Operators

Definition:  A mathematical operator is a symbol which stands for a mathematical operation that can be applied to a function.

For example, we have defined the Hamiltonian operator, %$\hat H$% as

%BEGINLATEX% \hat H = - \frac{ { \hbar }^{ 2} }{ 2m} { \nabla }^{ 2} + V}% %ENDLATEX%

which can be applied to the wavefunction %$\psi$% to obtain the energy value, i.e. %$\hat H \psi = E \psi$%.

Properties of an operator

Linear:   An operator is linear if

%MATHMODE{\hat A \left[ f(x) + g(x)\right ] = \hat A f(x) + \hat A g(x)}%

%MATHMODE{\hat A \left[ cf(x) \right ] = c\hat A f(x) }% where c is a constant.

Product of two operators, %$\hat C = \hat A \hat B$% is defined as

%MATHMODE{ \hat C \left[ f(x) \right ] = \hat A \hat B f(x) = \hat A \left[\hat B f(x) \right ] }%

Associative : Operator mulitiplcation is associative, i.e. %$\hat A \hat B \hat C = \hat A (\hat B \hat C) = (\hat A \hat B) \hat C$%.

Commute:  Two operators %$\hat A$% and %$\hat B$% are commute if and only if %$\hat A \hat B = \hat B \hat A$%.

Commutator of two operators %$\hat A$% and %$\hat B$% denoted by %$\left[\hat A, \hat B \right]$% is defined as

%MATHMODE{ \left[\hat A, \hat B \right] = \hat A \hat B - \hat B \hat A}%

so if %$\left[\hat A, \hat B \right] = 0$% then %$\hat A$% and %$\hat B$% commute.   There are a number of interesting consequences of the commutation property of operators in quantum mechanics.  They will be discussed in a later chapter.

Hermitian operators

In quantum mechanics we need to concern only with one type of operator called Hermitian operators.  Hermitian operators have three important properties:

1. A Hermitian operator %$\hat A$% has its own set of eigenfunctions and real eigenvalues, i.e.

%MATHMODE{\hat A { f}_{ i} = { a}_{ i} { f}_{ i}}%

%MATHMODE{ \left ( Operator\right ) \left ( function\right ) = Constant \left ( the \: same \: function\right ) }%

{%${ a}_{ i}$%} are the eigenvalues of the operator %$\hat A$% and are REAL, sometimes are called observables.

{%${ f}_{ i}$%} are the eigenfunctions of the operator %$\hat A$%.

2. Two eigenfunctions of a Hermitian operator with different eigenvalues are orthogonal to each other, i.e.

%MATHMODE{\hat A f = a f \; \; \; \; and \; \; \; \; \hat A g = bg}% where %$a \neq b$% then

%MATHMODE{ \int { f}^{ *}g d \tau = \int { g}^{ *} f d \tau = 0 }%

3. Eigenfunctions of a Hermitian operator form a complete set. Consequently, any well-behaved function can be exactly expanded in that set, namely

If {%${\psi}_{i} (x)$%} is the complete set of eigenfunctions of the Hermitian operator %$\hat H$%, then any well-behaved function f(x) can be expressed as

%MATHMODE{f(x) = \sum_{ n=1}^{ \infty } { a}_{ n} { \psi }_{ n}}% where {%${a}_{n}$%} are the expansion coefficients.

-- ThanhTruong - 12 Jul 2007