Hermitian Adjoint | Hermitian and Skew-Hermition

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In quantum mechanics, the Hermitian adjoint (adjoint or conjugate transpose) is a mathematical operation. It involves taking the transpose of a Wave function and then taking the complex conjugate of each entry.

The Hermitian adjoint is used to find the adjoint of operators and wavefunctions. The adjoint of an operator is denoted by a dagger symbol (†), while the adjoint of a wavefunction is denoted by an asterisk (*).

For an operator A, its adjoint A† can be obtained by taking the complex conjugate of each entry in A and then taking its transpose. This means that if A has elements a_ij, then the elements in A† are the complex conjugates of a_ij, but arranged in the reverse order of indices.

Mathematically, if A = [a_ij], then A† = [(a_ij)*]T, where (a_ij)* represents the complex conjugate of a_ij and T denotes the transpose operation.

For wavefunction ψ, its adjoint ψ† can be obtained by taking the complex conjugate of each coefficient in ψ. Mathematically, if ψ = [c_i], then ψ† = [(c_i)*].

 

It allows to define Hermitian operators. 

Hermitian Operator:

An operator A is said to be Hermitian if A† = A, which means that the adjoint of A is equal to A itself. Hermitian operators have real eigenvalues and their corresponding eigenvectors form an orthogonal basis.

If A† = - A, the Operator is skew-hermitian.

 

Inner Product and Norm of a Wavefunction:

The adjoint of a wavefunction is used to define the inner product and the norm of a wavefunction. The inner product of two wavefunctions ψ and φ is given by ⟨φ|ψ⟩, where ⟨φ| denotes the adjoint of φ. The norm of a wavefunction ψ is given by ||ψ|| = √⟨ψ|ψ⟩.