Hermitian conjugate of operators

In summary, we have found the hermitian conjugates for the given operators A and B, including the special case of A^\dagger A. These results can be used to simplify expressions involving A and B in quantum mechanics and other mathematical contexts.
  • #1
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Homework Statement



Find the hermitian conjugates, where A and B are operators.

a.) AB-BA
b.) AB+BA
c.) i(AB+BA)
d.) [itex] A^\dagger A [/itex]


Homework Equations



[itex] (AB)^\dagger =B^\dagger A^\dagger [/itex]



The Attempt at a Solution



Are they correct and can I simplify them more?

a.) [itex] (AB-BA)^\dagger = (AB)^\dagger - (BA)^\dagger= B^\dagger A^\dagger - A^\dagger B^\dagger = BA-AB = -(AB-BA) [/itex]

b.) [itex] (AB+BA)^\dagger = (AB)^\dagger + (BA)^\dagger= B^\dagger A^\dagger + A^\dagger B^\dagger = BA+AB [/itex]

c.) [itex] [i(AB+BA)]^\dagger = -i[(AB)^\dagger + (BA)^\dagger]= -i(B^\dagger A^\dagger + A^\dagger B^\dagger) = BA+AB [/itex]

d.) [itex] [A^\dagger A]^\dagger = A^\dagger (A^\dagger)^\dagger=A^\dagger A [/itex]
 
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  • #3
For point C.) [itex] -i(B^\dagger A^\dagger + A^\dagger B^\dagger) [/itex]
 
  • #4
Alright. The last equality is not true. That's what I meant.
 
  • #5
Is it ok until [itex] -i(B^\dagger A^\dagger + A^\dagger B^\dagger) [/itex]?
 
  • #6
Yes, it is.
 

1. What is the definition of the Hermitian conjugate of an operator?

The Hermitian conjugate of an operator A is denoted as A† and is defined as the transpose of the complex conjugate of A. In other words, the Hermitian conjugate of an operator is obtained by taking the complex conjugate of each element in the operator and then transposing the resulting matrix.

2. What is the significance of the Hermitian conjugate in quantum mechanics?

The Hermitian conjugate plays a crucial role in quantum mechanics as it allows us to define and calculate important quantities such as expectation values and probabilities. It also helps us determine if an operator is self-adjoint, which is necessary for the operator to have real eigenvalues.

3. How is the Hermitian conjugate related to the adjoint of an operator?

The Hermitian conjugate and the adjoint of an operator are closely related but are not exactly the same. The adjoint of an operator is obtained by taking the complex conjugate of each element in the operator and then transposing the resulting matrix. However, the adjoint only applies to operators that act on finite-dimensional vector spaces, while the Hermitian conjugate applies to operators on infinite-dimensional vector spaces.

4. Can the Hermitian conjugate of an operator be negative?

Yes, the Hermitian conjugate of an operator can be negative. This is because taking the complex conjugate of each element in the operator does not change the sign of the elements. However, the Hermitian conjugate of a self-adjoint operator will always be equal to the original operator, and therefore will not be negative.

5. How is the Hermitian conjugate of an operator used in the measurement process in quantum mechanics?

In quantum mechanics, the Hermitian conjugate of an operator is used to calculate the expectation value of an observable in a quantum state. The magnitude squared of the expectation value gives us the probability of measuring a particular value when the observable is measured in that state. This is known as the Born rule and is a fundamental concept in quantum mechanics.

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