First the bra vector dots into the state, giving the coefficient of in the state, then its multiplied by the unit vector, turning it back into a vector, with the right length to be a projection.An operator maps one vector into another vector, so this is an operator. The sum of the projection operators is 1, if we sum over a complete set of states, like the eigenstates of a Hermitian operator. What is meant by normalized projection operator? What is its physical meaning in quantum mechanics? I am pretty confused regarding the physical interpretation of both projection operator and normal.
$begingroup$I'm looking for an operator $hat P$ in $mathbb{R^3}$ such that $hat P^2=hat P$ that is also Hermitian
3 Answers
$begingroup$A standard example for a projection operator in $mathbb{R}^3$ is the projection onto the subspace spanned by a single vector:
Let $u = (u_1,u_2,u_3)^T in mathbb{R}^3$, then $P_u x = frac{x cdot u}{u cdot u} u$. Note that the linearity of the dot product makes $P_u$ linear, and $P_u u= u$.
Now $$P_u ( P_u x ) = P_u left( frac{x cdot u}{ u cdot u} u right) = frac{x cdot u}{ u cdot u} P_u u = frac{x cdot u}{ u cdot u} u$$ holds for all $x$, so $P_u^2 = P_u$.
Finally, $(P_u x) cdot y = frac{(x cdot u)(y cdot u)}{u cdot u} = x cdot (P_u y)$, so $P_u$ is self-adjoint (i.e. Hermitian).
We can express $P_u$ as a matrix using the standard basis in $mathbb{R}^3$. Let $e_1 = (1,0,0)^T$, $e_2 = (0,1,0)^T$ and $e_3=(0,0,1)^T$, and let $beta = { e_1, e_2, e_3}$ be the ordered set of basis vectors.
$$P_u e_i = frac{u_i}{ucdot u} u = frac{1}{u cdot u} (u_i u_1, u_i u_2, u_i u_3)^T.$$
Thus $$[P_u]_beta = frac{1}{u cdot u} begin{pmatrix} u_1 u_1 & u_2 u_1 & u_3 u_1 u_1 u_2 & u_2 u_2 & u_3 u_2u_1 u_3 & u_2 u_3 & u_3 u_3end{pmatrix}.$$
You can see directly from the matrix representation that the operator $P_u$ is self-adjoint. If you were so inclined you could also demonstrate through matrix multiplication that this matrix satisfies $[P_u]_beta = [P_u]_beta^2$.
JoelJoelThe matrix all of whose elements are equal to $1/3$ is Hermitian and a projection, though 'Hermitian' is silly in a real vector space.
Igor RivinIgor Rivin