1. The problem statement, all variables and given/known data
So this is part of a broader problem about the quantum harmonic oscillator, but there's one particular bit of mathematics I'm stuck on.
We have the differential equation:
y''(x) +(ε-x2) y = 0
And I'm told that we're to examine how y behaves as x tends towards infinity. I took this to mean that we can ignore the term in epsilon entirely.
We're also told that in this limit, we should obtain y = A xk e-x2/2 as the solution to the differential equation.
2. Relevant equations
3. The attempt at a solution
I'm not entirely sure how to go about solving the differential equation (bit rusty) but when I substitute in the given solution, the "xk" term doesn't cancel as I suspect that it should - unless of course you just set k = 0, but the next parts of the question require we prefix it with A x^k . I'm confused.
So this is part of a broader problem about the quantum harmonic oscillator, but there's one particular bit of mathematics I'm stuck on.
We have the differential equation:
y''(x) +(ε-x2) y = 0
And I'm told that we're to examine how y behaves as x tends towards infinity. I took this to mean that we can ignore the term in epsilon entirely.
We're also told that in this limit, we should obtain y = A xk e-x2/2 as the solution to the differential equation.
2. Relevant equations
3. The attempt at a solution
I'm not entirely sure how to go about solving the differential equation (bit rusty) but when I substitute in the given solution, the "xk" term doesn't cancel as I suspect that it should - unless of course you just set k = 0, but the next parts of the question require we prefix it with A x^k . I'm confused.
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