Aplicação obscena



No anexo ao artigo, são fornecidos materiais cortados: o caminho das partículas : integrais do caminho, um experimento de dupla fenda em átomos frios de néon, quadros de teletransporte de partículas e outras cenas de crueldade e natureza sexual.



Na minha opinião, não pode haver outra maneira de considerar o formalismo matemático logicamente consistente do que demonstrar o desvio de suas conseqüências da experiência ou provar que suas previsões não esgotam as possibilidades de observação.

Niels Bohr



Em conclusão, o autor gostaria de afirmar que reconhecemos apenas duas boas razões para rejeitar uma teoria que explica uma ampla gama de fenômenos. Em primeiro lugar, a teoria não é consistente internamente e, em segundo lugar, não concorda com os experimentos.

David Bohm

Anteriormente, aprendemos de onde a equação de Schrödinger vem, por que é retirada dali e como é feita por vários métodos.



iΨt=(22m2+V)Ψ



Agora, vamos dividi-lo em duas equações reais. Para fazer isso, imagine psi na forma polarΨ=ReiS/, :



St+(S)22m22m2RR+V=0R2t+(R2Sm)=0



S, R . , , , — -, ( ).



, U=S/m. , S -, - - :



S=i2lnΨΨ



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U=S/m=i2mlnΨΨ=mIm(ψψ)



-, :



x+=UΔt



using LinearAlgebra, SparseArrays, Plots

const ħ = 1.0546e-34 # J*s
const m = 9.1094e-31 # kg
const q = 1.6022e-19 # Kl
const nm = 1e-9 # m
const fs = 1e-12; # s

function wavefun()

    gauss(x) = exp( -(x-x0)^2 / (2*σ^2) + im*x*k )
    k = m*v/ħ

    #    
    Vm = spdiagm(0 => V )
    H = spdiagm(-1 => ones(Nx-1), 0 => -2ones(Nx), 1 => ones(Nx-1) )
    H *= ħ^2 / ( 2m*dx^2 )
    H += Vm

    A = I + 0.5im*dt/ħ * H #  
    B = I - 0.5im*dt/ħ * H # 
    psi = zeros(Complex, Nx, Nt)
    psi[:,1] = gauss.(x)

    for t = 1:Nt-1

        b = B * psi[:,t]
        #       A*psi = b
        psi[:,t+1] = A \ b
    end

    return psi
end


dx = 0.05nm # x step
dt = 0.01fs # t step
xlast = 400nm
Nt = 260 # Number of time steps
t = range(0, length = Nt, step = dt)

x = [0:dx:xlast;]
Nx = length(x) # Number of spatial steps

a1 = 200nm
a2 = 200.5nm # 5 
a3 = 205nm
a4 = 205.5nm 
#V = [ a1<xi<a2 ? 2q : 0.0 for xi in x ] # 2 eV 
V = [ a1<xi<a2 || a3<xi<a4 ? 2q : 0.0 for xi in x ] # 2 eV 

x0 = 80nm # Initial position
σ = 20nm # gauss width
v = -120nm/fs;

@time Psi = wavefun()
P = abs2.(Psi);

function ψ(xi, j)

    k = findfirst(el-> abs(el-xi)<dx, x)
    k == nothing && ( k = 1 )

    Psi[k, j]
end

function corpusculaz(n)

    X = zeros(n, Nt)
    X[:,1] = x0 .+ σ*randn(n)

    for j in 2:Nt, i in 1:n
        U = ħ/(2m*dx) * imag( ( ψ(X[i,j-1]+dx, j) -  ψ(X[i,j-1]-dx, j) ) /  ψ(X[i,j-1], j) )
        X[i,j] = X[i,j-1] - U*dt
    end

    X
end

@time X = corpusculaz(20);

plot(x, P[:, 1], legend = false)
plot!(x, P[:, 80], legend = false)
scatter!( X[:,1], zeros(20) )
scatter!( X[:,80], zeros(20) )




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, . Double-slit interference with ultracold metastable neon atoms — - .







using Random, Gnuplot, Statistics
#  
function trapez(f, a, b, n)
    h = (b - a)/n
    result = 0.5*(f(a) + f(b))
    for i in 1:n-1
        result += f(a + i*h)
    end
    result * h
end
#  
function rk4(f, x, y, h)
    k1 = h * f(x       , y        )
    k2 = h * f(x + 0.5h, y + 0.5k1)
    k3 = h * f(x + 0.5h, y + 0.5k2)
    k4 = h * f(x +    h, y +    k3)

    return y + (k1 + 2*(k2 + k3) + k4)/6.0
end


const ħ = 1.055e-34 # J*s 
const kB = 1.381e-23 # J*K-1
const g = 9.8 # m/s^2

const l1 = 76e-3 # m
const l2 = 113e-3 # m
const yh = 2.8e-3 # m
const d = 6e-6 # m
const b = 2e-6 # m
const a1 = 0.5(- d - b) # 
const a2 = 0.5(- d + b)
const b1 = 0.5(  d - b)
const b2 = 0.5(  d + b)

const m = 3.349e-26 # kg
const T = 2.5e-3 # K
const σv = sqrt(kB*T/m) #  
const σ₀ = 10e-6 # 10 mkm #      
const σz = 3e-4 # 0.3 mm #  z
const σk = m*σv / (ħ*√3) # 2e8 m/s #    

v₀ = zeros(3)
k₀ = zeros(3);


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k



1)ψ0(x,y,z;k0x,k0y,k0z)=ψ0x(x;k0x)ψ0y(z;k0y)ψ0z(z;k0z)=(2πσ02)1/4ex2/4σ02eik0xx×(2πσ02)1/4ey2/4σ02eik0yy×(2πσz2)1/4ez2/4σz2eik0zz



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Quantum Mechanics And Path Integrals, A. R. Hibbs, R. P. Feynman, , 3.6 .



,



K(β,tβ;α,tα)=Kx(xβ,tβ;xα,tα)Ky(yβ,tβ;yα,tα)Kz(zβ,tβ;zα,tα)



Kx(xβ,tβ;xα,tα)=(m2iπ(tβtα))1/2expim((xβxα)22(tβtα))Ky(yβ,tβ;yα,tα)=(m2iπ(tβtα))1/2expim((yβyα)22(tβtα))Kz(zβ,tβ;zα,tα)=(m2iπ(tβtα))1/2expim((zβzα)22(tβtα))×expim(g2(zβ+zα)(tβtα)g224(tβtα)3)



Z , . ,



ψ(β,t)=K(β,t;α,tα)ψ(α,tα)dxα,dyα,dzα



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ψx(x,t;k0x)=K(x,t;xα,0)ψ0(xα)dxαψx(x,t;k0x)=(2πs02(t))1/4exp[(xv0xt)24σ0s0(t)+ik0x(xv0xt)]ρx(x,t)=(2πε02(t))1/2exp[x22ε02(t)]



ε02(t)=σ02(t)+(htσvm)2σ02(t)=σ02+(ht2mσ02)2



, :



ε₀(t) = σ₀^2 + ( ħ/(2m*σ₀) * t )^2 + (ħ*t*σv/m)^2
s₀(t) = σ₀ + im*ħ/(2m*σ₀) * t

ρₓ(x, t) = exp( -x^2 / (2ε₀(t)) ) / sqrt( 2π*ε₀(t) )

t₁(v, z) = sqrt( 2*(l1-z)/g + (v/g)^2 ) - v/g
tf1 = t₁(v₀[3], 0) # s #    
X1 = range(-100, stop = 100, length = 100)*1e-6 
Z1 = range(0, stop = l1, length = 100)
Cron1 = range(0, stop = tf1, length = 100)

P1 = [ ρₓ(x, t) for t in Cron1, x in X1 ]
P1 /= maximum(P1); #   

@gp "set title 'Wavefun before the slits'" xlab="Z, mkm" ylab="X, mkm"
@gp :- 1000Z1 1000000X1 P1 "w image notit"




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ψx(x,t)=ψA+ψBψA=AKx(x,t;xa,t1)ψx(xa,t1,k0x)dxaψB=BKx(x,t;xb,t1)ψx(xb,t1,k0x)dxb





ρx(x,t)=+(2πσv2)1/2exp(k0x22σv2)|ψx(x,t;k0x)|2dk0x



, :



function ρₓ(xᵢ, tᵢ) 

    Kₓ(xb, tb, xa, ta) = sqrt( m/(2im*π*ħ*(tb-ta)) ) * exp( im*m*(xb-xa)^2 / (2ħ*(tb-ta)) )

    function subintrho(k) 

        ψₓ(x, t) = ( 2π*s₀(t) )^-0.25 * exp( -(x-ħ*k*t/m)^2 / (4σ₀*s₀(t)) + im*k*(x-ħ*k*t/m) )
        subintpsi(x) = Kₓ(xᵢ, tᵢ, x, tf1) * ψₓ(x, tf1)

        ψa = trapez(subintpsi, a1, a2, 200)
        ψb = trapez(subintpsi, b1, b2, 200)

        exp( -k^2 / (2σv^2) ) * abs2(ψa + ψb)
    end

    trapez(subintrho, -10σk, 10σk, 20) / sqrt( 2π*σv^2 )
end


t₂(v, z) = sqrt( 2*(l1+l2-z)/g + (v/g)^2 ) - v/g
tf2 = t₂(v₀[3], 0) # s #    
X2 = range(-800, stop = 800, length = 200)*1e-6
Z2 = range(l1, stop = l2, length = 100)
Cron2 = range(tf1, stop = tf2, length = 100)

@time P2 = [ ρₓ(x, t) for t in Cron2, x in X2 ];

P2 /= maximum(P2[4:end,:]);
@gp "set title 'Wavefun after the slits'" xlab="t, s" ylab="X, mkm"
@gp :- Cron2[4:end] 1000000X2 P2[4:end,:] "w image notit"


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v=Sm



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v(x,y,z,t)=Sm+logρ×sms=(0,0,/2)logρ×sm=2mρ(ρy,ρx,0)



, ,



dxdt=vx(x,t)=1mSx+2mρρy=v0x+(xv0xt)2t4m2σ02σ02(t)(yv0yt)2mσ02(t)dydt=vy(x,t)=1mSy2mρρx=v0y(yv0yt)2t4m2σ02σ02(t)+(xv0xt)2mσ02(t)dzdt=vz(z,t)=1mSz=v0z+gt+(zv0ztgt2/2)2t4m2σz2σz2(t)



x(t)=v0xt+x02+y02σ0(t)σ0cosφ(t)y(t)=v0yt+x02+y02σ0(t)σ0sinφ(t)z(t)=v0zt+12gt2+z0σz(t)σz



σ02(t)=σ02+(t2mσ0)2φ(t)=φ0+arctan(t2mσ02)cos(φ0)=x0x02+y02sin(φ0)=y0x02+y02sin(arctanx)=x1+x2cos(arctanx)=11+x2cos(a+b)=cosacosbsinasinbsin(a+b)=sinacosb+cosasinb



function xbs(t, xo, yo, vo)

    cosϕ = xo / sqrt( xo^2 + yo^2 )
    sinϕ = yo / sqrt( xo^2 + yo^2 )
    atnx = -ħ*t / (2m*σ₀^2)
    sinatnx = atnx / sqrt( atnx^2 + 1)
    cosatnx =    1 / sqrt( atnx^2 + 1)
    cosϕt = cosϕ * cosatnx - sinϕ * sinatnx
    #sinϕt = sinϕ*cosatnx + cosϕ*sinatnx

    σ₀t = sqrt( σ₀^2 + ( ħ*t / (2*m*σ₀) )^2 )

    vo*t + sqrt(xo^2 + yo^2) * σ₀t/σ₀ * cosϕt
end

@gp "set title 'Trajectories before slits'" xlab="t, s" ylab="X, mkm" # "set yrange [-10:10]"

@time for i in 1:80

    x0 = randn()*σ₀
    y0 = randn()*σ₀
    v0 = randn()*σv*1e-4

    prtclᵢ = [ xbs(t,x0,y0,v0) for t in Cron1 ]

    @gp :- Cron1 1000000prtclᵢ "with lines notit lw 2 lc rgb 'black'"
    # slits -4-> -2, 2->4 mkm
end




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vx(x,t)=1tt1[x+12(α2+βt2)(βtIm(C(x,t)H(x,t))+αRe(C(x,t)H(x,t))βtγx,t)]



18+

H(x,t)=XAbXA+bf(x,u,t)du+XBbXB+bf(x,u,t)duC(x,t)=[f(x,u,t)]u=XAbu=XA+b+[f(x,u,t)]u=XBbu=XB+bf(x,u,t)=exp[(α+iβt)u2+iγx,tu]α=14σ02(1+(t12mσ02)2)βt=m2(1tt1+1t1(1+(2mσ02t1)2))γx,t=mx(tt1)



function Uₓ(t, x)

    γ = -m*x / ( ħ*(t-tf1) )
    β = m/(2ħ) * ( 1/(t-tf1) + 1 / ( tf1*( 1 + (2m*σ₀^2 / (ħ*tf1) )^2 ) ) )
    α = -( 4σ₀^2 * ( 1 + ( ħ*tf1 / (2m*σ₀^2) )^2 ) )^-1

    f(x, u, t) = exp( ( α + im*β )*u^2 + im*γ*u )

    C = f(x, a2, t) - f(x, a1, t) + f(x, b2, t) - f(x, b1, t)
    H = trapez( u-> f(x, u, t) , a1, a2, 400) + trapez( u-> f(x, u, t) , b1, b2, 400)

    return 1/(t-tf1) * ( x - 0.5/(α^2 + β^2) * ( β*imag(C/H) + α*real(C/H) - β*γ ) )
end

init() = bitrand()[1] ? rand()*(b2 - b1) + b1 : rand()*(a2 - a1) + a1
myrng(a, b, N) = collect( range(a, stop = b, length = N ÷ 2) )




n = 1
xx = 0
ξ = 1e-5
while xx < Cron2[end]-Cron2[1]
    n+=1
    xx += (ξ*n)^2 
end
n

steps = [ (ξ*i)^2 for i in 1:n1 ]
Cronadapt = accumulate(+, steps, init = Cron2[1] )

function modelsolver(Np = 10, Nt = 100)

    #xo = [ init() for i = 1:Np ] # 
    xo = vcat( myrng(a2, a1, Np), myrng(b1, b2, Np) ) # 

    xpath = zeros(Nt, Np)
    xpath[1,:] = xo

    for i in 2:Nt, j in 1:Np
        xpath[i,j] = rk4(Uₓ, Cronadapt[i], xpath[i-1,j], steps[i] )
    end

    xpath
end


@time paths = modelsolver(20, n);

@gp "set title 'Trajectories after the slits'" xlab="t, s" ylab="X, mkm" "set yrange [-800:800]"
for i in 1:size(paths, 2)
    @gp :- Cronadapt 1e6paths[:,i] "with lines notit lw 1 lc rgb 'black'"
end


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function yas(t, xo, yo, vo)

    cosϕ = xo / sqrt( xo^2 + yo^2 )
    sinϕ = yo / sqrt( xo^2 + yo^2 )
    atnx = -ħ*t / (2m*σ₀^2)
    sinatnx = atnx / sqrt( atnx^2 + 1)
    cosatnx =    1 / sqrt( atnx^2 + 1)
    #cosϕt = cosϕ * cosatnx - sinϕ * sinatnx
    sinϕt = sinϕ*cosatnx + cosϕ*sinatnx

    σ₀t = sqrt( σ₀^2 + ( ħ*t / (2*m*σ₀) )^2 )

    vo*t + sqrt(xo^2 + yo^2) * σ₀t/σ₀ * sinϕt
end

function modelsolver2(Np = 10)

    Nt = length(Cronadapt)
    xo = [ init() for i = 1:Np ]
    #xo = vcat( myrng(a2, a1, Np), myrng(b1, b2, Np) )

    xcoord = copy(xo)
    ycoord = zeros(Np)

    for i in 2:Nt, j in 1:Np
        xcoord[j] = rk4(Uₓ, Cronadapt[i], xcoord[j], steps[i] )
    end

    for (i, x) in enumerate(xo)

        y0 = randn()*yh
        v0 = 0 #randn()*σv*1e-4

        ycoord[i] = yas(Cronadapt[end], x, y0, v0)
    end

    xcoord, ycoord
end

@time X, Y = modelsolver2(100);

@gp "set title 'Impacts on the screen'" xlab="X, mm" ylab="Y, mm"# "set xrange [-1:1]"
@gp :- 1000X 1000Y "with points notit pt 7 ps 0.5 lc rgb 'black'"






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