My understanding is that any wavefunction can be decomposed into a linear sum of basis vectors, which for momentum are something like sine waves and for position are delta functions. And then eigenstates are what you observe when you attempt to measure a property of the wavefunction…but I've also heard that you observe a basis vector…but eigenstates must be square integrable while basis vectors don't? I really just don't understand the difference between these two things and what actually happens to the wavefunction in real life when you make a measurement.

# [Physics] Difference between eigenstate and basis vector

hilbert-spacequantum mechanicsquantum-states

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## Best Answer

You can understand the distinction between a basis and a set of eigenstates by noting that the same concepts apply in a more simple setting, such as ordinary vectors and matrices in 3 dimensions.

A basis in 3 dimensions is any set of 3 linearly independent vectors. For convenience we would ordinarily choose them to be also mutually orthogonal and of unit size.

An eigenvector of a 3 x 3 matrix is any vector such that the matrix acting on the vector gives a multiple of that vector. A 3x3 matrix will ordinarily have this action for 3 vectors, and if the matrix is Hermitian then the vectors will be mutually orthogonal if their eigenvalues are distinct. Thus the set of eigenvectors can be used to form a basis. Equally, given any basis one can construct a matrix whose eigenvectors are the members of that basis.

All the above carries over directly to quantum mechanics. The matrix is one way of representing an operator; the vectors now may have any number of components and may have complex values.

When learning these conceptual ideas, an added level of difficulty comes in when dealing with eigenstates of either momentum or position, because then you get strictly unphysical cases such as states with infinite or zero spatial extent. It is not physically possible for a system to move from whatever state it is in to one with either infinite extent or zero extent. Equally, it is not possible to perform a measurement of either momentum or position with perfect precision. But we often invoke the notion of a delta function as a convenient mathematical tool, and physically possible states (with finite spatial extent) can conveniently be Fourier analysed as a sum of sin waves. In such cases the set of eigenstates form a continuum, and so does any basis. The wavefunction can be thought of as a vector with an infinite number of components.