Quantum hamiltonian with an expoenntial potetial.

[zetafunction]

given the Schroedinger equation with an exponential potential

$$-D^{2}y(x)+ae^{bx}y(x)-E_{n}y(x)= 0$$

with the boudnary conditons $$y(0)=0=y(\infty)$$

is this solvable ?? what would be the energies and eigenfunction ? thanks.

 

[Simon Bridge]

$V(x)=ae^{bx}$ represents an infinite barrier when approached from the left ... if b>0.
Where did you get those boundary conditions from?

$V(x=0)=ae^{0}=a$ if E>a, then y(0) need not be zero.
You do need y(x) to be continuous where it crosses the barrier.

 

[zetafunction]

um i forgot .. $$y(0)=0$$ assume there is an infinite potential barrier so the wave function must be 0 at the origin.

 

[Simon Bridge]

Oh well, that one will have solutions that look a lot like the 1/x potential - at least, for the lower energies.

For simplicity, measure energy from the bottom of the well so $V(0<x)=a(e^{bx}-1)$ ... positive values of E will include bound states - so the answer to your question is: yes - solutions exist, and the SE for this potential should be solvable.

If you want a rigorous proof of solvability you'll have to ask a mathematician ;)
Getting the analytical solution will probably be a bit of a pain... but it usually is.
How did this come up?