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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 !
Fourier Transforms:
F.T
ikxexfdxxfkF )(2
1)]([)(
Inverse F.T.
ikxekFdkkFxf )(2
1)]([)( 1
(x)
xx0
39
Properties of Fourier transforms:
Linearity: [f(x)+g(x)] = [f(x)] + [g(x)] = F(k) + G(k)
Complex Conjugation: [f*(x)] = F*(-k)
Integration : f1*(x)f2(x) dx = F1*(k)F2(k) dk
|f(x)|2 dx = |F(k)|2 dk
Most of you know FT’s as a way to extract the frequency spectrum from a
complicated time trace (see applet).
Why is this useful in QM ?
One example (there are more!):
We have seen that “localization” of a free particle involves adding up
many plane waves with many different momenta:
n
xipn
necx
/
2
1)(
But p is continuously variable!
exchange discrete for , get a “wave packet”:
/)(2
1)( ipxepFdpx
draw Gaussian-like curve to visualize!
40
F(p) = p-dependent “weighting factor” analogous to cn, i.e., this is the
momentum distribution of the wave packet.
F(p) is the Fourier transform of (x). If we (somehow) know how (x)
looks, we can get to F(p) that way.
/)(2
1)( ipxexdxpF
Note “-“ sign of exponent
From knowing the momentum distribution F(p), we use the inverse Fourier
transform (like above)
Note:
1. We get the “QM” form of the Fourier transform from the general
form, if we set p=ħk and )(1
)(1
)(
pFkFpF
2. (x) and F(p) contain the same information
3. (x) is the wave function in position space. |(x)|2 = P(x)
draw little diagram
4. F(p) is the wave function in momentum space. |F(p)|2 = P(p)
draw little diagram
41
Example: free particle with momentum of exactly p0 (=plane wave)
)()(2
1)(
2
1)(
2
1)(
0/)(/
/
0
0
ppexdxexdxpF
ex
xppiipx
xip
P(p) = |F(p)|2=0 unless p=p0
Learning goals slide
42
5. Wave function space, Dirac notation, Hermitian operators
We have met operators acting on wave functions, let’s now look at the wave
functions themselves:
One-particle wave function space
(r,t)*(r,t) = probability of finding particle with wavefunction at point
r in space at time t.
Better: (r,t)*(r,t) d3r = probability of finding particle in the volume
element d3r = dx dy dz around r.
Physically meaningful one-particle wave functions must satisfy
normalization condition
1),(23 trrd
, when integrated over all space
square-integrable, or quadratically integrable
space of square-integrable functions is called L2.
Usually, WF’s must also be everywhere defined, continuous, infinitely
differentiable), let’s call them “well behaved”.
They form a subspace of L2, let’s call it F.
43
Note:
(a) F is a linear vector space: if F, then is also F.
(b) A scalar product exists on F. (Analogy in R3: baba),( )
Definition: We associate a complex number with
)()(*, 3 rrrd (always converges if ,F)
Properties:
o (,) = (,)*
o (, 11 + 22) = 1(,1) + 2(,2)
o (11 + 22,) = 1*(1,) + 2*(2,)
(linear w. r. to 2nd function, antilinear w. r. to 1st)
If (,) = 0, we say that and are orthogonal
(analogy to vectors in R3).
(c) The scalar product of a function with itself
233 )()()(*, rrdrrrd
is always real and positive. It is only zero, if (r)=0 for all r.
, is called the norm of .
44
Dirac notation (bracket notation):
In three dimensional space: r=(x, y, z)
Change the axes different coordinates (but the same point!):
r=(r1, r2, r3)
Geometrical vector concept and vector calculation
avoid referring to a specific system of axes
makes life simpler.
Similar for description of QM system:
Quantum state characterized by state vector in state space .
Any element of is called a ket vector or a ket (symbol: |>, e.g., |>).
(r) F |> r.
Scalar Product:
There is a dual space * belonging to , on which we define another set of
vectors, called bra vectors <|.
45
Scalar product of two ket vectors
(|>, |>) = <|> = d *
(bra and ket come from “bracket”)
Rules for the scalar product in bracket notation:
<|> = <|>*
<|11 + 22> = 1 <|1> + 2 <|2>
<11 + 22|> = 1* <1|> + 2* <2|>
<|> real, positive; zero only if |> = 0
A linear operator A^ acting on a ket |> produces another ket:
|> = A^|>
----------------------- End Lecture 6 (09/04/2015) -----------------------
46
Last hour:
We get the “QM” form of the Fourier transform from the general form,
if we set p=ħk and )(
1)(
1)(
pFkFpF
(x) and F(p) contain the same information
The WF space F is a linear vector space.
A scalar product exists on F, with )()(*, 3 rrrd
If (,) = 0, we say that and are orthogonal
Dirac notation (bracket notation): Quantum state characterized by state
vector (“ket”) in state space . Every function F is mapped onto a
ket |> . There is a dual space * belonging to , on which we define
another set of vectors, called bra vectors <|. Scalar product of two ket
vectors (|>, |>) = <|> = d *
A linear operator A^ acting on a ket |> produces another ket: |> =A^|>
47
Consider a countable set of function in F, labeled by an index j
(j = 1,2,3,..). If
<uj|uk> = 1 for j=k, 0 for jk
then we call {uj} orthonormal.
This condition can also be written as
<uj|uk> = jk (jk is called the Kronecker delta)
If every function in F can be expanded in one (and only one) way in terms of
the ui:
)()( rucrj
jj
we call {uj} a basis.
Analogy from R3: (1,0,0), (0,1,0), and (0,0,1) constitute a basis in R3.
How do we get the components ci of a wave function?
)()(,
),(,,
*3 rrurduc
ccuucucuu
uc
jjj
kj
kjjj
jkjj
jjkk
jjj
If , are wave functions, we can write:
48
jjj
kjjkkj
kjkjkj
kkk
jjj
kkk
jjj
cb
cbuucbucub
uc
ub
*
,
*
,
*
,
),(,,
What about plane waves?
/
2
1)( ipx
p exv
= plane wave w/ momentum p F because L2
Remember “wave packet”:
/)(2
1)( ipxepfdpx
Let’s write this a bit different:
)()(),()(
)()()(
* xxvdxvpF
xvpFdpx
pp
p
Note:
1. vp(x) is a plane wave, i.e., not square integrable!
2. Compare
49
)()()( xvpFdpx p )()( rucrj
jj
)()(),()( * xxvdxvpF pp
)()(, *3 rrurduc jjj
continuous basis discrete basis
We see that f(p) is the analog of cj and that vp is the
analog of uj !
3. It looks as if plane waves form a basis! However, vp cannot represent a
physical state! They are just mathematical constructs to describe a
wave function.
------------------------ End Lecture 7 (09/10/14) -------------------------
50
Adjoint operators:
Definition:
)(*)](ˆ[)(ˆ)(* xgxfAdxgAxfd
or
<f|Âg> = <Â+f|g>
Note that  acts on g, Â+ acts on f. Â+ is called adjoint operator of Â.
Hermitian operators:
Definition:
An operator  is called Hermitian (or symmetric) if
<f|Âg> = <Âf|g>
or d means generalized “integration over all spatial coordinates”
)(*)](ˆ[)(ˆ)(* xgxfAdxgAxfd
Why is that important? All operators representing observables in QM are
Hermitian!
51
Implications of an operator being Hermitian:
1. Â+ = Â (self-adjoint) Proof in Levine’s book
2. If Ân = nn (n is eigenvalue of n w.r.t Â)
<n|Ân> = <n|nn> = n <n|n>
also = <Ân|n> = <nn| n> = n* <n|n>
n = n* n is real number
3. All eigenfunctions of a Hermitian operator with different eigenvalues are
orthogonal (see homework).
4. The eigenfunctions of a Hermitian operator form a basis. Special
example: If we can solve the TISE, the solutions (EF’s of H^) form a
basis!
Learning goals slide
52
Last hour:
If every element of state space can be expanded in one (and only one)
way in terms of a set of countable, orthonormal functions ui:
)()( rucrj
jj
we call {uj} a discrete basis.
If every element of state space can be written in terms of functions with
a continuously varying parameter p as
)()()( xvpFdpx p , we call the
p a continuous basis.
Adjoint operator Â+: <f|Âg> = <Â+f|g>
Hermitian operator: <f|Âg> = <Âf|g>
All operators representing observables in QM are Hermitian!
Hermitian Operators are self-adjoint: Â+ = Â
All eigenvalues of Hermitian operators are real
All eigenfunctions of a Hermitian operator with different eigenvalues are
orthogonal
The eigenfunctions of a Hermitian operator form a basis
53
6. Average values:
We know how to predict probabilities for
1. position |(x)|2
2. momentum |f(p)|2
We can predict average values (expectation values) for measurements which
depend on x or p:
Position:
)()()( * xxxdxxxPdxx
Kinetic energy:
)(2
ˆ)(
2)(
2
2*
22
pfm
ppfdp
m
ppPdp
m
pE
In general (Ô is Hermitian!):
)(ˆ)(ˆ * xOxdO
In Dirac notation, we write this as
<Ô> = <|Ô|>
Why?
If is eigenfunction of Ô, the measurement yields the corresponding
eigenvalue.
54
If is not EF of Ô, we can expand in terms of the eigenfunctions:
n nnn nn cc ***;
n nn
n nnn
mnn m mmn
m mmmn nn
m mmn nn
cO
cc
dcc
ccd
cOcdOdO
2
*
**
**
***
ˆ
ˆˆˆ
In other words, the probability of measuring a particular eigenvalue in a set
of many identical systems is given by |cn|2. The average value is the weighted
average of the EV’s (weighted by their probability!)
Note:
A common definition for uncertainties is
2222 ; pppxxx
From this definition, one can obtain the rigorous formulation of Heisenberg’s
uncertainty principle xp ≥ ½ ħ
Clicker Question “Ehrenfest’s theorem”
Time evolution of average values:
55
t
AdAdAdAd
dt
d
dt
Ad ˆˆˆˆ
ˆ****
From the T.D.S.E: ** ˆ1
:ˆ1 H
iH
i
t
AHAd
iAHd
idt
Ad ˆˆˆ1ˆ)ˆ(
1ˆ
**
H^ is Hermitian, therefore
t
AAH
i
t
AHAd
iAHd
idt
Ad
ˆ
]ˆ,ˆ[
ˆˆˆ1ˆˆ1
ˆ**
If [H^, Â] = 0 and  f(t), then <Â> is a constant of motion, independent of
time.
Ehrenfest’s Theorem: “QM averages obey classical mechanics“
ppxxppm
i
m
ppxx
m
ppi
xVxTi
t
xxH
i
dt
xd
ˆˆˆˆˆˆ22
ˆˆˆˆ
2
ˆˆ
]ˆ,ˆ[]ˆ,ˆ[ˆ
]ˆ,ˆ[
Use ixppxpx ˆˆˆˆ]ˆ,ˆ[
ixppx ˆˆˆˆ and ipxxp ˆˆˆˆ
56
Bring x^ between the p^’s:
m
ppixpipxp
m
i
dt
xd ˆˆ)ˆˆ()ˆˆ(ˆ
2
----------------- Skipped ------------------
2
2
*
*
*
)(
)()(
)(
)()()(
)()(
]ˆ,ˆ[]ˆ,ˆ[ˆ
]ˆ,ˆ[
dt
xdmxF
xFx
xV
dxx
xV
i
i
dxxVxixi
xVxi
xVi
dxxVxixi
xVi
pVpTi
t
ppH
i
dt
pd
Note: Careful! <F(x)> F(<x>)
except for F=0
or F=const.
or F=kx
--------------------------- End Skipped ----------------------------
57
---------------------- END Lecture 08 ---- 09/11/15 ------------------