# The Heisenberg picture of quantum mechanics

14 Jul

In an earlier post we introduced the Schrödinger picture of quantum mechanics, which can be summarized as follows: the state of a quantum system is described by a unit vector $latex \psi$ in some Hilbert space $latex L^2(X)$ (up to multiplication by a constant), and time evolution is given by

$latex \displaystyle \psi \mapsto e^{ \frac{H}{i \hbar} t} \psi$

where $latex H$ is a self-adjoint operator on $latex L^2(X)$ called the Hamiltonian. Observables are given by other self-adjoint operators $latex F$, and at least in the case when $latex F$ has discrete spectrum measurement can be described as follows: if $latex \psi_k$ is a unit eigenvector of $latex F$ with eigenvalue $latex F_k$, then $latex F$ takes the value $latex F_k$ upon measurement with probability $latex \left| \langle \psi, \psi_k \rangle \right|^2$; moreover, the state vector $latex \psi$ is projected onto $latex \psi_k$.

The Heisenberg picture is an alternate way…

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