25 3. Operators, Eigenfunctions, Eigenvalues What is an operator? Define: Operator  is a rule that transforms a function f(x) of a given function space into another well-defined function g(x) of the same function space. Example: Square integrable functions: If |f(x)| 2 dx exists, then |g(x)| 2 dx = |Âf(x)| 2 dx also exists. Note: 1. An operator always influences functions written to its right hand side. 2. Multiplication of an operator with a constant: (aÂ)f a(Âf) (a C) 3.  is called a linear operator if  (c 1 f 1 + c 2 f 2 ) = c 1 (Âf 1 ) + c 2 (Âf 2 ) (c 1 , c 2 C) 4. Sum of two operators: ( + Bˆ) f Âf + B^f
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3. Operators, Eigenfunctions, Eigenvalues
What is an operator?
Define:
Operator  is a rule that transforms a function f(x) of a given function
space into another well-defined function g(x) of the same function space.
Example:
Square integrable functions:
If |f(x)|2dx exists, then
|g(x)|2dx = |Âf(x)|2dx also exists.
Note:
1. An operator always influences functions written to its right hand side.
2. Multiplication of an operator with a constant:
(aÂ)f a(Âf) (a C)
3. Â is called a linear operator if
 (c1f1 + c2f2) = c1(Âf1) + c2(Âf2) (c1, c2 C)
4. Sum of two operators:
(Â + Bˆ) f Âf + B^f
26
5. Product of an operator:
Pf ÂB^ f Â(B^f)
Important: ÂB^ f B^ Âf !!!!
6. The difference
[Â,B^] ÂB^ - B^Â
is called Commutator. If [Â,B^] = 0, we say that  and B^ commute.
Commutator Examples: Let’s define the operators D^=d/dx and â as the
multiplication with a (a C). Then
[D^,â] = d/dx∙a – a∙d/dx = 0
But: [D^,x] = d/dx x – x d/dx = 1 0
D^ = d/dx is a very important operator!
Recall: In order to satisfy the correspondence principle, there must be
formal parallels between quantum theory and classical theory.
27
Example: Remember Schrödinger equation for a free particle:
QM CM
Exmt
i
2
22
2
p2/2m = T = E
Note:
1. Operator of the kinetic energy (for 1-dim. problems):
m
p
xmT
2
ˆ
2
2
2
22
2. Operator for linear momentum: xi
px
in 3D:
ip̂
28
Define:
If  is an operator and  = a (same on both sides!)
we call “eigenfunction” of Â
we call a “eigenvalue” of with respect to Â
Examples:
a. consider a 1-dim. free particle, and assume
/
2
1)( ipxex
Linear momentum:
)(2
1
2
1)(ˆ // xpepe
xixp ipxipx
x
(x) is an eigenfunction of px^ with the eigenvalue p
b. Particle in a potential V(x,t)
Classical physics: kin. E. + pot. E. = Etotal
p2/2m + V = Etotal
Schrödinger equation:
),(),(),(),(2
),(2
22
txEtxtxVtxxm
txt
i
With
29
2
22
2ˆ
xmT
kinetic energy
V^ = V(x,t) potential
Hˆ = T^ + V^
),(),(ˆ),( txEtxHtxt
i
H^ is called Hamilton operator or Hamiltonian, in analogy to the Hamiltonian
function (Sir Rowan Hamilton, 1833) in classical theoretical mechanics.
For a free particle: V(x,t) = const. = V0
0
2//
2
22
22
1),(
2
1
2ˆ V
m
petxVe
xmH ipxipx
is eigenfunction of H^ with eigenvalue (p2/2m + V0)
Note:
1. Every measurable physical property of a system is described by an
operator acting in the state space of that system. Operating on a wave
function is the QM mechanism for measurement:
H^ = (Energy)
2. A physically measurable property in a QM system is called an observable
30
3. If the wave function is an eigenfunction of Â, measurement yields the
eigenvalue A
Are and in the relation  = unique w/ respect to each other?
Each eigenfunction has only one eigenvalue
If there are two eigenfunctions of an operator  that have the same
eigenvalue, the eigenvalue is called degenerate
------------------------ End Lecture 4 (09/03/14) --------------------------
31
Last hour:
Operators are rules transforming a function f(x) of a given function
space into another well-defined function g(x) of the same function space.
(aÂ)f a(Âf) (a C)
Linear operators: Â (c1f1 + c2f2) = c1(Âf1) + c2(Âf2) (c1, c2 C
(Â + Bˆ) f Âf + B^f
ÂB^ f Â(B^f) In general ÂB^ f B^ Âf
Commutator: [Â,B^] ÂB^ - B^Â
Operator of kinetic energy: m
p
xmT
2
ˆ
2
2
2
22
or m
p
mT
2
ˆ
2
22
Operator of linear momentum: xi
px
or
ip̂
If  = a we call “eigenfunction” of Â, “eigenvalue” of with
respect to Â
Example: plane waves are EFs of px^
Hamiltonian: )(2
ˆ2
ˆˆˆˆ2
222
xVxm
Vm
pVTH
Every measurable physical property of a system (=”observable”) is
described by an operator acting in the state space of that system.
32
Operating on a wave function is the QM mechanism for measurement
33
Measurements influence the wave function:
measure first p, then x measure first x, then p:
)(')()()(ˆˆ
)(')()(ˆˆ
xxi
xi
xxxi
xxp
xi
xxxi
xxpx
There is no set of wave functions that can be eigenfunctions of both x^
and p^. This is a consequence of [x^,p^] = iħ 0
Remember HUP: We cannot measure x,p at the same time with infinite
accuracy. In the language of Q.M.: Measurements of observables
corresponding to non-commuting operators interfere with one another
[x^,p^] = iħ 0, but [p^, T^] = 0, [x^,pz] = 0, etc.
If [Â,B^] = 0 and is eigenfunction of Â, B^ is also eigenfunction of Â.
Eigenfunctions of H^:
Stationary States (requires time-independent H^)
Consider time-dependent S.E.
iħ (x,t)/t = H^ (x,t)
Trick: look for special solutions with (x,t) = (x)∙f(t)
34
Separation of variables (x,t) = (x)∙f(t), insert into T.D.S.E.
)()()()()()()(
2)()(
2
22
tfxEtfxxVx
tfx
mtfx
ti
divide by (x,t)
ExVx
x
mtf
tfi )(
)(
)("
2)(
)( 2
E = indep. of x and t = const.
Consequences:
i) From left hand side: f(t) = e-iEt/ħ
|(x,t)|2 = |(x)|2 time independent probabilities
ii) From right hand side: H^(x) = E (x) time independent S.E.
stationary states (x)
Learning goals slide
35
4. The Dirac Delta Function and Fourier Transforms:
The Dirac Delta Function: Slide Dirac
The delta function (r – r0) is a strange but useful function:
In one dimension:
(x-x0) = for x = x0
(x-x0) = 0 for x x0
(x-x0) has the dimension of a length-1.
1)(
)()()(
0
00
dxxx
xfdxxxxf
(x)
xx0
Analogous in 3 dimensions:
(r – r0) = for r = r0
(r – r0) = 0 for r r0
36
(r-r0) has the dimension of a volume-1 !
1)(
)()()(
30
03
0
rdrr
rfrdrrrf
Clicker question: How can we construct a function?
functions come in many shapes and colors:
QMin usedmost 2
1
otherwise 0 a/2,x a/2-for 1
lim
functionr rectangula)/sin(
lim
ondistributi normal a oflimit 1
lim
/)'(
0
0
/
0
22
xxip
a
a
ax
a
edp
a
x
ax
ea
Why is the function useful in QM ?
Point charge q at r0: (r) = q (r – r0)
Localization of a particle using infinitely many plane waves with varying
momentum (i.e. p = )
More uses later
37
Fourier Transforms: Slide Fourier
For a function f(x), we call
ikxexfdxxfkF )(2
1)]([)(
the Fourier transform of f(x).
The inverse procedure (inverse Fourier transform) gives us
ikxekFdkkFxf )(2
1)]([)( 1
(note the + sign in the exp !)
------------------------- End Lecture 5 (09/02/2015) ---------------------
38
Last hour:
Non-commuting operators cannot have the same eigenfunctions
Eigenfunctions of H^ have time-independent probabilities
Solutions of TISE are “stationary states”
The time-dependence of any WF is given by the TDSE
The Dirac Delta Function: In one dimension:
)()()( 00 xfdxxxxf
[(x-x0)] = m-1
Analogous in 3 dimensions:
)()()( 03
0 rfrdrrrf ; (r-r0) has the dimension of a volume-1 !