In QM we require that an operator acting on a state vector gives the corresponding observable multiplied by the vector.(adsbygoogle = window.adsbygoogle || []).push({});

Spin up can be represented by the state vector [tex] \left( \begin{array}{c} 1 \\ 0 \end{array} \right) [/tex], while spin down can be represented by [tex] \left( \begin{array}{c} 0 \\ 1 \end{array} \right) [/tex]

As I understand the Hamiltonian is represented by an infinite dimensional matrix, because there is an infinite number of energy eigenstates. My question is, how can we satisfy both

[tex] \hat{H} \left | \psi \right \rangle = E \left | \psi \right \rangle [/tex]

and

[tex] \hat{L_{z}} \left | \psi \right \rangle = m\hbar \left | \psi \right \rangle[/tex]

when in one case [tex] \left | \psi \right \rangle [/tex] is a 2 dimensional vector, and in the other it is an infinite dimensional vector.

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# Vectors in Hilbert space

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