@@ -231,8 +231,7 @@ Having demonstrated that we can represent state vectors and operators in term of
-*Equality*: two operators are said to be equal if their corresponding operator matrix elements are equal, for instance $\hat A = \hat B$ if $A_{ij}=B_{ij}$ for all possible values of $i$ and $j$.
-*Identity operator*: the unit (or identity) operator is represented by $\hat1$ and satisfies $\hat1|\psi\rangle=|\psi\rangle$ for all possible state vectors $|\psi\rangle$. The diagonal elements of the matrix representation of the identity operator $\hat1$ are all unity while the off-diagonal elements vanish: $\hat1_{ij}=\delta_{ij}$. This means that for a $n$-dimensional Hilbert space the unit operator is the $n$-dimensional identity matrix. Note that the unit operator has the same form in all representations irrespective of the specific
choice of basis states.
-*Identity operator*: the unit (or identity) operator is represented by $\hat1$ and satisfies $\hat1|\psi\rangle=|\psi\rangle$ for all possible state vectors $|\psi\rangle$. The diagonal elements of the matrix representation of the identity operator $\hat1$ are all unity while the off-diagonal elements vanish: $\hat1_{ij}=\delta_{ij}$. This means that for a $n$-dimensional Hilbert space the unit operator is the $n$-dimensional identity matrix. Note that the unit operator has the same form in all representations irrespective of the specific choice of basis states.
-*Null operator*: the zero operator $\hat0$ is such as $\hat{0}|\psi\rangle=0$ for all $\psi$. Its matrix elements are all zero, $\hat0_{ij}=0$.
which as you might recall is nothing but the standard rule for matrix multiplication. So once we express operators in their matrix representation, we can multiply them by following standard matrix multiplication.
-*Commutator*: in the same way as matrix multiplication is not commutative, also operator multiplication is {\it not commutative}:
-*Commutator*: in the same way as matrix multiplication is not commutative, also operator multiplication is *not commutative*:
$$
\hat{A}\hat{B}\;\neq\hat{B}\hat{A}
$$
...
...
@@ -269,13 +268,12 @@ As discussed above, Hermitian operators play a central role in the physical int
We can now provide the explicit expression of the Hermitian operators in the matrix representation.
Let us assume that we have an operator $\hat{A}$ in an $n$-dimensional Hilbert space
with matrix elements $A_{ij}$ defined with respect to a set of orthonormal basis states, $\{ |psi_i\rangle; i=1,2,\ldots\,n\}$.
with matrix elements $A_{ij}$ defined with respect to a set of orthonormal basis states, $\{ |\phi_i\rangle; i=1,2,\ldots\,n\}$.
From its matrix representation one can construct a new operator by taking the transpose and complex conjugate of the original matrix, that is:
This new matrix corresponds to the matrix representation of a new operator that will be denoted by $\hat{A}^\dagger$. This new operator is called the *Hermitian adjoint* of
the operator $\hat{A}$.
This new matrix corresponds to the matrix representation of a new operator that will be denoted by $\hat{A}^\dagger$. This new operator is called the *Hermitian adjoint* of the operator $\hat{A}$.
Hermitian operators are such that they are *the same* as their adjoint version.
That is, at the operator level we can write that Hermitian operators satisfy $\hat{A}=\hat{A}^\dagger$.
Any operator which satisfies this condition will be an Hermitian operator. Note that the Hermiticity of an operator is a property which is independent of the specific choice of basis.
