In Sakurai's Modern Physics, the author says, "... consider an outer product acting on a ket: (1.2.32). Because of the associative axiom, we can regard this equally well as as (1.2.33), where [itex]\left<\alpha|\gamma\right>[/itex] is just a number. Thus the outer product acting on a ket is just another ket; in other words, [itex]\left|\beta\right>\left<|\alpha\right|[/itex] can be regarded as an operator. Because (1.2.32) and (1.2.33) are equal, we may as well omit the dots and let [itex]\left|\beta\right>\left<\alpha|\gamma\right>[/itex] standing for the operator [itex]\left|\beta\right>\left<\alpha\right|[/itex] acting on [itex]\left|\gamma\right>[/itex] or, equivalently, the number [itex]\left<\alpha|\gamma\right>[/itex] multiplying [itex]\left|\beta\right>[/itex]. (On the other hand, if (1.2.33) is written as [itex]\left(\left<\alpha|\gamma\right>\right)\cdot\left|\beta\right>[/itex], we cannot afford to omit the dot and brackets because the resulting expression would look illegal.) Notice that the operator [itex]\left|\beta\right>\left<\alpha\right|[/itex] rotates [itex]\left|\gamma\right>[/itex] into the direction [itex]\left|\beta\right>[/itex] . It is easy to see that if (1.2.34) then (1.2.35), which is left as an exercise.
The Attempt at a Solution
I know that the definition of an adjoint involves taking the complex conjugate of the tranpose of a complex-vallued quantity. I can't just turn all the bras into kets and all the kets into bras, because then I end up with an inner product [itex]\left<\alpha|\beta\right>[/itex], which isn't right since the outer product is an operator (a matrix). What am I missing? Thanks!
Edit: I have noticed that this may be relevant. The dual correspondence principle.
If I start with all the [itex]\beta[/itex] coefficients being zero, I should just get that the [itex]\alpha[/itex] ket has a corresponding bra. Can I then "multiply" both sides of the equation with [itex]\left<\beta\right|[/itex]? I guess I can't because then I end up with the same issue. I would get an inner product.