The Exercise

A quantum system is prepared in an eigenstate $|\psi\rangle$ of an observable $M$, with eigenvalue $m$, and $M$ is measured on it. The exercise asks for the average value of $M$ in this state and for the standard deviation of the outcomes about that average.

A Quantum System

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ A pure state $|\psi\rangle$ of $S$ is a unit vector in the state space of $S$, $$\langle\psi|\psi\rangle = 1.$$(1)^✦

An observable $M$ on $S$ is a self adjoint operator on the state space of $S$, $$M^\dagger = M,$$(2)^✦ and its eigenvalues, all real, are the values a measurement of $M$ can return. A pure state $|\psi\rangle$ of $S$ is an eigenstate of an observable $M$ on $S$ with eigenvalue $m$, a real number, when $$M|\psi\rangle = m|\psi\rangle.$$(3)^✦

Average and Spread

The expectation value $\langle M\rangle_\psi$ of an observable $M$ on $S$ in a pure state $|\psi\rangle$ of $S$ is the inner product of $|\psi\rangle$ with its image under $M$, $$\langle M\rangle_\psi = \langle\psi|M|\psi\rangle,$$(4)^✦ a real number since $M$ is self adjoint.

The square $M^2$ of an observable $M$ on $S$ is the observable on $S$ obtained by applying $M$ twice.^✦ With it, the variance $[\Delta(M)]^2$ of $M$ in a pure state $|\psi\rangle$ of $S$ is the real number $$[\Delta(M)]^2 = \langle M^2\rangle_\psi - \langle M\rangle_\psi^2,$$(5) and the standard deviation $\Delta(M)$ of $M$ in $|\psi\rangle$ is its square root.

The Claim

What Is Checked

The claim is stated over a quantum system $S$, an observable $M$ on $S$ and a pure state $|\psi\rangle$ of $S$ that is an eigenstate of $M$ with eigenvalue $m$, through the expectation value (4) and the standard deviation, the square root of the variance (5).

About the expectation value (4) and the variance (5) the library proves that the square of a self adjoint operator is self adjoint, so that $M^2$ satisfies (2).