Understanding the Adjoint Operator in Physics

[kuan]

Dear All,

I am right now facing a problem of the adjoint operator.

From the mathematical point of view, if [tex] T [/tex] is a linear operator mapping from Hilbert space [tex] H [/tex] to another Hilbert space [tex] H' [/tex], there exist an unique adjoint operator [tex] T^{\dagger} [/tex] mapping from [tex] H' [/tex] back to [tex] H [/tex].

However I am right now struggling in understanding the physical meaning of the adjoint operator (not necessarily self-adjoint). The only thing I can think is that the operator [tex] T [/tex] and [tex] T^{\dagger}[/tex] have the same eigenvalues. Does this reveals some kind of symmetry? Could some one give me a bit hint for that?

Many thanks

 

[Mechzero]

For anyone in the future looking at this question, here's the main gist of the definition of an adjoint operator.
Assuming an orthonormal set of bra and ket states, if A is a an operator, then it will have the function

A = [itex]\sum[/itex]_i |b_i> <a_i|
so that when we operate on some state |a_j>, we get the corresponding |b_j>
A|a_j> = |b_j>
This is by definition of orthonormality, that <a_n|a_n>=1 and <a_n|a_m>=0 (same for b).

Meanwhile, for the very same set of states A^† is the exact definition, with the a and b reversed.
A^† = [itex]\sum[/itex]_i |a_i> <b_i|


As far as physical interpretation, well, we know that these bras and kets inform us of states of a system. When we observe the system (such as with the momentum operator -i[itex]\hbar[/itex][itex]\frac{d}{dx}[/itex]), we may find particular eigenvalues and corresponding eigenfunction that serve as a basis of probable outcomes within the set of states. The adjoint operator serves the purpose of assisting in the mathematical representation of the scenario, giving complex conjugates counterparts to state constants.

I'm still a bit in the beginning of a grad course in Quantum I, if anyone can add this, by all means go for it. If I acquire more relevant info from the course/elsewhere later on I will share.