# Weird Schrodinger solution by DifferentialEquations.jl versus time interval

**URL:** <https://discourse.julialang.org/t/weird-schrodinger-solution-by-differentialequations-jl-versus-time-interval/70542>\
**Category:** Modelling & Simulations\
**Tags:** diffeq, ode, differentialequation\
**Created:** [October 28, 2021, 10:13am UTC](https://discourse.julialang.org/t/weird-schrodinger-solution-by-differentialequations-jl-versus-time-interval/70542 "2021-10-28T10:13:05Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![TravellingLight6](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/travellinglight6/32/30408_2.png) [@TravellingLight6](https://discourse.julialang.org/u/TravellingLight6)\
**Post date:** [October 28, 2021, 10:13am UTC](https://discourse.julialang.org/t/weird-schrodinger-solution-by-differentialequations-jl-versus-time-interval/70542/1 "2021-10-28T10:13:05Z")

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Suppose we want to solve a simple schrodinger equation under columb potential,namely,  
\Psi''=(V-E)\Psi and V(r)=-\frac{1}{r},  
The wavefunction u(r) of ground state is: \sim a\times r\cdot e^{-br} which should be the exact solution no matter how far the interval.But the numerical solution turns to be weired. For very large r\_{max}, the wavefunction goes to large again for a certain point.

The codes goes as following:

```julia
 using DifferentialEquations,Plots,Roots
    function columbode!(du,u,p,r)
        v(r)=-1/r
        du[1]=u[2]
        du[2]=(v(r)-p)*u[1]
    end
    
    function plotf(rmax)
        u0=[0.0,1.0]
        rspan=(1e-10,rmax)
        prob(e)=ODEProblem(columbode!,u0,rspan,e)
        solf(e)=solve(prob(e), Tsit5(), reltol=1e-8, abstol=1e-8)
        e_eigen=find_zero(e->solf(e).u[end][1],(-1.0,-0.1))
        plot(solf(e_eigen),vars=[1])
    end
    
    plot(plotf(50.0),plotf(60.0),plotf(70.0),plotf(80.0),layout=(2,2))

```

The result is as following:  
 ![捕获](https://global.discourse-cdn.com/julialang/original/3X/e/5/e5c60526a349599d9fe2dfe8d5265c565aa23d23.png)

Any idea on how to solve this problem?By the way, I don’t want to use callback to set the wavefunction to artificial zero after I need to find the root of wavefunction with respect to trial energy. Thanks!

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<div class="post-metadata">

**Author:** ![antoine-levitt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/antoine-levitt/32/4008_2.png) [@antoine-levitt](https://discourse.julialang.org/u/antoine-levitt)\
**Post date:** [October 28, 2021, 11:20am UTC](https://discourse.julialang.org/t/weird-schrodinger-solution-by-differentialequations-jl-versus-time-interval/70542/2 "2021-10-28T11:20:49Z")

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The outwards equation is very unstable (if you mess up even a little bit on E or u0, the solution will blow up). Either tighten the tolerance further, use a small interval or possibly use a more stable variant of this method (look up “shooting methods”)

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<div class="post-metadata">

**Author:** ![TravellingLight6](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/travellinglight6/32/30408_2.png) [@TravellingLight6](https://discourse.julialang.org/u/TravellingLight6)\
**Post date:** [October 28, 2021, 12:53pm UTC](https://discourse.julialang.org/t/weird-schrodinger-solution-by-differentialequations-jl-versus-time-interval/70542/3 "2021-10-28T12:53:16Z")

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I’ve give the BVP problem a try but when the enery is not the eigen-energy, the wavefunction can be very hash. I don’t know if there’s any idea to deal with this like shooting and how to judge if the shape of wave function is resonable. Thanks for your quick comment!