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ECE 3060: Introduction to Quantum and Statistical Mechanics
Handout 3
Operators, Expectation Values, Eigenvalues and Eigenfunctions
Further Reading:
Senturia, Chaps. 5 and 6 (pp. 65110).
3.1
Physical quantities as operators in quantum mechanics
We have learned the fundamental postulates in quantum mechanics in the last Chapter.
These postulates, though reasonable from the waveparticle duality and asymptotically mapped
into the classical conservation laws of momentum and energy, cannot be proven directly.
However, we will just accept the postulates now as is, and complete the mathematical formalism
of quantum mechanics. Up to now, the observable operators we know include position
x
, time
t
,
momentum
x
i
p
∂
∂
−
=
h
ˆ
, total energy
t
i
E
∂
∂
=
h
ˆ
, the kinetic energy
2
2
2
2
2
ˆ
ˆ
∇
−
=
=
m
m
p
K
h
, the
potential energy
V(x)
, and the Hamiltonian
)
(
2
ˆ
2
2
2
x
V
x
m
H
+
∂
∂
−
=
h
.
There are other operators we
will introduce later in the semester, some are observables such as angular momentum, but some
are nonobservables such as damping.
If we have confusion whether we are denoting a number or an operator, we often (but not
always) will add a head symbol for the operator.
For example,
⋅
=
x
x
ˆ
will read as “the position
operator
x
ˆ is defined as the variable
x
times the operand”.
Please also notice that when we write
down the operator equations, say,
ψ
E
ψ
H
ˆ
ˆ
=
, we do NOT imply that
E
H
ˆ
ˆ
=
at all, but just that
the results AFTER the operation on
ψ
is equal.
For the Schrodinger equation of
ψ
E
ψ
H
ˆ
ˆ
=
, we
can see that for
V(x)=0, if
is a plane wave function of
e
i(kx 
ω
t)
and,
m
k
k
ω
2
/
)
(
2
h
=
then the
equality
ψ
E
ψ
H
ˆ
ˆ
=
is true.
The extension can be made to the eigenvalue equation where one of the operators is
simply a constant
c
.
If for a specific operator
Q
ˆ
, so that
ψ
a
ψ
Q
=
ˆ
, we will call
a
is the
eigenvalue with the corresponding eigenfunction
.
You have seen the importance of the
eigenvalue and eigenfunction in the basic postulates in the previous Chapter.
(Exercise 3.1)
When we state
ψ
p
ψ
p
=
ˆ
with
x
i
p
∂
∂
−
=
/
ˆ
h
, what do we mean?
(Exercise 3.2)
What is the difference when we state
ψ
E
ψ
H
ˆ
ˆ
=
and
ψ
E
ψ
H
=
ˆ
, when
E
ˆ
is an
operator defined by
t
i
E
∂
∂
=
h
ˆ
and
E
is a real number?
3.2
Duality of operators in the real and reciprocal space
Before we start more formal introductions on the operators in quantum mechanics, we
will first clear up the concepts for operators in the real space
x
and reciprocal space
k
.
This will
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be convenient later, since some operators are easier to calculate in one of the spaces than the
other.
We will use the momentum operators as an example.
For a wave packet with
ψ
(x,t)
expressed as the superposition of the planewave function for
V(x) = 0
:
∫
∞
∞
−
−
=
π
dk
e
k
A
t
x
ψ
t
ω
kx
i
2
)
(
)
,
(
)
(
(
3
.
1
)
where
A(k)
is the Fourier transform of the wave function at
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 '05
 TANG
 wave function, Hermitian, Sir William Rowan Hamilton

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