Consider the time-independent Schroedinger equation phi′′(x)+V(x) phi (x)=lambda phi (x), x in R.

Aden Lambert

Aden Lambert

Answered question

2022-11-17

Potential such that the time-independent Schroedinger equation has an explicit solution
Consider the time-independent Schroedinger equation
ϕ ( x ) + V ( x ) ϕ ( x ) = λ ϕ ( x )
For testing my numerics, I would like to now:
For which choices of V are there explicit expressions for the solutions ( ϕ , λ )?
For example, if V ( x ) = x 2 , then the solutions are products of Hermite polynomials and e x p ( x 2 ) (up to correct scales) and arithmetic progressions (the same works in higher dimensions with the analogue potential).

Answer & Explanation

Kristen Garza

Kristen Garza

Beginner2022-11-18Added 13 answers

e x is defined everywhere, but for arccos ( t ) you want 1 t 1. Now e x > 0 > 1, but what do you need to get e x 1?

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