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Can the wave-function of any particle in any basis be written as a matrix?

If no, how can we explain this, where the Hamiltonian $H$ in U is a QM operator that can be written as a linear transformation therefore a matrix. And if we take the exponential matrix of H, which gives us another matrix. So surely, we can write Ψ as a matrix. Right??

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The state can be written (actually is) as a vector upon which the time evolution operator (or the hamiltoninan) operates. The i-th component of the vector is the coefficient corresponding to i-th basis vector.

Thus in a given basis $\vec{v_1}$ and $\vec{v_2}$ the state $\vec{v}=a\vec{v_1}+b\vec{v_2}$ can be represented as a vector $(a\;\; b)$.

The position/momentum space, being continous, require particular care though.

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The wavefunction is a linear span of eigenvectors of the differential equations (which is why it is more convenient to study the eigenvectors). So yes, it is a matrix in the base of eigenvectors.

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