Encountering Method error when using abs2 function on a Complex{Num}

New to Julia
1

I have published my code here: GitHub - VikramD2k/Hydrogen-Atom-Simulator: This computes and plots the probability density of any orbital of the hydrogen atom
and I am also pasting my code down below

##
using Symbolics
using Plots
using Statistics
##
function Legrande(l::Int, cosθ::Num) 
    if l == 0
        legrande = 1
    else
        D_cst_l = Differential(cosθ)^l
        legrande = (1 / (2^l * factorial(l))) * D_cst_l((cosθ^2 - 1)^l)
    end
    return expand_derivatives(legrande)
end
##
function AssoLegrande(l::Int, m::Int, cst::Float64)
    @variables cosθ
    if m == 0
        asslegrande = (-1)^m * (1 - cosθ^2)^(m / 2) * Legrande(l, cosθ)
    else
        D_cst_m = Differential(cosθ)^m
        asslegrande = (-1)^m * (1 - cosθ^2)^(m / 2) * D_cst_m(Legrande(l, cosθ))
    end
    asslegrande = expand_derivatives(asslegrande)
    asslegrande = substitute(asslegrande, cosθ => cst)
    return asslegrande
end
##
function Laguerre(q::Int, x::Num)
    if q == 0
        leg = 1
    else
        Dx_q = Differential(x)^q
        leg = exp(x) * Dx_q(exp(-x) * x^q) / factorial(q)
    end
    return expand_derivatives(leg)
end
##
function AssoLaguerre(p::Int, q::Int, k::Float64)
    @variables ρ
    if p == 0
        asso_lag = (-1)^p * Laguerre(p + q, ρ)
    else
        Dρ_p = Differential(ρ)^p
        asso_lag = (-1)^p * Dρ_p(Laguerre(p + q, ρ))
    end
    asso_lag = expand_derivatives(asso_lag)
    asso_lag = substitute(asso_lag, ρ => k)
    return asso_lag
end
##
function Radial(n::Int, l::Int, r::Float64)
    a = 0.529 * 10^-10  # Radius of hydrogen atom in meters
    r_val = 2 * r / (n * a)
    radial = sqrt((2 / (n * a))^3 * factorial(n - l - 1) / (factorial(n + l) * 2 * n)) * exp(-r / (n * a)) * (2 * r / (n * a))^l
    radial = radial * AssoLaguerre(2 * l + 1, n - l - 1, r_val)
    radial = expand_derivatives(radial)
    return radial
end
##
function Angular(l::Int, m::Int, cst::Float64, phi::Float64)
    i = Complex{Float64}(0, 1)
    angular = sqrt((2 * l + 1) * factorial(l - m) / (4 * π * factorial(l + m))) * AssoLegrande(l, m, cst) * exp(i * m * phi)
    return angular
end
##
function get_prob_density(n, l, m, rₘᵢₙ, rₘₐₓ, r_steps, θ_steps, φ_steps)
    x = zeros(Total_steps)
    y = zeros(Total_steps)
    z = zeros(Total_steps)
    ψ² = zeros(Total_steps)  # Initializing placeholder for |ψ|²
    counter = 1
    for r in range(rₘᵢₙ, rₘₐₓ, length=r_steps)
        for θ in range(0, π, length=θ_steps)
            for ϕ in range(0, 2 * π, length=φ_steps)
                ψ = eval(Radial(n, l, r) * Angular(l, m, cos(θ), ϕ)) # Here we got probability amplitudes at the point (r,θ,φ)
                print("ψ is ", ψ, " Type of ψ is ", typeof(ψ), "\n")
                ψ²[counter] = abs2(ψ)
                x[counter] = r * sin(θ) * cos(ϕ)
                y[counter] = r * sin(θ) * sin(ϕ)
                z[counter] = r * cos(θ)
                counter = counter + 1
            end
        end
    end
    ψ²_sum = sum(ψ²)  # This should be close to 1
    ψ² = ψ² / ψ²_sum  # Now it is normalized
    return ψ², x, y, z
end
##
function plot_probs(Total_steps::Int, ψ², x, y, z)
    X, Y, Z = [], [], []
    c = 0

    for i in 1:Total_steps
        if ψ²[i] < 1.1 && ψ²[i] > 0.9
            c += 1
            push!(X, x[i])
            push!(Y, y[i])
            push!(Z, z[i])
        end
    end
    Plots.scatter3d(X, Y, Z, markersize=2, color=:red)
    title = "n is $n, l is $l, m is $m"
    title!(title)
    xlabel!("x")
    ylabel!("y")
    zlabel!("z")
end
##
# With the above all the required functions are defined
a = 0.529 * 10^-10  # Radius of hydrogen atom in meters
rₘᵢₙ = 0.5 * a
rₘₐₓ = 50 * a
r_steps = 10
θ_steps = 10
φ_steps = 10
Total_steps = r_steps * θ_steps * φ_steps
n = 1   # principle quantum number
l = 0   # azimuthal quantum number
m = 0   # magnetic quantum number
##
prob_density, x, y, z =get_prob_density(n, l, m, rₘᵢₙ, rₘₐₓ, r_steps, θ_steps, φ_steps)
plot_probs(Total_steps, prob_density, x, y, z)
##

the error I am encountering is the following

ψ is 8.893942444146948e14 Type of ψ is Complex{Num}
1-element ExceptionStack:
LoadError: MethodError: no method matching Float64(::Num)

Closest candidates are:
  (::Type{T})(::Real, ::RoundingMode) where T<:AbstractFloat
   @ Base rounding.jl:207
  (::Type{T})(::T) where T<:Number
   @ Core boot.jl:792
  Float64(::IrrationalConstants.Sqrt2)
   @ IrrationalConstants C:\Users\vikra\.julia\packages\IrrationalConstants\vp5v4\src\macro.jl:112
  ...

Stacktrace:
  [1] convert(::Type{Float64}, x::Num)
    @ Base .\number.jl:7
  [2] setindex!(A::Vector{Float64}, x::Num, i1::Int64)
    @ Base .\array.jl:1021
  [3] get_prob_density(n::Int64, l::Int64, m::Int64, rₘᵢₙ::Float64, rₘₐₓ::Float64, r_steps::Int64, θ_steps::Int64, φ_steps::Int64)
    @ Main c:\Users\vikra\OneDrive\Desktop\Julia Programmes\Hydrogen Atom\Hydrogen_atom.jl:78
  [4] top-level scope
    @ c:\Users\vikra\OneDrive\Desktop\Julia Programmes\Hydrogen Atom\Hydrogen_atom.jl:123
  [5] eval
    @ .\boot.jl:385 [inlined]
  [6] include_string(mapexpr::typeof(identity), mod::Module, code::String, filename::String)
    @ Base .\loading.jl:2076
  [7] include_string(m::Module, txt::String, fname::String)
    @ Base .\loading.jl:2086
  [8] invokelatest(::Any, ::Any, ::Vararg{Any}; kwargs::@Kwargs{})
    @ Base .\essentials.jl:892
  [9] invokelatest(::Any, ::Any, ::Vararg{Any})
    @ Base .\essentials.jl:889
 [10] inlineeval(m::Module, code::String, code_line::Int64, code_column::Int64, file::String; softscope::Bool)
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\eval.jl:271
 [11] (::VSCodeServer.var"#69#74"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\eval.jl:181
 [12] withpath(f::VSCodeServer.var"#69#74"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams}, path::String)
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\repl.jl:276
 [13] (::VSCodeServer.var"#68#73"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\eval.jl:179
 [14] hideprompt(f::VSCodeServer.var"#68#73"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\repl.jl:38
 [15] (::VSCodeServer.var"#67#72"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\eval.jl:150
 [16] with_logstate(f::Function, logstate::Any)
    @ Base.CoreLogging .\logging.jl:515
 [17] with_logger
    @ .\logging.jl:627 [inlined]
 [18] (::VSCodeServer.var"#66#71"{VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\eval.jl:263
 [19] #invokelatest#2
    @ .\essentials.jl:892 [inlined]
 [20] invokelatest(::Any)
    @ Base .\essentials.jl:889
 [21] (::VSCodeServer.var"#64#65")()
    @ VSCodeServer c:\Users\vikra\.vscode\extensions\julialang.language-julia-1.79.2\scripts\packages\VSCodeServer\src\eval.jl:34
in expression starting at c:\Users\vikra\OneDrive\Desktop\Julia Programmes\Hydrogen Atom\Hydrogen_atom.jl:123

I think the abs2(ψ) is unable to convert the Complex{Num} type ψ to a float.
I have no clue how to fix the above issue. Also tell me if I my analysis is correct.
additional info:
I am new to julia and I have migrated this code from MATLAB. I have gotten great suggestions from @mikmoore related to this code in a previous topic (Getting Stack Overflow error when running Derivative(x)^n) . I have incorporated all the changes they suggested.

2

The issue is not with abs2. From the stacktrace, we have the lines

  [1] convert(::Type{Float64}, x::Num)
    @ Base .\number.jl:7
  [2] setindex!(A::Vector{Float64}, x::Num, i1::Int64)
    @ Base .\array.jl:1021

We see here that we are attempting to store a Num into a Vector{Float64} (via setindex!, which is used for x[i] = v syntax). To do this, we see that to do this we attempt to convert a Num to a Float64 so that we can store it, but the MethodError indicates this operation tries Float64(::Num) which is not defined.

A MWE is

using Symbolics
x = zeros(Float64, 1)
v = Num(1.0)
x[1] = v # MethodError

We need to tell the system how to do this conversion. The docs indicate that Symbolics.value can be used to extract the value from a Num.

So a fix is to instead use

x[1] = Symbolics.value(v)

In general, try not to post big code for small problems. You should attempt to localize the problem and create a small example showing the error (this can be difficult initially but gets easier the more familiar you become with the language). I provide an example of doing this above.

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3

Or allocate an array of Num rather than an array of Float64, if you want to store Num (symbolic) values.

The basic point is that you need to think about what type(s) you want to store in an array or other container — an array of Float64 and an array of Num are quite different things.

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