Every operator has a partner, its adjoint, which does the same job seen from the bra side of a bracket. For a matrix it is the conjugate transpose: swap rows and columns and conjugate every entry. The adjoint is what lets an operator be moved from one side of a bracket to the other, and operators equal to their own adjoint are the subject of the next lessons.

Reading a sentence in a mirror gives the words in reverse order with every letter flipped. The adjoint of a product is the same: the factors come in reverse order, each one flipped.

The adjoint $\hat{A}^{\dagger}$ is the operator that gives the same bracket when it acts on the bra instead of the ket. In an orthonormal basis its matrix is the conjugate transpose of the matrix of $\hat{A}$. The same dagger turned a ket into its bra in the first chapter, and that is no coincidence: a ket is a one-column matrix.

• $(\hat{A}^{\dagger})^{\dagger}=\hat{A}$, and .

Move the operators across the bracket one at a time: first $\hat{A}$, which is on the outside, then $\hat{B}$. They arrive on the bra side in the reverse order, each wearing a dagger. Comparing with the definition of the adjoint of the product settles it, because a bracket that agrees for every pair of vectors fixes the operator.

Start from the product acting on the ket side.

Move $\hat{A}$, the outer operator, to the bra side using its adjoint.

Now move $\hat{B}$, which lands to the left of $\hat{A}^{\dagger}$.

By definition, the adjoint of the product does the same job in one move.

The two agree for every $\phi$ and $\psi$, so they are the same operator.

The adjoint’s matrix swaps rows with columns and conjugates every entry. So its entry in row $m$ and column $n$ comes from row $n$ and column $m$ of the original.

Transposing alone is not enough: every entry is also conjugated. Real entries are unchanged; imaginary parts change sign.

A complex number multiplying an operator comes out of the adjoint conjugated, just as it did when a ket became a bra.

The adjoint of a product is the product of the adjoints in the opposite order, as with the transpose or the inverse of a product of matrices.

Great work! You now know how to:

• write the adjoint of a matrix as its conjugate transpose
• move an operator across a bracket with its adjoint
• take the adjoint of numbers, sums, products and ket-bras