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r4 - 22 Jul 2007 - 07:07:51 - TuongHuynhYou are here: TWiki >  Main Web > LectureNotes > UndergraduatePChem > Chapter4

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

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

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

\[\hat A \left[ f(x) + g(x)\right ] = \hat A f(x) + \hat A g(x)\]

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

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

\[ \hat C \left[ f(x) \right ] = \hat A \hat B f(x) = \hat A \left[\hat B f(x) \right ] \]

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

\[ \left[\hat A, \hat B \right] = \hat A \hat B - \hat B \hat A\]

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.

\[\hat A { f}_{ i} = { a}_{ i} { f}_{ i}\]

\[ \left ( Operator\right ) \left ( function\right ) = Constant \left ( the \: same \: function\right ) \]

{${ a}_{ i}$} 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.

\[\hat A f = a f \; \; \; \; and \; \; \; \; \hat A g = bg\]
where $a \neq b$ then

\[ \int { f}^{ *}g d \tau = \int { g}^{ *} f d \tau = 0 \]

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

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

 

-- ThanhTruong - 12 Jul 2007

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