# [Physics] Adjoint of a Wave Function

quantum mechanics

Why is the adjoint of a function simply it's complex conjugate? Normally with a vector we consider the adjoint to be the transpose (And the conjugate? I don't know why), so does this concept carry through to these functions? Should I imagine the conjugate of a wave function to actually be a column vector? Further what does it mean for a function to be a row vector, versus a column vector. Do they live in a completely separate space? Are our operators always square?

Further what does it mean for a function to be a row vector, versus a column vector

Recall that a vector is a geometric object and is distinct from the components of a vector which are just numbers.

A particular component of a vector is given by the contraction of the vector with a basis one-form,or dual vector, to give a number:

$v_x = \vec v \cdot \hat e_x$

The value of the wavefunction at a particular value of x is a number given by the contraction of the state, a ket, with a basis bra:

$\psi(x) = \langle x|\psi \rangle$

So, to be clear:

• the "column vector" ket is $|\psi\rangle$
• the "row vector" bra is $\langle\psi|$

These are abstract objects, they are not numbers.

• the "component" of $|\psi\rangle$ "in the x direction" is $\psi(x)$.

This is a (complex) number.