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let be the Schroedinguer equation

$$ - \frac{d^{2}}{dx^{2}}y(x)+ae^{cx}y(x)=E_{n} $$ (1)

here a and c are constants.

i know how to solve it from

but what is the condtion to get the energies ?? on the interval $ [0. \infty) $ or in other interval if you wish

the solution to (1) i know that is given in terms of the Bessel function but i have problems to get the energy quatnizatio condition i have tried a get a nonsense like

$$ J_{\sqrt{E_{n}}}(b)=0 $$ for a certain real number b but is this true ??

of course i know that semiclassically the energies satisfy

$$ 2\pi n \sim \int_{0}^{\infty} \sqrt{E_{n}-ae^{cx}} $$

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Related: exponential potential $\exp(|x|)$. – Emilio Pisanty Aug 24 '13 at 2:49
up vote 2 down vote accepted

To find the bound states for the potential

$$V(x) ~=~\left\{\begin{array}{ccc}ae^{cx} &\text{for}& x>0, \\ \infty&\text{for}& x\leq 0, \end{array} \right.$$

where $a,c>0$ are two positive constants, one should solve the time-independent Schrödinger eq. with the two boundary conditions

$$ \psi(x=0)~=~0 \qquad \text{and} \qquad \lim_{x \to \infty}\psi(x)~=~0.$$

This boundary value problem does only have solutions for certain discrete values of the energy $E$.

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