Download 4 Operators

4
Operators
The time-independent Schrödinger equation is given by
!
"
!2 2
−
∇ + V (r) ψ(r) = Ĥψ(r) = Eψ(r)
2m
(140)
The terms in the brackets are called the Hamiltonian and in quantum mechanics is an operator. An operator transform one function into another
function. One example is the derivative operator
D̂f (x) = f ! (x)
(141)
We now consider some rules of operators. The sum, differences and product
of two operators  and B̂ is given by
(Â + B̂)f (x) = Âf (x) + B̂f (x)
(Â − B̂)f (x) = Âf (x) − B̂f (x)
ÂB̂f (x) = Â[B̂f (x)]
(142)
(143)
(144)
and obey the associate law of multiplication
Â(B̂ Ĉ) = (ÂB̂)Ĉ
(145)
Lets consider an example where  = d/dx and B̂ = x̂. First we evaluate ÂB̂
ÂB̂ =
d
[xf (x)] = f (x) + xf ! (x) = (1 + x̂D̂)f (x)
dx
(146)
second lets evaluate B̂ Â
B̂ Âf (x) = x̂[
d
f (x)] = xf ! (x)
dx
(147)
thus we see that for operators ÂB̂ is not always B̂ Â. It is convenient to
define the commutator of two operators as
[Â, B̂] = ÂB̂ − B̂ Â
(148)
if ÂB̂ = B̂ Â then the commutator is zero and the two operators are said to
commute. Examples
"
!
d
d
d
=3 − 3=0
(149)
3,
dx
dx dx
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and
!
"
d
, x = D̂x − xD̂ = 1
dx
(150)
Lets consider the commutator between the position operator x and the momentum operator p̂ − −i!d/dx
[p̂, x] = p̂x − xp̂ = −i!(
d
d
x − x ) = −i!
dx
dx
(151)
The square of an operator is defined as
Â2 = ÂÂ
(152)
and the exponential of an operator as
exp(Â) = 1 + Â +
4.1
Â2 Â3
+
+···
2!
3!
(153)
Linear operators
An important class of operators are linear operators. An operator  is said
to be linear if and only if it has the following two properties
Â[f (x) + g(x)] = Âf (x) + Âg(x)
(154)
Â[cf (x)] = cÂf (x)
(155)
and
where c is a constant and f (x) and g(x) are functions. We’ll consider two
examples first is the D̂ and the second is Â2 = ()2 . For the differential
operator we see that
(d/dx)[f (x) + g(x)] = (d/dx)f (x) + (d/dx)g(x)
(d/dx)[cf (x)] = c(d/dx)f (x)
(156)
(157)
and therefore the operator is linear. What about the square operator
Â2 [f (x) + g(x)] = (f (x) + g(x))2 #= Â2 f (x) + Â2 g(x)
(158)
and, thus, is nonlinear. It turns out that almost every operators in quantum
mechanics are linear operators.
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4.2
Eigenfunctions and eigenvalues
An eigenfunction of an operator  is a function which when the operator
works on it return the functions times a constant
Âf (x) = af (x)
(159)
where a is called the eigenvalue. As an example, what would be an eigenfunction of the differentiation operator? Either cosine, sine or exponentials
(d/dx)(f x) = (d/dx) exp(−ax) = −a exp(ax) = −af (x)
(160)
Another example would be the Schrödinger equations for a particle in a 1D
box
d2 f (x) 2m
+ 2 Ex f (x) = 0
(161)
dx2
!
which we have already explored.
4.3
Operators in quantum mechanics
In quantum mechanics, physical observables (e.g., energy, momentum, position, etc.) are represented mathematically by operators. For instance, the
operator corresponding to energy is the Hamiltonian operator
!
"
!2 2
−
∇ + V (r) ψ(r) = Ĥψ(r) = Eψ(r)
(162)
2m
The operators are found be writing down the classical expression for the
property of interest and then substitute
x → x̂ = x ·
∂
px →= p̂ = −i!
∂x
(163)
(164)
The classical expression for the Hamiltonian of a system (which is just a
reformulation of Newton’s equations) is
H =T +V =
px2
+ V (x)
2m
(165)
using the substitution we get
Ĥ = T̂ + V̂ = −
!2 d 2
+ V̂ (x)
2m dx2
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(166)
and as we see that all these operators are linear. The properties of a system
is related to the eigenvalues of the operator
Ĥψi = Ei ψi
(167)
where the energy Ei is the eigenvalues of the Hamiltonian. Quantum mechanics postulates that a measurement of a property A most give one of the
eigenvalues of the operator Â. As an example consider a stationary state
Ψ(x, t) = exp(−iEt/!)ψ(x)
(168)
operating on this wavefunction with the Hamiltonian gives
ĤΨ(x, t) = exp(−iEt/!)Ĥ = exp(−iEt/!)E exp(−iEt/!) = EΨ(x, t)
(169)
and, thus, stationary states are eigenfunctions of the Hamiltonian.
4.4
Degeneray
If we have n independent wavefuctions each having the same eigenvalue a
Ĥψi = aψi i = 1, 2, 3, · · ·
(170)
The every linear combination of these wavefunctions
#
φ=
ci ψi
(171)
i
will also be an eigenfunction of the operator with the same eigenvalues since
#
#
#
#
Ĥφ = Ĥ
ci ψi =
ci Ĥψi =
ci aψi = a
ci ψi = aφ
(172)
i
4.5
i
i
i
Expectation values of an operator
In quantum mechanics the average value (or expectation value) of an property
is given by
$
ψ ∗ (r)Âφ(r)dr
(173)
if the wave function is normalized otherwise
% ∗
ψ (r)Âφ(r)dr
%A& = % ∗
ψ (r)φ(r)dr
(174)
%A& =
24
Thus is a system is in an eigenstate of the operator Â, i.e.
Âψ = aψ
(175)
where ψ is assumed normalized, then the expectation value is given by
$
%A& =
ψ ∗ (r)Âφ(r)dr
(176)
$
=
ψ ∗ (r)aφ(r)dr
(177)
$
= a ψ ∗ (r)φ(r)dr
(178)
= a
(179)
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