Conceptual problem with perturbation theory

[eljose]

-Ok..Let,s be the Hamiltonian $$H=H_0 +W$$ in one dimension where W is a "weak" term so we can apply perturbation theory.

-The "problem" comes when we need to calculate the eigenvalues and eigenfunction of H0 of course we set the system in an "imaginary potential well of width L" so we have the set of eigen-values-functions:

$$E_n =P^{2}/2m$$ $$p=(n\pi \hbar)/L$$ $$\Phi(x) =e^{in\pi x/L}$$

the problem is...what is the value of L?...so when doing calculations..what,s the value of the width of our "imaginary" well..if we set L--->oo then the Energies and Wave functions tend all to 0.

 

[Perturbation]

-Ok..Let,s be the Hamiltonian $$H=H_0 +W$$ in one dimension where W is a "weak" term so we can apply perturbation theory.

-The "problem" comes when we need to calculate the eigenvalues and eigenfunction of H0 of course we set the system in an "imaginary potential well of width L" so we have the set of eigen-values-functions:

$$E_n =P^{2}/2m$$ $$p=(n\pi \hbar)/L$$ $$\Phi(x) =e^{in\pi x/L}$$

the problem is...what is the value of L?...so when doing calculations..what,s the value of the width of our "imaginary" well..if we set L--->oo then the Energies and Wave functions tend all to 0.

For one, when you set $L\rightarrow\infty$, $\Phi_n$ does not tend to zero, it tends to 1, because that's a complex exponential,

$$e^{in\pi\tfrac{x}{L}}=\cos\left(n\pi\tfrac{x}{L}\right)+i\sin\left(n\pi\tfrac{x}{L}\right)$$.

Since each $\Phi_n(x) =e^{in\pi x/L}$ is linearly dependant you can write the wave-function as a Fourier series with these functions as a basis. If L tends to infinity your sum of plane wave bases becomes a Fourier transform of the continuous variable p, rather than the discrete sum of p's (or n's). This is just the same procedure one goes through in generalising the Fourier series of functions with finite period to those with infinite period.

 

[denian]

This is a valid concern when using perturbation theory. In order for perturbation theory to work, the perturbation term W must be "small" compared to the unperturbed Hamiltonian H0. This means that the perturbation term should be much smaller than the energy scales of the system. In the case of the imaginary potential well, the width L determines the energy scale of the system. If L is too large, then the perturbation term W may not be small enough and perturbation theory may not be valid.

One solution to this problem is to choose a value of L that is physically reasonable for the system being studied. For example, if the system is a physical well with a certain width, then that width can be used in the calculations. Another approach is to use a variational method, where the width L is treated as a parameter that is optimized to give the best approximation to the true solution.

Overall, the conceptual problem with perturbation theory is that it relies on the perturbation term being small, and this may not always be the case in realistic systems. Careful consideration must be given to choosing appropriate values for parameters such as L in order to ensure the validity of perturbation theory.