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**1. In the part of the analytical solution of the harmonic oscillator, p38 in my version of the book a certain mathematical trick is used. The position part of the wave function in function of a dimensionless variable z, is given by**

##\frac{d^{2}\Psi(z)}{dz^{2}}=(z^{2}-K) \Psi##

In the limit of z going to +- infinity the solution to this ODE (bearing in mind that we want the function to be normalizable)

##\Psi(z)=Ae^{\frac{-z^{2}}{2}}##

Then he says that for any z now the function can be written as

##\Psi(z)=h(z)e^{\frac{-z^{2}}{2}}##

##\frac{d^{2}\Psi(z)}{dz^{2}}=(z^{2}-K) \Psi##

In the limit of z going to +- infinity the solution to this ODE (bearing in mind that we want the function to be normalizable)

##\Psi(z)=Ae^{\frac{-z^{2}}{2}}##

Then he says that for any z now the function can be written as

##\Psi(z)=h(z)e^{\frac{-z^{2}}{2}}##

My question. How certain can I be, only from this reasoning that ##h(z)## does converge? A informal reasoning along the lines:

Well as z goes to the limit of +- infinity the behaviour of ##\Psi(z)## has to be that of the earlier mentioned exponential. This can only be true if ##h(z)## doesn't diverge because even if ##h(z)=z## for example, it would change the behaviour in the limit to ##\Psi(z)=ze^{\frac{-z^{2}}{2}}##

My question. How certain can I be, only from this reasoning that ##h(z)## does converge? A informal reasoning along the lines:

Well as z goes to the limit of +- infinity the behaviour of ##\Psi(z)## has to be that of the earlier mentioned exponential. This can only be true if ##h(z)## doesn't diverge because even if ##h(z)=z## for example, it would change the behaviour in the limit to ##\Psi(z)=ze^{\frac{-z^{2}}{2}}##

This was an informal reasoning that can have gaps in it. Can I be sure that this is 100% true in a formal way as well?This was an informal reasoning that can have gaps in it. Can I be sure that this is 100% true in a formal way as well?

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