A Power series in quantum mechanics

gremory
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Power series to solving the infinite square well
Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for ##\psi(x)## wouldn't give me the quantised energies. My solution was
$$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$, with ##k = \frac{\sqrt{2mE}}{\hbar}##. I want to know what's the catch when doing this because we need the boundary conditions to quantize the energies and with this solution i don't see how i would get that.
 
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\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))=(\cos(x) + \sin(x)) \sum^{\infty}_{n=0} k^{2n}=\psi(x,k)

so something seems wrong. Where are the edges of infinite well ?
 
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Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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