Testing Hermiticity: How to Prove it?

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In summary, to determine if an operator is Hermitian, it must be equal to its complex conjugate. This applies even if the operator is not in matrix form, such as in the case of L_x, the x component of angular momentum. In this case, we can find the complex conjugate by using the differential operator d/dx, which is anti-Hermitian. Additionally, if two operators are Hermitian and commute, then their product is also Hermitian. The concept of complex conjugates and conjugate imaginaries is also important to keep in mind when working with bra and ket vectors. Overall, this information can be found in P. Dirac's book on quantum mechanics.
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Lorna
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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
 
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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>
 
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  • #3
That was very helpful, thanks a LOT!
 

1. What is hermiticity and why is it important in testing?

Hermiticity is a property in quantum mechanics that describes the symmetry of a system. It is important in testing because it ensures that the results of a measurement are accurate and consistent.

2. How do I know if an operator is hermitian?

An operator is hermitian if it is equal to its adjoint, which is the complex conjugate of the operator. In other words, the operator and its adjoint have the same matrix representation.

3. What are the steps in testing hermiticity?

The first step is to determine the operator's adjoint by taking the complex conjugate of its matrix representation. Then, compare the original operator to its adjoint to see if they are equal. If they are equal, the operator is hermitian. Additionally, you can use the eigenvalue equation to check if all eigenvalues are real numbers, as hermitian operators have only real eigenvalues.

4. Can a non-hermitian operator still produce valid results?

Yes, a non-hermitian operator can still produce valid results, but it may not accurately reflect the physical reality of the system. Hermiticity ensures that the results of a measurement are consistent, so a non-hermitian operator may lead to inconsistent or incorrect results.

5. How is hermiticity related to the conservation of probability?

Hermiticity is directly related to the conservation of probability in quantum mechanics. This is because hermitian operators have real eigenvalues, which correspond to the probabilities of different outcomes in a measurement. If an operator is not hermitian, the eigenvalues may be complex numbers, which do not represent probabilities and violate the conservation of probability.

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