exponential potential quantization

Asked Apr 01 '13 at 16:00

Active Apr 01 '13 at 16:08

Viewed 99 times

What are the energies $E_{n}$ of the Schroedinger operator

$$ -\frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)=E_{n}y(x) $$

for some real and positive 'a' and 'b' with the Boundary conditions $ y(0)=0=y(\infty) $?

Working a bit I get the quantization condition

$$ J_{E_{n}/4}(\sqrt{-a})=0,$$

($ J_{a}(x) $ is a Bessel function) but unfortunately this equation yields only imaginary roots.

edited Apr 01 '13 at 16:08

Michael

asked Apr 01 '13 at 16:00

Did you mean $e^{bx}$ ? – user10001 Apr 01 '13 at 16:06
re yes ,. Michael has corrected the equation. – Jose Javier Garcia Apr 01 '13 at 16:09
Please check my edits are what you meant, particularly re: the order of the Bessel function. – Michael Apr 01 '13 at 16:09
possible duplicate of exponential potential $ \exp(|x|) $ – Emilio Pisanty Aug 24 '13 at 02:56

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