@ -147,7 +148,7 @@ Array(H) | |||
| # !!! tip "Exercise 6" | |||
| # Increase the size of the problem, and compare the time spent | |||
| # by DFTK's internal diagonalization algorithms to a full diagonalization of `Array(H)`. | |||
| # *Hint:* The `@time` macro is handy for this task. | |||
| # *Hint:* The `@benchmark` macro is handy for this task. | |||
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| # *Hint:* The `@benchmark` macro is handy for this task. | |
| # *Hint:* The `@benchmark` macro (from the [BenchmarkTools](https://github.com/JuliaCI/BenchmarkTools.jl) package) is handy for this task. Note that there are some subtleties with global variables (see the BenchmarkTools docs for details). E.g. to use it to benchmark a function like `eigen(H)` run it as | |
| # ``` | |
| # using BenchmarkTools | |
| # @benchmark eigen($H) | |
| # ``` | |
| # (note the `$`). |
docs/src/guide/discretisation.jl
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| @@ -198,22 +199,121 @@ Nrange = 10:10:100 | |||
| plot(Nrange, abs.(fconv.(Nrange) .- fconv(200)); yaxis=:log, legend=false) | |||
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| # ### Exercise 2 | |||
| # !!! note "TODO" | |||
| # This solution has not yet been written. Any help with a PR is appreciated. | |||
| # ```math | |||
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No need to indent this and other math blocks.
docs/src/guide/discretisation.jl
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| # !!! note "TODO" | ||
| # This solution has not yet been written. Any help with a PR is appreciated. | ||
| # ```math | ||
| # \bullet \langle e_G, e_{G'}\rangle = ∫_0^{2π} e_G^\ast(x) e_{G'}(x) d x = 1/2π ∫_0^{2π} e^i(G'-G)x d x |
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I thought I would put a bullet for each part of the equation, should I remove it ?
docs/src/guide/discretisation.jl
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| # if G≠G', since the function is periodic over $[0, 2\pi]$: | ||
| # ```math | ||
| # \langle e_G, e_{G'}\rangle = \frac 1 {i2π(G'-G)} ∫_0^{2π} (e^{ix})^{G'-G} d x = 0 | ||
| # ``` | ||
| # if G=G': | ||
| # ```math | ||
| # \langle e_G, e_{G'}\rangle = \frac 1 {2π} ∫_0^{2π} d x = 1 | ||
| # ``` | ||
| # Therefore: |
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Which function ? I think that's a bit wordy
Suggested change
| # if G≠G', since the function is periodic over $[0, 2\pi]$: | |
| # ```math | |
| # \langle e_G, e_{G'}\rangle = \frac 1 {i2π(G'-G)} ∫_0^{2π} (e^{ix})^{G'-G} d x = 0 | |
| # ``` | |
| # if G=G': | |
| # ```math | |
| # \langle e_G, e_{G'}\rangle = \frac 1 {2π} ∫_0^{2π} d x = 1 | |
| # ``` | |
| # Therefore: | |
| # Since `e^{iy}` is a periodic function with period $2\pi$, $\int_0^{2\pi} e^{i m y} dy = \delta_{0,m}$. Therefore: |
docs/src/guide/discretisation.jl
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| # ```math | ||
| # \langle e_G, H e_{G'}\rangle = \frac 1 2 ∫_0^{2π} e_G^\ast(x) H e_{G'}(x) d x \left(|G|^2 \delta_{G,G'} + \delta_{G, G'+1} + \delta_{G, G'-1}\right). | ||
| # ``` |
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This is the final result, right ? I would not repeat that here. People can read it in the question.
mfherbst
changed the title
mfherbst
reviewed
Aug 20, 2024
| using LinearAlgebra | ||
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| function build_plane_waves_matrix_cos(N::Integer) | ||
| # Plane wave approximation to -½Δ |
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When you put in-line comments in code segments, they need to start with ## ... Ask me later why if you are curious
| function build_plane_waves_matrix_cos(N::Integer) | ||
| # Plane wave approximation to -½Δ | ||
| G = [float((-N + i)^2) for i in 0:2N] | ||
| #using results from exercice 2 for the case of cos potential, the following matrix is built for the Hamiltonian |
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finite differences: boundary conditions and formula fix
ex 2: * missing in formula
ex 4: put in square box
sol 1: formula fix