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At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform.

That is, if your energy eigenfunctions are complete, $$ \sum_n \psi_n(x) \psi_n^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full.

That is, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi_n(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the SHO.

In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one. If it made you feel better, you might label the eigenvectors by $|E_n\rangle$ instead of $|\psi_n\rangle$.

At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform, $$ |p\rangle= \int\!\!dx~{e^{ixp/\hbar}\over\sqrt{2\pi\hbar}} |x\rangle, $$ ($=e^{ip\hat x/\hbar} |p\!=\!0\rangle$.)

That is, if your orthonormal energy eigenfunctions are complete, $$ \sum_n \psi_n(x) \psi_n^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full.

To wit, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi_n(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the SHO.

In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one. If it made you feel better, you might label the eigenvectors by $|E_n\rangle$ instead of $|\psi_n\rangle$.

At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform.

That is, if your energy eigenfunctions are complete, $$ \sum_n \psi(x) \psi^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full.

That is, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the SHO.

In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one. If it made you feel better, you might label the eigenfunctions by $|E_n\rangle$ instead of $|\psi_n\rangle$.

At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform.

That is, if your energy eigenfunctions are complete, $$ \sum_n \psi_n(x) \psi_n^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full.

That is, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi_n(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the SHO.

In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one. If it made you feel better, you might label the eigenvectors by $|E_n\rangle$ instead of $|\psi_n\rangle$.

At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform.

That is, if your eigenfunctions are complete, $$ \sum_n \psi(x) \psi^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full.

That is, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the SHO.

In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one...

At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform.

That is, if your energy eigenfunctions are complete, $$ \sum_n \psi(x) \psi^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full.

That is, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the SHO.

In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one. If it made you feel better, you might label the eigenfunctions by $|E_n\rangle$ instead of $|\psi_n\rangle$.