Testing Hermiticity: How to Prove it?

[Lorna]

How do we know that a given operator is Hermitian. I know that for an operator A to be Hermitian then A=A+. But I don't know how to apply this on something which is not in a matrix form. For example I want to know if L_x (x component of angular momentum) is Hermitian and I have no idea how to start. Do I just find the complex conjugate of it because how would I find the T of it? I know that L_x = YP_z - ZP_y = -i*hbar(Yd/dz - Zd/dy)

thanks

 

[omyojj]

L_x consists of y and p_z (by multiplication, y and p_z are real(Hermitian) and commute
If a,b are hermitian and [a,b]=0 then ab is hermitian..)


For p_x = -i*hbar * d/dx, (p_x)+=+i*hbar * -(d/dx) = p_x
(differential operator d/dx is anti-Hermitian, just think of doing integration by parts when you calculate a triple product including d/dx with physically acceptible boundary condition)

y+ = y is trivial..--> they are hermitian..

that is to say..for any vector (ket and bra) Q> and P>

<P|p_x|Q> = (<Q|p_x|P>)* = <P|(p_x)+|Q>
by def

and p_x and y commutes..[p_x, y] = 0

then the required result follows..

Note:

P. Dirac`s book on QM..

We shall use the words 'conjugate complex' to refer to numbers and other complex quantities which can be split up into real and pure imaginary parts,
and the words 'conjugate imaginary' for bra and ket vectors, which cannot..

conjugate imaginary of |P> is <P|
conjugate imaginary of a|P> is <P|a+

complex conjugate of <P|Q> is <Q|P>

 

[Lorna]

That was very helpful, thanks a LOT!