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The way I've understood it is that eigenfunction of an operator are the different states which the actual wavefunction can take when the property/observable corresponding to the given operator is being observed.

For the Hamiltonian operator, the eigenfunctions are forms the wavefunction will take when energy is being measured. The probability that we get a specific state corresponds to the coefficient from fourier decomposition of the wavefunction.

I've also read/interpreted/misinterpreted that upon measurement of position, a wavefunction will collapse to a single spike with a specific value of position (determinate state of the position observable).

With all of the above in mind, is it correct to think that the eigenstates of the position operator are continuous, because the probability density function is continuous and we can get a spike anywhere?

Also, does this mean that the wavefunction takes the form of a spike when position is being observed, but a sine wave when the energy is being observed?

Edit: by continuous eigenstates I want to say that there is an infinite number of them.

The way I've understood it is that eigenfunction of an operator are the different states which the actual wavefunction can take when the property/observable corresponding to the given operator is being observed.

For the Hamiltonian operator, the eigenfunctions are forms the wavefunction will take when energy is being measured. The probability that we get a specific state corresponds to the coefficient from fourier decomposition of the wavefunction.

I've also read/interpreted/misinterpreted that upon measurement of position, a wavefunction will collapse to a single spike with a specific value of position (determinate state of the position observable).

With all of the above in mind, is it correct to think that the eigenstates of the position operator are continuous, because the probability density function is continuous and we can get a spike anywhere?

Also, does this mean that the wavefunction takes the form of a spike when position is being observed, but a sine wave when the energy is being observed?

The way I've understood it is that eigenfunction of an operator are the different states which the actual wavefunction can take when the property/observable corresponding to the given operator is being observed.

For the Hamiltonian operator, the eigenfunctions are forms the wavefunction will take when energy is being measured. The probability that we get a specific state corresponds to the coefficient from fourier decomposition of the wavefunction.

I've also read/interpreted/misinterpreted that upon measurement of position, a wavefunction will collapse to a single spike with a specific value of position (determinate state of the position observable).

With all of the above in mind, is it correct to think that the eigenstates of the position operator are continuous, because the probability density function is continuous and we can get a spike anywhere?

Also, does this mean that the wavefunction takes the form of a spike when position is being observed, but a sine wave when the energy is being observed?

Edit: by continuous eigenstates I want to say that there is an infinite number of them.

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Are eigenstates of the position operator continuous?

The way I've understood it is that eigenfunction of an operator are the different states which the actual wavefunction can take when the property/observable corresponding to the given operator is being observed.

For the Hamiltonian operator, the eigenfunctions are forms the wavefunction will take when energy is being measured. The probability that we get a specific state corresponds to the coefficient from fourier decomposition of the wavefunction.

I've also read/interpreted/misinterpreted that upon measurement of position, a wavefunction will collapse to a single spike with a specific value of position (determinate state of the position observable).

With all of the above in mind, is it correct to think that the eigenstates of the position operator are continuous, because the probability density function is continuous and we can get a spike anywhere?

Also, does this mean that the wavefunction takes the form of a spike when position is being observed, but a sine wave when the energy is being observed?