Principles of Quantum Mechanics · Problem 1.3.2

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · gram-schmidt-orthogonalization

Problem

Show how to go from the basis

|I\rangle = \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \qquad |II\rangle = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \qquad |III\rangle = \begin{bmatrix} 0 \\ 2 \\ 5 \end{bmatrix}

to the orthonormal basis

|1\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \qquad |2\rangle = \begin{bmatrix} 0 \\ 1/\sqrt{5} \\ 2/\sqrt{5} \end{bmatrix} \qquad |3\rangle = \begin{bmatrix} 0 \\ -2/\sqrt{5} \\ 1/\sqrt{5} \end{bmatrix}