Operators in Quantum Mechanics

Operators:

An operator A can be considered a rule that transforms one function f(x) into a new function Af(x).

An operator in a vector space is a mapping between two vectors in that space. More precisely, this mapping is a transformation that takes a vector as an input and produces another vector as the output.

Example: If an operator A acts on vector v₁ in a vector space to transform it into another vector v₂ in the same space, this is written as:

A |v₁ > |v₂ >

A measurable property of a physical system is denoted as observable in quantum mechanics. An operator is often used to represent an observable.

Rules for Operators:

There are certain rules for working with operators that are collectively known as algebra for operators:

(A + B) | φ> = A|φ> + B|φ>
(AB)|ψ> = A{B|ψ>]

Linear Operator:

An operator can transform a vector v_i in a vector space V to another vector v_j that may belong to a vector space other than V. In general, the operator could be of any form. But in quantum mechanics, only a class of operators known as linear operators are of particular interest. If operator A acts on vector v_i in the vector space V to transform it into another vector v₂ also in the vector space V, then operator A is referred to as a linear operator.

A | v₁ >=|v₂ >; for all v₁, v₂ ε V

A linear operator A has the following properties:

1. A(u + v) = Au + Av, for all u, v ∈ V
2. A(cu) = cAu, for all c ∈ F and u ∈ V

Commutator:

It should be noted that the order in which two operators C and D are multiplied is important because in general:

CD ≠ DC

Identity Operator:

The identity operator written I, is a matrix in which all items on the main diagonal are ones and all remaining items are zeroes. Thus an identity matrix can be specified as:

I_ii = 1, and
I_ij = 0 if i ≠ j

Adjoint Operator:

One of the most important operations in complex linear algebra is the Hermitian conjugate or adjoint of a linear operator. The adjoint of an operator A denoted as A⁺ is denoted by taking the complex conjugate of all of its entries and then interchanging the rows and columns.

Hermitian Operator:

A class of operators of special importance is the Hermitian operators because the eigenvalues of a Hermitian operator are the possible values of the observables. This implies that measured values are real numbers, not complex numbers.

In quantum mechanics, operators that are equal to their adjoints are called Hermitian or self-adjoint. In other words, an operator A is Hermitian if and only if satisfies the following equation:

A⁺ = A

Unitary Operator:

An operator represented by an n x n matrix A is unitary if it produces an identity matrix multiplied by its conjugate transpose:

AA⁺ = I

In other words, A is a unitary matrix if its conjugate transpose is equal to its inverse:

A⁺ = A^-1

Properties of Unitary Operator:

1. A unitary operator preserves the linear product of any two vectors u and v in the vector space V.

2. The operator is invertible. Notice that the matrix corresponding to the operator is invertible if and only if the matrix isn’t singular. The determinant of a singular matrix is zero.

3. The eigenfunctions are orthonormal and they are complete.

4. Eigenvalues need not be real.