A projector keeps the part of a state that lies in a chosen subspace and throws the rest away. Applying it twice does nothing more than applying it once. The ket-bra of a normalised state is the simplest example, and projectors are how the next chapter will say what a measurement does to a state.

The shadow of a stick on the floor is its projection. The shadow of the shadow is the same shadow, and a stick lying on the floor already is its own shadow.

The projector $\hat{P}$ onto the dashed line keeps the part of $\psi$ along it; $\hat{1}-\hat{P}$ keeps the part at right angles. The two parts add up to $\psi$, and projecting $\hat{P}\psi$ again changes nothing.

An operator $\hat{P}$ is an orthogonal projector when $\hat{P}^{2}=\hat{P}$ and . The projector onto the subspace spanned by orthonormal vectors is the sum of their ket-bras; is then the projector onto everything orthogonal to that subspace.

• For a normalised state, $\ket{\phi}\bra{\phi}$ projects onto the line through $\ket{\phi}$: it sends $\ket{\psi}$ to $⟨\varphi$.

Apply the projector twice to an eigenvector. Once gives $\lambda$ times the vector, twice gives $\lambda^{2}$ times it, and since projecting twice is the same as projecting once the two agree. A number equal to its own square is 0 or 1: a vector is either kept whole or removed entirely.

An eigenvector, so $\ket{v}\neq0$.

Apply $\hat{P}$ again and use linearity.

But $\hat{P}^{2}=\hat{P}$.

Since $\ket{v}\neq0$, the number $\lambda^{2}-\lambda$ must vanish.

The squared length of what a projector keeps is the bracket of the state with the projector acting on it. For the ket-bra of a basis state, that is the modulus squared of one coefficient.

An orthogonal projector squares to itself and equals its own adjoint. The ket-bra of a normalised state always qualifies.

Projectors onto orthogonal subspaces add up to the projector onto the whole of both. The completeness relation is the extreme case: the projectors onto all the basis states add to the identity.

In a basis adapted to the subspace a projector is diagonal with ones for the kept directions and zeros for the rest, so its trace counts the kept dimensions.

Great work! You now know how to:

• recognise a projector by squaring it
• compute the kept part and its squared length
• prove that a projector’s eigenvalues are 0 and 1
• use the complementary projector