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-rw-r--r--hw7/.DS_Storebin0 -> 8196 bytes
-rw-r--r--hw7/5.6-face.pngbin0 -> 76602 bytes
-rw-r--r--hw7/5.6-fx.pngbin0 -> 38040 bytes
-rw-r--r--hw7/5.6-fy.pngbin0 -> 64046 bytes
-rw-r--r--hw7/5.6-phi-field.pngbin0 -> 25715 bytes
-rw-r--r--hw7/5.6-phi.pngbin0 -> 54394 bytes
-rw-r--r--hw7/5.6-test.pngbin0 -> 26702 bytes
-rw-r--r--hw7/5.6.jl192
-rw-r--r--hw7/5.6.pngbin0 -> 86224 bytes
-rw-r--r--hw7/8-12.jl191
-rw-r--r--hw7/8-15-d.pngbin0 -> 30807 bytes
-rw-r--r--hw7/8-15-scales-d.pngbin0 -> 28596 bytes
-rw-r--r--hw7/8-15-scales.pngbin0 -> 31626 bytes
-rw-r--r--hw7/8-15.jl274
-rw-r--r--hw7/8-15.pngbin0 -> 31061 bytes
-rw-r--r--hw7/BoundStates.jl73
-rw-r--r--hw7/Laplacians.jl145
-rw-r--r--hw7/RadialBoundStates.jl59
-rw-r--r--hw7/mvsb.svg55
-rw-r--r--hw7/mvsb2.svg46
-rw-r--r--hw7/tightbinding (1).jl24
21 files changed, 1059 insertions, 0 deletions
diff --git a/hw7/.DS_Store b/hw7/.DS_Store
new file mode 100644
index 0000000..9b9f923
--- /dev/null
+++ b/hw7/.DS_Store
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diff --git a/hw7/5.6-face.png b/hw7/5.6-face.png
new file mode 100644
index 0000000..b02d8f9
--- /dev/null
+++ b/hw7/5.6-face.png
Binary files differ
diff --git a/hw7/5.6-fx.png b/hw7/5.6-fx.png
new file mode 100644
index 0000000..a445d09
--- /dev/null
+++ b/hw7/5.6-fx.png
Binary files differ
diff --git a/hw7/5.6-fy.png b/hw7/5.6-fy.png
new file mode 100644
index 0000000..56c38f5
--- /dev/null
+++ b/hw7/5.6-fy.png
Binary files differ
diff --git a/hw7/5.6-phi-field.png b/hw7/5.6-phi-field.png
new file mode 100644
index 0000000..3556e4b
--- /dev/null
+++ b/hw7/5.6-phi-field.png
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diff --git a/hw7/5.6-phi.png b/hw7/5.6-phi.png
new file mode 100644
index 0000000..93b9775
--- /dev/null
+++ b/hw7/5.6-phi.png
Binary files differ
diff --git a/hw7/5.6-test.png b/hw7/5.6-test.png
new file mode 100644
index 0000000..58328e6
--- /dev/null
+++ b/hw7/5.6-test.png
Binary files differ
diff --git a/hw7/5.6.jl b/hw7/5.6.jl
new file mode 100644
index 0000000..e6f6c8a
--- /dev/null
+++ b/hw7/5.6.jl
@@ -0,0 +1,192 @@
+using Pkg
+Pkg.add("Plots")
+using Plots
+
+function del2_5(a::Matrix{Float64}, dx = 1.0)
+ #=
+ Returns a finite-difference approximation of the laplacian of the array a,
+ with lattice spacing dx, using the five-point stencil:
+
+ 0 1 0
+ 1 -4 1
+ 0 1 0
+ =#
+
+ del2 = zeros(size(a))
+ del2[2:end-1, 2:end-1] .= (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] +
+ a[3:end, 2:end-1] + a[1:end-2, 2:end-1] -
+ 4.0 * a[2:end-1, 2:end-1]) ./ (dx^2)
+ return del2
+end
+
+function del2_9(a::Matrix{Float64}, dx = 1.0)
+ #=
+ Returns a finite-difference approximation of the laplacian of the array a,
+ with lattice spacing dx, using the nine-point stencil:
+
+ 1/6 2/3 1/6
+ 2/3 -10/3 2/3
+ 1/6 2/3 1/6
+ =#
+
+ del2 = zeros(size(a))
+ del2[2:end-1, 2:end-1] .= (4.0 * (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] +
+ a[3:end, 2:end-1] + a[1:end-2, 2:end-1]) +
+ (a[1:end-2, 1:end-2] + a[1:end-2, 3:end] +
+ a[3:end, 1:end-2] + a[3:end, 3:end]) -
+ 20.0 * a[2:end-1, 2:end-1]) ./ (6.0 * dx^2)
+ return del2
+end
+
+function invDel2_5(b::Matrix{Float64}, dx = 1.0, N = 10000)
+ #=
+ Relaxes over N steps to a discrete approximation of the inverse laplacian
+ of the source term b, each step is a weighted average over the four neighboring
+ points and the source. This is the Jacobi algorithm.
+ =#
+
+ invDel2 = zeros(size(b))
+ newInvDel2 = zeros(size(b))
+
+ for m in 1:N
+ newInvDel2[2:end-1, 2:end-1] .= 0.25 * (invDel2[2:end-1, 3:end] +
+ invDel2[2:end-1, 1:end-2] +
+ invDel2[3:end, 2:end-1] +
+ invDel2[1:end-2, 2:end-1] -
+ (dx^2) * b[2:end-1, 2:end-1])
+ invDel2 .= newInvDel2
+ end
+
+ diff = del2_5(invDel2, dx) - b
+ diffSq = diff .* diff
+ error = sqrt(sum(diffSq))
+
+ println("\nerror = ", error)
+
+ return invDel2
+end
+
+
+# # Define variables
+# L = 10
+# dx = 1.0
+
+# # Parabola of revolution has constant Laplacian
+# phi = zeros((L, L))
+# for i ∈ 1:L
+# x = (i + 0.5 - 0.5 * L) * dx
+# for j ∈ 1:L
+# y = (j + 0.5 - 0.5 * L) * dx
+# phi[i, j] = x^2 + y^2
+# end
+# end
+
+# println(size(phi))
+
+# println(del2_5(phi, dx))
+
+# println("\n\n")
+
+# println(del2_9(phi, dx))
+
+# Electrostatics example: uniformly charged cylinder (rho = 1) of radius R
+L = 100
+dx = 1.0
+R = 1.0
+R2 = R^2
+
+# Charge distribution
+rho = zeros((L, L))
+for i ∈ 1:L
+ y = (i - 1 + 0.5 - 0.5 * L) * dx
+ println("y = ", y)
+ for j ∈ 1:L
+ x = (j - 1 + 0.5 - 0.5 * L) * dx
+ # this is the rod here
+ if y < 0 && abs(x) < R
+ rho[i, j] = 1.0 # high voltage
+ elseif y > 0 && abs(x) < R
+ rho[i, j] = -1.0 # conducting plane
+ else
+ rho[i, j] = 0.0
+ end
+ end
+end
+
+rhoPlot = plot(rho[Int(L / 2), :])
+
+# Exact (analytical) electric potential
+# phi = zeros((L, L))
+# for i ∈ 1:L
+# x = (i + 0.5 - 0.5 * L) * dx
+# for j ∈ 1:L
+# y = (j + 0.5 - 0.5 * L) * dx
+# r2 = x^2 + y^2
+# if r2 < R2
+# phi[i, j] = -π * r2
+# else
+# phi[i, j] = -π * (R2 + R2 * log(r2 / R2))
+# end
+# end
+# end
+
+# phiPlot = plot(phi[Int(L / 2), :])
+
+# Charge density obtained from exact potential
+# rho = -1.0 / (4.0 * π) * del2_5(phi, dx)
+# rhoPlotLattice = plot(rho[Int(L / 2), :])
+
+function calc_field(phi::Matrix{Float64}, dx = 1.0)
+ E_x = zeros(size(phi))
+ E_y = zeros(size(phi))
+
+ E_x[2:end-1, 2:end-1] .= -(phi[3:end, 2:end-1] - phi[1:end-2, 2:end-1]) / (2 * dx)
+ E_y[2:end-1, 2:end-1] .= -(phi[2:end-1, 3:end] - phi[2:end-1, 1:end-2]) / (2 * dx)
+
+ # for i ∈ 2:size(phi, 1)-1
+ # for j ∈ 2:size(phi, 2)-1
+ # E_x[i, j] = -(phi[i+1, j] - phi[i-1, j]) / (2 * dx)
+ # E_y[i, j] = -(phi[i, j+1] - phi[i, j-1]) / (2 * dx)
+ # end
+ # end
+
+ return E_x, E_y
+end
+
+phi = invDel2_5(-4.0 * π * rho, dx, 1000)
+# phi = invDelSOR(-4.0 * π * rho, dx, 500)
+phi .-= phi[Int(L / 2), Int(L / 2)]
+
+phiPlotInvDel = plot(phi[Int(L / 2), :])
+
+# phi_contour = contourf(phi, title = "Electric Potential", color = :viridis, aspect_ratio = :equal, colorbar_title = "V")
+# savefig(phi_contour, "hw7/5.6-phi.png")
+
+# plot the arrows of the magnatic field on the contour plot
+xxs = [x for x in 1:L for y in 1:L]
+xxy = [y for x in 1:L for y in 1:L]
+d_x, d_y = calc_field(phi, dx)
+
+function df(i, j)
+ norm = d_x[Int64(i)] * d_x[Int64(i)] + d_y[Int64(j)] * d_y[Int64(j)]
+ if norm == 0
+ return (0, 0)
+ end
+ return (d_x[Int64(i)] / norm, d_y[Int64(j)] / norm)
+end
+
+quiver(xxs, xxy, quiver = df, aspect_ratio = :equal, legend = false)
+savefig("hw7/5.6-test.png")
+
+
+# d_x, d_y = calc_field(phi, dx)
+# field_y = contourf(d_y, title = "Electric Field (y)", color = :viridis, aspect_ratio = :equal, colorbar_title = "V/m")
+
+# field_x = contourf(d_x, title = "Electric Field (x)", color = :viridis, aspect_ratio = :equal, colorbar_title = "V/m")
+
+
+# savefig(field_y, "hw7/5.6-fy.png")
+# savefig(field_x, "hw7/5.6-fx.png")
+
+#plot(rhoPlot, rhoPlotLattice)
+#plot(phiPlot, phiPlotInvDel)
diff --git a/hw7/5.6.png b/hw7/5.6.png
new file mode 100644
index 0000000..f17824a
--- /dev/null
+++ b/hw7/5.6.png
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diff --git a/hw7/8-12.jl b/hw7/8-12.jl
new file mode 100644
index 0000000..29e18ec
--- /dev/null
+++ b/hw7/8-12.jl
@@ -0,0 +1,191 @@
+#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia
+
+using Statistics
+using Plots
+
+function wrap_index(i::Int, l::Int)::Int
+ wrap = (i - 1) % l + 1
+ return (wrap <= 0) ? l + wrap : wrap
+end
+
+mutable struct Ising2D
+ l::Int
+ n::Int
+ temperature::Float64
+ w::Vector{Float64} # Boltzmann weights
+ state::Matrix
+ energy::Float64
+ magnetization::Int
+ mc_steps::Int
+ accepted_moves::Int
+ energy_array::Vector{Float64}
+ magnetization_array::Vector{Int}
+ H::Float64
+end
+
+Ising2D(l::Int, temperature::Float64, H=1.0) = begin
+ n = l^2
+ w = zeros(9)
+ w[9] = exp(-8.0 / temperature)
+ w[5] = exp(-4.0 / temperature)
+ state = ones(Int, l, l) # initially all spins up
+ energy = Float64(-2 * n + 2 * H * n)
+ magnetization = n
+ return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0,
+ Int[], Int[], H)
+end
+
+function reset!(ising::Ising2D)
+ ising.mc_steps = 0
+ ising.accepted_moves = 0
+ ising.energy_array = Int[]
+ ising.magnetization_array = Int[]
+end
+
+function mc_step!(ising::Ising2D)
+ l::Int = ising.l
+ n::Int = ising.n
+ w = ising.w
+
+ state = ising.state
+ accepted_moves = ising.accepted_moves
+ energy = ising.energy
+ magnetization = ising.magnetization
+
+ random_positions = l * rand(2 * n)
+ random_array = rand(n)
+
+ for k in 1:n
+ i = trunc(Int, random_positions[2 * k - 1]) + 1
+ j = trunc(Int, random_positions[2 * k]) + 1
+
+ changed_spins = state[i, j] * (state[i % l + 1, j] +
+ state[wrap_index(i - 1, l), j] + state[i, j % l + 1] +
+ state[i, wrap_index(j - 1, l)])
+ de = 2 * changed_spins + 2 * ising.H * state[i, j]
+
+ if de <= 0 || rand() < exp(-de / ising.temperature)
+ accepted_moves += 1
+ new_spin = - state[i, j] # flip spin
+ state[i, j] = new_spin
+
+ # add the effects of the new spin
+ energy += de
+ magnetization += 2 * new_spin
+ end
+
+ end
+
+ ising.state = state
+ ising.accepted_moves = accepted_moves
+ ising.energy = energy
+ ising.magnetization = magnetization
+
+ append!(ising.energy_array, ising.energy)
+ append!(ising.magnetization_array, ising.magnetization)
+ ising.mc_steps = ising.mc_steps + 1
+end
+
+function steps!(ising::Ising2D, num::Int=100)
+ for i in 1:num
+ mc_step!(ising)
+ end
+end
+
+function mean_energy(ising::Ising2D)
+ return mean(ising.energy_array) / ising.n
+end
+
+function specific_heat(ising::Ising2D)
+ return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n
+end
+
+function mean_magnetization(ising::Ising2D)
+ return mean(ising.magnetization_array) / ising.n
+end
+
+function susceptibility(ising::Ising2D)
+ return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n)
+end
+
+function observables(ising::Ising2D)
+ printstyled("Temperature\t\t", bold=true)
+ print(ising.temperature); print("\n")
+
+ printstyled("Mean Energy\t\t", bold=true)
+ print(mean_energy(ising)); print("\n")
+
+ printstyled("Mean Magnetiz.\t\t", bold=true)
+ print(mean_magnetization(ising)); print("\n")
+
+ printstyled("Specific Heat\t\t", bold=true)
+ print(specific_heat(ising)); print("\n")
+
+ printstyled("Susceptibility\t\t", bold=true)
+ print(susceptibility(ising)); print("\n")
+
+ printstyled("MC Steps\t\t", bold=true)
+ print(ising.mc_steps); print("\n")
+ printstyled("Accepted Moves\t\t", bold=true)
+ print(ising.accepted_moves); print("\n")
+end
+
+
+function plot_ising(state::Matrix{Int})
+ pos = Tuple.(findall(>(0), state))
+ neg = Tuple.(findall(<(0), state))
+ scatter(pos, markersize=5)
+ scatter!(neg, markersize=5)
+end
+
+function find_m(H::Float64, l::Int, num::Int, T::Float64)
+ m = Ising2D(l, T, H)
+ steps!(m, num)
+ print("T = $T\n")
+ print("H = $H\n")
+ print("Mean Energy: $(mean_energy(m))\n")
+ print("Mean Magnetization: $(mean_magnetization(m))\n\n")
+ return mean_magnetization(m)
+end
+
+function map_h_to_m(H_range::Vector{Float64}, l::Int, num::Int, T::Float64)
+ m = []
+ for H in H_range
+ push!(m, find_m(H, l, num, T))
+ end
+ return m
+end
+
+function do_linear_regression(x::Vector{Float64}, y::Vector{Float64})
+ n = length(x)
+ x̄ = mean(x)
+ ȳ = mean(y)
+ Σxy = sum((x .- x̄) .* (y .- ȳ))
+ Σx² = sum((x .- x̄) .^ 2)
+ b = Σxy / Σx²
+ a = ȳ - b * x̄
+ return a, b
+end
+
+function plot_log_of_m_and_h(H_range::Vector{Float64}, l::Int, num::Int, T=2.27)
+ m = map_h_to_m(H_range, l, num, T)
+ p = scatter(H_range, m, label="M vs H", xlabel="H", ylabel="M", title="Magnetization (M) vs Field (B) for Ising Model at T_c", scale=:ln)
+
+ # get the linear regression of the log
+ log_h = log.(H_range)
+ log_m = log.(m)
+ a, b = do_linear_regression(log_h, log_m)
+ println("a: $a, b: $b")
+ # plot the linear regression
+ plot!(p, H_range, exp.(a) .* H_range .^ b, label="linear regression = $(round(a, digits=3)) + $(round(b, digits=3))x", line=:dash, color=:red)
+
+ return p
+end
+
+# textbook rec
+h_range = .02:.02:.2
+h_range = collect(h_range)
+T = 2.27 # T_c for this system
+side = 64
+steps = 5000
+plot_log_of_m_and_h(h_range, side, steps) \ No newline at end of file
diff --git a/hw7/8-15-d.png b/hw7/8-15-d.png
new file mode 100644
index 0000000..eb64f61
--- /dev/null
+++ b/hw7/8-15-d.png
Binary files differ
diff --git a/hw7/8-15-scales-d.png b/hw7/8-15-scales-d.png
new file mode 100644
index 0000000..5e4f2e4
--- /dev/null
+++ b/hw7/8-15-scales-d.png
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diff --git a/hw7/8-15-scales.png b/hw7/8-15-scales.png
new file mode 100644
index 0000000..2256167
--- /dev/null
+++ b/hw7/8-15-scales.png
Binary files differ
diff --git a/hw7/8-15.jl b/hw7/8-15.jl
new file mode 100644
index 0000000..5c833be
--- /dev/null
+++ b/hw7/8-15.jl
@@ -0,0 +1,274 @@
+#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia
+
+using Statistics
+using Plots
+
+function wrap_index(i::Int, l::Int)::Int
+ wrap = (i - 1) % l + 1
+ return (wrap <= 0) ? l + wrap : wrap
+end
+
+mutable struct Ising2D
+ l::Int
+ n::Int
+ temperature::Float64
+ w::Vector{Float64} # Boltzmann weights
+ state::Matrix
+ energy::Float64
+ magnetization::Int
+ mc_steps::Int
+ accepted_moves::Int
+ energy_array::Vector{Float64}
+ magnetization_array::Vector{Int}
+ H::Float64
+end
+
+Ising2D(l::Int, temperature::Float64, H = 1.0) = begin
+ n = l^2
+ w = zeros(9)
+ w[9] = exp(-8.0 / temperature)
+ w[5] = exp(-4.0 / temperature)
+ state = ones(Int, l, l) # initially all spins up
+ energy = Float64(-2 * n + 2 * H * n)
+ magnetization = n
+ return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0,
+ Int[], Int[], H)
+end
+
+function reset!(ising::Ising2D)
+ ising.mc_steps = 0
+ ising.accepted_moves = 0
+ ising.energy_array = Int[]
+ ising.magnetization_array = Int[]
+end
+
+function mc_step!(ising::Ising2D)
+ l::Int = ising.l
+ n::Int = ising.n
+ w = ising.w
+
+ state = ising.state
+ accepted_moves = ising.accepted_moves
+ energy = ising.energy
+ magnetization = ising.magnetization
+
+ random_positions = l * rand(2 * n)
+ random_array = rand(n)
+
+ for k in 1:n
+ i = trunc(Int, random_positions[2*k-1]) + 1
+ j = trunc(Int, random_positions[2*k]) + 1
+
+ changed_spins = state[i, j] * (state[i%l+1, j] +
+ state[wrap_index(i - 1, l), j] + state[i, j%l+1] +
+ state[i, wrap_index(j - 1, l)])
+ de = 2 * changed_spins + 2 * ising.H * state[i, j]
+
+ if de <= 0 || rand() < exp(-de / ising.temperature)
+ accepted_moves += 1
+ new_spin = -state[i, j] # flip spin
+ state[i, j] = new_spin
+
+ # add the effects of the new spin
+ energy += de
+ magnetization += 2 * new_spin
+ end
+
+ end
+
+ ising.state = state
+ ising.accepted_moves = accepted_moves
+ ising.energy = energy
+ ising.magnetization = magnetization
+
+ append!(ising.energy_array, ising.energy)
+ append!(ising.magnetization_array, ising.magnetization)
+ ising.mc_steps = ising.mc_steps + 1
+end
+
+function steps!(ising::Ising2D, num::Int = 100)
+ for i in 1:num
+ mc_step!(ising)
+ end
+end
+
+function mean_energy(ising::Ising2D)
+ return mean(ising.energy_array) / ising.n
+end
+
+function specific_heat(ising::Ising2D)
+ return (std(ising.energy_array) / ising.temperature)^2 / ising.n
+end
+
+function mean_magnetization(ising::Ising2D)
+ return mean(ising.magnetization_array) / ising.n
+end
+
+function susceptibility(ising::Ising2D)
+ return (std(ising.magnetization_array))^2 / (ising.temperature * ising.n)
+end
+
+function observables(ising::Ising2D)
+ printstyled("Temperature\t\t", bold = true)
+ print(ising.temperature)
+ print("\n")
+
+ printstyled("Mean Energy\t\t", bold = true)
+ print(mean_energy(ising))
+ print("\n")
+
+ printstyled("Mean Magnetiz.\t\t", bold = true)
+ print(mean_magnetization(ising))
+ print("\n")
+
+ printstyled("Specific Heat\t\t", bold = true)
+ print(specific_heat(ising))
+ print("\n")
+
+ printstyled("Susceptibility\t\t", bold = true)
+ print(susceptibility(ising))
+ print("\n")
+
+ printstyled("MC Steps\t\t", bold = true)
+ print(ising.mc_steps)
+ print("\n")
+ printstyled("Accepted Moves\t\t", bold = true)
+ print(ising.accepted_moves)
+ print("\n")
+end
+
+
+function plot_ising(state::Matrix{Int})
+ pos = Tuple.(findall(>(0), state))
+ neg = Tuple.(findall(<(0), state))
+ scatter(pos, markersize = 5)
+ scatter!(neg, markersize = 5)
+end
+
+function h_to_m(H::Float64, l::Int, num::Int, T::Float64)
+ m = Ising2D(l, T, H)
+ steps!(m, num)
+ print("T = $T\n")
+ print("H = $H\n")
+ print("Mean Energy: $(mean_energy(m))\n")
+ print("Mean Magnetization: $(mean_magnetization(m))\n\n")
+ return mean_magnetization(m)
+end
+
+function map_h_to_m(H_range::Vector{Float64}, l::Int, num::Int, T::Float64)
+ m = []
+ for H in H_range
+ push!(m, h_to_m(H, l, num, T))
+ end
+ return m
+end
+
+function do_linear_regression(x::Vector{Float64}, y::Vector{Float64})
+ n = length(x)
+ x̄ = mean(x)
+ ȳ = mean(y)
+ Σxy = sum((x .- x̄) .* (y .- ȳ))
+ Σx² = sum((x .- x̄) .^ 2)
+ b = Σxy / Σx²
+ a = ȳ - b * x̄
+ return a, b
+end
+
+function plot_log_of_m_and_h(H_range::Vector{Float64}, l::Int, num::Int, T = 2.27)
+ m = map_h_to_m(H_range, l, num, T)
+ p = scatter(H_range, m, label = "M vs H", xlabel = "H", ylabel = "M", title = "Magnetization (M) vs Field (B) for Ising Model at T_c", scale = :ln)
+
+ # get the linear regression of the log
+ log_h = log.(H_range)
+ log_m = log.(m)
+ a, b = do_linear_regression(log_h, log_m)
+ println("a: $a, b: $b")
+ # plot the linear regression
+ plot!(p, H_range, exp.(a) .* H_range .^ b, label = "linear regression = $(round(a, digits=3)) + $(round(b, digits=3))x", line = :dash, color = :red)
+
+ return p
+end
+
+function t_to_m(T::Float64, l::Int, num::Int, H::Float64)
+ m = Ising2D(l, T, H)
+ steps!(m, num)
+ print("T = $T\n")
+ print("H = $H\n")
+ print("Mean Energy: $(mean_energy(m))\n")
+ print("Mean Sus: $(mean_magnetization(m))\n\n")
+ return susceptibility(m)
+end
+
+function plot_m_over_t(plt, T_range::Vector{Float64}, l::Int, num::Int, H = 0.0)
+ m = []
+ for T in T_range
+ push!(m, t_to_m(T, l, num, H))
+ end
+ p = scatter!(plt, T_range, m, label = "H = $H", xlabel = "T", ylabel = "X", title = "Susceptibility (X) vs Temperature (T) on Ising Model")
+
+ return p, m
+end
+
+function plot_m_over_t_and_h(T_range::Vector{Float64}, l::Int, num::Int, H_range::Vector{Float64})
+ plt = Plots.scatter()
+ h = []
+ for H in H_range
+ p, m = plot_m_over_t(plt, T_range, l, num, H)
+ push!(h, m)
+ end
+
+ # plot the critical temp as a vertical line
+ plot!(plt, [2.27, 2.27], [-0.01, 30], label = "T_c = 2.27", line = :dash, color = :red)
+
+ return plt, h
+end
+
+function plot_scales(data, t_range, h_range)
+ x1 = []
+ y1 = []
+ x2 = []
+ y2 = []
+ for i in 1:length(h_range)
+ h = h_range[i]
+ for j in 1:length(t_range)
+ t = t_range[j]
+
+ m = data[i][j]
+
+ scaled_t = abs((t - 2.27) / 2.27)
+ scaled_m = m * (scaled_t^(7.0 / 4.0))
+ scaled_h = h / (scaled_t^(15.0 / 8.0))
+ if scaled_h > 30
+ continue # dont add if too large
+ end
+
+ if t > 2.27
+ push!(x1, scaled_h)
+ push!(y1, scaled_m)
+ else
+ push!(x2, scaled_h)
+ push!(y2, scaled_m)
+ end
+ end
+ end
+
+ tmp = scatter(x1, y1, label = "T > T_c", xlabel = "h / abs(t)^(15/8)", ylabel = "X * abs(t)^(7/4)", title = "Susceptibility (X) vs Field (H) for Ising Model")
+ scatter!(tmp, x2, y2, label = "T < T_c", xlabel = "h / abs(t)^(15/8)", ylabel = "X * abs(t)^(7/4)", title = "Susceptibility (X) vs Field (H) for Ising Model")
+ return tmp
+end
+
+h_range = 0.01:0.01:0.05
+h_range = collect(h_range)
+t_range = 1.5:0.1:3.0
+t_range = collect(t_range)
+println("t_range: $t_range")
+side = 20
+steps = 3000
+plt, data = plot_m_over_t_and_h(t_range, side, steps, h_range)
+
+savefig(plt, "hw7/8-15.png")
+
+p = plot_scales(data, t_range, h_range)
+savefig(p, "hw7/8-15-scales.png")
+
+
diff --git a/hw7/8-15.png b/hw7/8-15.png
new file mode 100644
index 0000000..a5a4b28
--- /dev/null
+++ b/hw7/8-15.png
Binary files differ
diff --git a/hw7/BoundStates.jl b/hw7/BoundStates.jl
new file mode 100644
index 0000000..386a5da
--- /dev/null
+++ b/hw7/BoundStates.jl
@@ -0,0 +1,73 @@
+#!/usr/bin/env julia
+
+"""Find eigenstates and eigenenergies of 1-D quantum problems"""
+
+using LinearAlgebra
+using Plots
+
+N = 1000 # number of lattice points
+L = 20.0 # x runs from -L/2 to L/2
+dx = L / N
+
+D = zeros(N, N) # discrete laplacian operator
+V = zeros(N, N) # potential
+
+for i in 1:N
+ D[i, i] = -2.0
+end
+
+for i in 1:N-1
+ D[i, i+1] = 1.0
+ D[i+1, i] = 1.0
+end
+
+#println("\nLattice Laplacian operator")
+#println(D)
+
+function potential(x)
+ """ The potential energy"""
+ #return 0.0 # particle in a box
+ #return 0.5 * x^2 # SHO with the spring constant k = 1
+ #return -6.0 * x^2 + 8.0 * x^6 # potential with zero ground state energy
+ #return 0.1 * x^4 - 2.0 * x^2 + 0.0 * x # double-well potential
+ return 8*x^6 - 8*x^4 - 4*x^2 + 1 # another double-well potential
+end
+
+for i in 1:N
+ x = (i + 0.5) * dx - 0.5*L # coordinates of lattice points
+ V[i, i] = potential(x)
+end
+
+H = -0.5 * dx^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1
+
+#println("\nMatrix elements of Hamiltonian = ")
+#println(H)
+
+e, v = eigen(H) # diagonalize Hamiltonian
+
+println("\nGround state energy = ", e[1])
+println("\n1st excited state energy = ", e[2])
+println("\n2nd excited state energy = ", e[3])
+println("\n3rd excited state energy = ", e[4])
+println("\n4th excited state energy = ", e[5])
+
+
+gs(x) = exp(-0.5*x^2) # Gaussian that is exact ground state of SHO
+
+
+plot(potential)
+
+
+plot(v[:,1])
+#plot(v[:,2])
+#plot(gs)
+
+#=
+eList = zeros(0)
+for i in 1:20
+ push!(eList, e[i])
+end
+
+
+bar(eList, orientation = :horizontal)
+=#
diff --git a/hw7/Laplacians.jl b/hw7/Laplacians.jl
new file mode 100644
index 0000000..10f0d48
--- /dev/null
+++ b/hw7/Laplacians.jl
@@ -0,0 +1,145 @@
+using Pkg
+Pkg.add("Plots")
+using Plots
+
+function del2_5(a::Matrix{Float64}, dx=1.0)
+ #=
+ Returns a finite-difference approximation of the laplacian of the array a,
+ with lattice spacing dx, using the five-point stencil:
+
+ 0 1 0
+ 1 -4 1
+ 0 1 0
+ =#
+
+ del2 = zeros(size(a))
+ del2[2:end-1, 2:end-1] .= (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] +
+ a[3:end, 2:end-1] + a[1:end-2, 2:end-1] -
+ 4.0 * a[2:end-1, 2:end-1]) ./ (dx^2)
+ return del2
+end
+
+function del2_9(a::Matrix{Float64}, dx=1.0)
+ #=
+ Returns a finite-difference approximation of the laplacian of the array a,
+ with lattice spacing dx, using the nine-point stencil:
+
+ 1/6 2/3 1/6
+ 2/3 -10/3 2/3
+ 1/6 2/3 1/6
+ =#
+
+ del2 = zeros(size(a))
+ del2[2:end-1, 2:end-1] .= (4.0 * (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] +
+ a[3:end, 2:end-1] + a[1:end-2, 2:end-1]) +
+ (a[1:end-2, 1:end-2] + a[1:end-2, 3:end] +
+ a[3:end, 1:end-2] + a[3:end, 3:end]) -
+ 20.0 * a[2:end-1, 2:end-1]) ./ (6.0 * dx^2)
+ return del2
+end
+
+function invDel2_5(b::Matrix{Float64}, dx=1.0, N=10000)
+ #=
+ Relaxes over N steps to a discrete approximation of the inverse laplacian
+ of the source term b, each step is a weighted average over the four neighboring
+ points and the source. This is the Jacobi algorithm.
+ =#
+
+ invDel2 = zeros(size(b))
+ newInvDel2 = zeros(size(b))
+
+ for m in 1:N
+ newInvDel2[2:end-1, 2:end-1] .= 0.25 * (invDel2[2:end-1, 3:end] +
+ invDel2[2:end-1, 1:end-2] +
+ invDel2[3:end, 2:end-1] +
+ invDel2[1:end-2, 2:end-1] -
+ (dx^2) * b[2:end-1, 2:end-1])
+ invDel2 .= newInvDel2
+ end
+
+ diff = del2_5(invDel2, dx) - b
+ diffSq = diff .* diff
+ error = sqrt(sum(diffSq))
+
+ println("\nerror = ", error)
+
+ return invDel2
+end
+
+
+# Define variables
+L = 10
+dx = 1.0
+
+# Parabola of revolution has constant Laplacian
+phi = zeros((L, L))
+for i = 1:L
+ x = (i + 0.5 - 0.5 * L) * dx
+ for j = 1:L
+ y = (j + 0.5 - 0.5 * L) * dx
+ phi[i, j] = x^2 + y^2
+ end
+end
+
+println(size(phi))
+
+println(del2_5(phi, dx))
+
+println("\n\n")
+
+println(del2_9(phi, dx))
+
+# Electrostatics example: uniformly charged cylinder (rho = 1) of radius R
+L = 100
+dx = 1.0
+R = 20.0
+R2 = R^2
+
+# Charge distribution
+rho = zeros((L, L))
+for i = 1:L
+ x = (i + 0.5 - 0.5 * L) * dx
+ for j = 1:L
+ y = (j + 0.5 - 0.5 * L) * dx
+ r2 = x^2 + y^2
+ if r2 < R2
+ rho[i, j] = 1.0
+ else
+ rho[i, j] = 0.0
+ end
+ end
+end
+
+rhoPlot = plot(rho[Int(L/2), :])
+
+# Exact (analytical) electric potential
+phi = zeros((L, L))
+for i = 1:L
+ x = (i + 0.5 - 0.5 * L) * dx
+ for j = 1:L
+ y = (j + 0.5 - 0.5 * L) * dx
+ r2 = x^2 + y^2
+ if r2 < R2
+ phi[i, j] = -π * r2
+ else
+ phi[i, j] = -π * (R2 + R2*log(r2/R2))
+ end
+ end
+end
+
+phiPlot = plot(phi[Int(L/2), :])
+
+# Charge density obtained from exact potential
+rho = -1.0/(4.0 * π) * del2_5(phi, dx)
+rhoPlotLattice = plot(rho[Int(L/2), :])
+
+phi = invDel2_5(-4.0 * π * rho, dx, 20000)
+#phi = invDelSOR(-4.0 * π * rho, dx, 500)
+phi .-= phi[Int(L/2), Int(L/2)]
+
+phiPlotInvDel = plot(phi[Int(L/2), :])
+
+contourf(phi)
+
+#plot(rhoPlot, rhoPlotLattice)
+#plot(phiPlot, phiPlotInvDel)
diff --git a/hw7/RadialBoundStates.jl b/hw7/RadialBoundStates.jl
new file mode 100644
index 0000000..afb91b0
--- /dev/null
+++ b/hw7/RadialBoundStates.jl
@@ -0,0 +1,59 @@
+#!/usr/bin/env julia
+
+"""Find eigenstates and eigenenergies of central potential problems"""
+
+using LinearAlgebra
+using Plots
+
+N = 5000 # number of lattice points
+L = 20.0 # r runs from 0 to L
+dr = L / N
+
+D = zeros(N, N) # discrete radial 2nd derivative operator
+V = zeros(N, N) # potential
+
+for i in 1:N
+ D[i, i] = -2.0
+end
+
+for i in 1:N-1
+ D[i, i+1] = 1.0
+ D[i+1, i] = 1.0
+end
+
+#println("\nLattice Laplacian operator")
+#println(D)
+
+function potential(r, ℓ = 0)
+ """ The potential energy"""
+ #return 0.5 * ell * (ℓ+1.0) * pow(r, -2.0) # V=0: Free particle in spherical coordinates
+
+ return -1.0/r + 0.5 * ℓ * (ℓ+1.0) * r^(-2.0) # Hydrogen atom
+
+ #return -r^(-1.1) + 0.5 * ℓ * (ℓ+1.0) * r^(-2.0) # modified Coulomb potential
+end
+
+for i in 1:N
+ r = (i + 0.5) * dr # radial coordinates of lattice points
+ V[i, i] = potential(r, 0)
+end
+
+H = -0.5 * dr^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1
+
+#println("\nMatrix elements of Hamiltonian = ")
+#println(H)
+
+e, v = eigen(H) # diagonalize Hamiltonian
+
+println("\nGround state energy = ", e[1])
+println("\n1st excited state energy = ", e[2])
+println("\n2nd excited state energy = ", e[3])
+println("\n3rd excited state energy = ", e[4])
+println("\n4th excited state energy = ", e[5])
+
+
+plot(potential)
+
+
+plot(v[:,1])
+#plot(v[:,2])
diff --git a/hw7/mvsb.svg b/hw7/mvsb.svg
new file mode 100644
index 0000000..9e4afbb
--- /dev/null
+++ b/hw7/mvsb.svg
@@ -0,0 +1,55 @@
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+<defs>
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diff --git a/hw7/tightbinding (1).jl b/hw7/tightbinding (1).jl
new file mode 100644
index 0000000..0d1cc8c
--- /dev/null
+++ b/hw7/tightbinding (1).jl
@@ -0,0 +1,24 @@
+using LinearAlgebra
+
+# Define the tight binding Hamiltonian
+function tight_binding_hamiltonian(n_sites::Int64, t::Float64)
+ H = zeros(n_sites, n_sites)
+ for i in 1:(n_sites-1)
+ H[i, i+1] = t
+ H[i+1, i] = t
+ end
+ H
+end
+
+# Find the eigenenergies of the Hamiltonian
+function eigenenergies(H::Array{Float64, 2})
+ eigvals(H)
+end
+
+n_sites = 10 # number of sites in the chain
+t = 1.0 # hopping parameter
+H = tight_binding_hamiltonian(n_sites, t)
+energies = eigenenergies(H)
+println(energies)
+
+
generated by cgit v1.2.3-70-g09d2 (git 2.46.0) at 2025-10-06 03:10:45 +0000