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The state space is the space of square integrable functions $L^2(\mathbb{R}^3)$. (Edit: in the following I added reference to the radial wavefunction) One way to get a basis that spans this space is to combine (i.e. take the tensor product of) the angular momentum basis $|l,m\rangle$ and some basis for the radial wavefunction (such as the eigenfunctions of the Hydrogen atom). Such a basis determines the cardinality of the state space to be countably infinite.

However, sometimes it is convenient to consider a larger space: the space of all continuous functions $C(\mathbb{R}^3)$, which is spanned by the position basis $|x\rangle$, and therefore has a larger cardinality (Edit: this space is analogous to the rigged Hilbert space referred to this answer). This space contains functions that have an infinite norm, and therefore cannot represent quantum states, but they can still be useful in our calculations.
Take for example the wavefunction $e^{i k x}$. It is part of $C(\mathbb{R}^3)$, but not part of $L^2(\mathbb{R}^3)$ since it cannot be normalized. Nevertheless, it is very useful to use this wavefunction in calculations, for example in scattering theory. Once we know how this wavefunction evolves over time, we can use the linearity of Quantum mechanics to know how any superposition of such wavefunctions evolve over time. Using the Fourier transform we can express any function in $L^2(\mathbb{R}^3)$ as a linear combination of functions of the form $e^{i k x}$, and therefore we have solved the time evolution of any wavefunction in $L^2(\mathbb{R}^3)$.

In conclusion, using a function space that has a larger cardinality is useful, as long as we remember that in the end of the calculation we need to return to the state space we started with.

The state space is the space of square integrable functions $L^2(\mathbb{R}^3)$. (Edit: in the following I added reference to the radial wavefunction) One way to get a basis that spans this space is to combine (i.e. take the tensor product of) the angular momentum basis $|l,m\rangle$ and some basis for the radial wavefunction (such as the eigenfunctions of the Hydrogen atom). Such a basis determines the cardinality of the state space to be countably infinite.

However, sometimes it is convenient to consider a larger space: the space of all continuous functions $C(\mathbb{R}^3)$, which is spanned by the position basis $|x\rangle$, and therefore has a larger cardinality (Edit: this space is analogous to the rigged Hilbert space referred to this answer). This space contains functions that have an infinite norm, and therefore cannot represent quantum states, but they can still be useful in our calculations.
Take for example the wavefunction $e^{i k x}$. It is part of $C(\mathbb{R}^3)$, but not part of $L^2(\mathbb{R}^3)$ since it cannot be normalized. Nevertheless, it is very useful to use this wavefunction in calculations, for example in scattering theory. Once we know how this wavefunction evolves over time, we can use the linearity of Quantum mechanics to know how any superposition of such wavefunctions evolve over time. Using the Fourier transform we can express any function in $L^2(\mathbb{R}^3)$ as a linear combination of functions of the form $e^{i k x}$, and therefore we have solved the time evolution of any wavefunction in $L^2(\mathbb{R}^3)$.

In conclusion, using a function space that has a larger cardinality is useful, as long as we remember that in the end of the calculation we need to return to the state space we started with.

The state space is the space of square integrable functions $L^2(\mathbb{R}^3)$. (Edit: in the following I added reference to the radial wavefunction) One way to get a basis that spans this space is to combine (i.e. take the tensor product of) the angular momentum basis $|l,m\rangle$ and some basis for the radial wavefunction (such as the eigenfunctions of the Hydrogen atom). Such a basis determines the cardinality of the state space to be countably infinie.

However, sometimes it is convenient to consider a larger space: the space of all continuous functions $C(\mathbb{R}^3)$, which is spanned by the position basis $|x\rangle$, and therefore has a larger cardinality (Edit: this space is analogous to the rigged Hilbert space referred to this answer). This space contains functions that have an infinite norm, and therefore cannot represent quantum states, but they can still be useful in our calculations.
Take for example the wavefunction $e^{i k x}$. It is part of $C(\mathbb{R}^3)$, but not part of $L^2(\mathbb{R}^3)$ since it cannot be normalized. Nevertheless, it is very useful to use this wavefunction in calculations, for example in scattering theory. Once we know how this wavefunction evolves over time, we can use the linearity of Quantum mechanics to know how any superposition of such wavefunctions evolve over time. Using the Fourier transform we can express any function in $L^2(\mathbb{R}^3)$ as a linear combination of functions of the form $e^{i k x}$, and therefore we have solved the time evolution of any wavefunction in $L^2(\mathbb{R}^3)$.

In conclusion, using a function space that has a larger cardinality is useful, as long as we remember that in the end of the calculation we need to return to the state space we started with.

The state space is the space of square integrable functions $L^2(\mathbb{R}^3)$. (Edit: in the following I added reference to the radial wavefunction) One way to get a basis that spans this space is to combine (i.e. take the tensor product of) the angular momentum basis $|l,m\rangle$ and some basis for the radial wavefunction (such as the eigenfunctions of the Hydrogen atom). Such a basis determines the cardinality of the state space to be countably infinite.

However, sometimes it is convenient to consider a larger space: the space of all continuous functions $C(\mathbb{R}^3)$, which is spanned by the position basis $|x\rangle$, and therefore has a larger cardinality (Edit: this space is analogous to the rigged Hilbert space referred to this answer). This space contains functions that have an infinite norm, and therefore cannot represent quantum states, but they can still be useful in our calculations.
Take for example the wavefunction $e^{i k x}$. It is part of $C(\mathbb{R}^3)$, but not part of $L^2(\mathbb{R}^3)$ since it cannot be normalized. Nevertheless, it is very useful to use this wavefunction in calculations, for example in scattering theory. Once we know how this wavefunction evolves over time, we can use the linearity of Quantum mechanics to know how any superposition of such wavefunctions evolve over time. Using the Fourier transform we can express any function in $L^2(\mathbb{R}^3)$ as a linear combination of functions of the form $e^{i k x}$, and therefore we have solved the time evolution of any wavefunction in $L^2(\mathbb{R}^3)$.

In conclusion, using a function space that has a larger cardinality is useful, as long as we remember that in the end of the calculation we need to return to the state space we started with.

The state space is the space of square integrable functions $L^2(\mathbb{R}^3)$. One basis that spans this space is the angular momentum basis $|l,m\rangle$, and this is what determines the cardinality of the state space.

However, sometimes it is convenient to consider a larger space: the space of all continuous functions $C(\mathbb{R}^3)$, which is spanned by the position basis $|x\rangle$, and therefore has a larger cardinality. This space contains functions that have an infinite norm, and therefore cannot represent quantum states, but they can still be useful in our calculations.
Take for example the wavefunction $e^{i k x}$. It is part of $C(\mathbb{R}^3)$, but not part of $L^2(\mathbb{R}^3)$ since it cannot be normalized. Nevertheless, it is very useful to use this wavefunction in calculations, for example in scattering theory. Once we know how this wavefunction evolves over time, we can use the linearity of Quantum mechanics to know how any superposition of such wavefunctions evolve over time. Using the Fourier transform we can express any function in $L^2(\mathbb{R}^3)$ as a linear combination of functions of the form $e^{i k x}$, and therefore we have solved the time evolution of any wavefunction in $L^2(\mathbb{R}^3)$.

In conclusion, using a function space that has a larger cardinality is useful, as long as we remember that in the end of the calculation we need to return to the state space we started with.

The state space is the space of square integrable functions $L^2(\mathbb{R}^3)$. (Edit: in the following I added reference to the radial wavefunction) One way to get a basis that spans this space is to combine (i.e. take the tensor product of) the angular momentum basis $|l,m\rangle$ and some basis for the radial wavefunction (such as the eigenfunctions of the Hydrogen atom). Such a basis determines the cardinality of the state space to be countably infinie.

However, sometimes it is convenient to consider a larger space: the space of all continuous functions $C(\mathbb{R}^3)$, which is spanned by the position basis $|x\rangle$, and therefore has a larger cardinality (Edit: this space is analogous to the rigged Hilbert space referred to this answer). This space contains functions that have an infinite norm, and therefore cannot represent quantum states, but they can still be useful in our calculations.
Take for example the wavefunction $e^{i k x}$. It is part of $C(\mathbb{R}^3)$, but not part of $L^2(\mathbb{R}^3)$ since it cannot be normalized. Nevertheless, it is very useful to use this wavefunction in calculations, for example in scattering theory. Once we know how this wavefunction evolves over time, we can use the linearity of Quantum mechanics to know how any superposition of such wavefunctions evolve over time. Using the Fourier transform we can express any function in $L^2(\mathbb{R}^3)$ as a linear combination of functions of the form $e^{i k x}$, and therefore we have solved the time evolution of any wavefunction in $L^2(\mathbb{R}^3)$.

In conclusion, using a function space that has a larger cardinality is useful, as long as we remember that in the end of the calculation we need to return to the state space we started with.