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all. I am a high-school student who has recently familiarized himself with linear algebra and is looking to understand quantum computing. So, I bought the classic textbook "Quantum Computation and Quantum Information" by Nielsen and Chuang.

In the book, I came across the problem: find the eigenvalues and eigenvectors of the Pauli matrices.

I started out with the Pauli-X matrix and correctly found its eigenvalues to be 1 and -1.

When I set about finding the eigenvectors (using the standard methods of linear algebra), however, I found that for an eigenvalue of 1, any scalar multiple of (1 1) would do, and for -1, it could be any scalar multiple of (-1 1).

So suppose that a qubit has a Hamiltonian of hωX (this is an example in the book). Its energy eigenvalues are hω and -hω, and its energy eigenstates are the same as the unit eigenvectors of X.

But based on the results of my above calculation, there are two options per eigenvalue. For hω they are $\frac{1}{\sqrt2}(|0\rangle + |1\rangle)$ and $\frac{1}{\sqrt2}(-|0\rangle -|1\rangle)$, and for -hω they are $\frac{1}{\sqrt2}(|0\rangle - |1\rangle)$ and $\frac{1}{\sqrt2}(-|0\rangle + |1\rangle)$. For each case, aren't these states distinct? Are they both right? The textbook only acknowledges the first state in each case.

Thank you very much. I hope other people on this site are as quick to respond as you! – QFTUNIverse – 2019-08-04T17:17:54.343