Hermiticity and expectation value

In summary, the conversation discusses alternative ways of checking for hermicity and the property (AB)^\dagger=B^\dagger A^\dagger is mentioned as a valid method. The context of bounded and unbounded operators is also brought up, along with the need for a rigorous proof of the property discussed.
  • #1
holden
30
0
is there a better way to check for hermicity than doing expecation values? for example, what if you had xp (operators) - px (operators), or pxp (operators again); how can you tell if these combos are hermetian or not, without going through the clumsy integration (that doesn't give a solid result, as far as i can tell)? thanks.
 
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  • #2
[tex](AB+CD)^\dagger = B^\dagger A^\dagger + D^\dagger C^\dagger[/tex]
 
  • #3
heh, thanks. i suck.
 
  • #4
although, actually, i still have a question.. where does that come from, is it just an accepted property? i can't seem to derive it or find a place where it has been derived.
 
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  • #5
[tex]\left< (AB)x,y \right> = \left< A(Bx),y \right> = \left< Bx, A^\dagger y \right> = \left< x, (B^\dagger A^\dagger) y \right>[/tex]

By definition, this shows that [tex](AB)^\dagger=B^\dagger A^\dagger[/tex].
 
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  • #6
What Euclid wrote is valid only for bounded operators. The "x" and "p_x" operators are unbounded, so care is needed in order not to write some garbage.

Daniel.
 
  • #7
However, in the context of what holden wrote, what Euclid stated is valid enough, because holden wants to check for hermicity and hermitian operators are symmetric and bounded.

On the other note, I want to see a somewhat rigorous proof that
[tex]\left(A+B\right)^{\dagger} = A^{\dagger}+B^{\dagger} [/tex]

All the sources say this property, but they don't bother proving it.
 

What does it mean for an operator to be Hermitian?

For an operator to be Hermitian, it means that it is equal to its own adjoint. In other words, the operator is equal to the complex conjugate of its transpose. This property is important in quantum mechanics because it ensures that the eigenvalues of the operator are real numbers and the corresponding eigenvectors are orthogonal.

What is the significance of Hermiticity in quantum mechanics?

Hermiticity is a fundamental property in quantum mechanics because it guarantees that the observables in a quantum system have real-valued eigenvalues. This allows for the measurement of physical quantities, such as energy or position, to have well-defined values. Additionally, Hermiticity ensures that the system's wavefunction remains normalized over time.

How is the expectation value related to Hermitian operators?

The expectation value of a Hermitian operator is the average value of the observable in a given quantum state. This value is calculated by taking the inner product of the state vector with the operator applied to the state vector. The resulting value is always a real number, which is the expectation value of the observable in that state.

Can a non-Hermitian operator have a real expectation value?

No, a non-Hermitian operator cannot have a real expectation value. This is because the expectation value is calculated using the inner product, which involves taking the complex conjugate of the operator. If the operator is not Hermitian, then the inner product will result in a complex number, and the expectation value will also be complex.

How can one determine if an operator is Hermitian?

To determine if an operator is Hermitian, one can check if it is equal to its own adjoint. This can be done by taking the transpose of the operator and then taking the complex conjugate of each element in the matrix. If the resulting matrix is equal to the original operator, then it is Hermitian. Another way is to check if all of its eigenvalues are real numbers. If so, then the operator is Hermitian.

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