Ben Gurion University of the Negev
www.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter
Physics 3 for Electrical Engineering
Lecturers: Daniel Rohrlich, Ron Folman
Teaching Assistants: Daniel Ariad, Barukh Dolgin
Week 7. Quantum mechanics – scalar product of wave functions •
Hermitian operators, eigenvalues and eigenfunctions • expectation
values • eigenfunction expansions • Dirac’s -function •
commutators • generalized uncertainty principle
Sources: Merzbacher (2nd edition) Chap. 8;
Merzbacher (3rd edition) Chap. 3 Sects. 3-4, Chap. 4 Sects. 1-4,
Chap. 10 Sect. 5 and Appendix 1.
We have seen a few solutions of Schrödinger’s equation, and at
the same time we are still trying to understand it.
Let’s take a fresh look at the time-independent Schrödinger
equation:
Eψ  Hˆ ψ ,
where
in 1D.
2
2


Hˆ  

V
(
x
)
2m x 2
We have seen a few solutions of Schrödinger’s equation, and at
the same time we are still trying to understand it.
Let’s take a fresh look at the time-independent Schrödinger
equation:
Eψ  Hˆ ψ ,
where
2
2


Hˆ  

V
(
x
)
2m x 2
in 1D. The solutions are
Ei ψi  Hˆ ψi
What kind of equation is that?
.
Similarly, if we define
then the solutions

pˆ  i
,
x
ψk  e
ikx
,
of the free time-independent Schrödinger equation satisfy
pˆ ψk  kψk
What kind of equation is that?
.
One more: Let P̂ be the parity operation, defined by
Pˆ ψ( x)  ψ( x) ,
then the solutions
ψn
for a symmetric 1D square well satisfy
Pˆ ψn  ψn
What kind of equation is that?
.
Hermitian operators, eigenvalues and eigenfunctions
Hˆ , pˆ , Pˆ are examples of linear operators:
Aˆ c1ψ1  c2 ψ 2   c1 Aˆ ψ1  c2 Aˆ ψ 2 .
If Aˆ ψ k  ak ψ k , for some ak and ψ k , then we call ak
the eigenvalue of  and we call ψ k the (corresponding)
eigenvector of Â.
Hermitian operators, eigenvalues and eigenfunctions
Hˆ , pˆ , Pˆ are examples of linear operators:
Aˆ c1ψ1  c2 ψ 2   c1 Aˆ ψ1  c2 Aˆ ψ 2 .
If Aˆ ψ k  ak ψ k , for some ak and ψ k , then we call ak
the eigenvalue of  and we call ψ k the (corresponding)
eigenvector of Â. Dirac defined a “ket” vector notation:
Aˆ ψ k  ak ψ k
or even Aˆ ak  ak ak .
Hermitian operators, eigenvalues and eigenfunctions
Hˆ , pˆ , Pˆ are examples of linear operators:
Aˆ c1ψ1  c2 ψ 2   c1 Aˆ ψ1  c2 Aˆ ψ 2 .
If Aˆ ψ k  ak ψ k , for some ak and ψ k , then we call ak
the eigenvalue of  and we call ψ k the (corresponding)
eigenvector of Â. Dirac defined a “ket” vector notation:
Aˆ ψ k  ak ψ k
or even Aˆ ak  ak ak . He also defined
a “bra” vector which look like this: ψk .
Scalar product of wave functions
A vector space has a “scalar product” of two wave functions
ψ(x) and φ(x); it is

*
φψ 
 φ(x)
ψ( x) dx
.
in Dirac’s “bra-ket” notation.
Note that the scalar product of ψ(x) and φ(x) depends on their
order.
Scalar product of wave functions
A vector space has a “scalar product” of two wave functions
ψ(x) and φ(x); it is

*
φψ 
 φ(x)
ψ( x) dx
.
in Dirac’s “bra-ket” notation.
Note that the scalar product of ψ(x) and φ(x) depends on their
order.
The scalar product of ψ(x) with itself is
Scalar product of wave functions
A vector space has a “scalar product” of two wave functions
ψ(x) and φ(x); it is

*
φψ 
 φ(x)
ψ( x) dx
.
in Dirac’s “bra-ket” notation.
Note that the scalar product of ψ(x) and φ(x) depends on their
order.
The scalar product of ψ(x) with itself is
ψψ 


| ψ( x) |2 dx  1 .
A property of the scalar product:
ψφ 
 φ ψ *
.
A property of the scalar product:
ψφ 
 φ ψ *
.
Proof:
ψφ 


ψ( x) * φ( x) dx
 
*
*
φ( x) ψ( x) dx 
 
 




 φ ψ *.
Hˆ , pˆ , Pˆ are not only linear operators, they are also physical
operators (“observables”) that represent physical quantities.
Therefore their eigenvalues must be real.
The set of all eigenvectors of an observable forms the basis of
a vector space. This vector space contains all possible states of
the system. It is complete.
Hˆ , pˆ , Pˆ are not only linear operators, they are also physical
operators (“observables”) that represent physical quantities.
Therefore their eigenvalues must be real.
The set of all eigenvectors of an observable forms the basis of
a vector space. This vector space contains all possible states of
the system. It is complete.
Example: The eigenvectors of p̂ are all the functions eikx.
Any function on the line can be written as a linear combination
of these eigenvectors (Fourier analysis on a line).
Hˆ , pˆ , Pˆ are not only linear operators, they are also physical
operators (“observables”) that represent physical quantities.
Therefore their eigenvalues must be real.
The set of all eigenvectors of an observable forms the basis of
a vector space. This vector space contains all possible states of
the system. It is complete.
Example: The energy eigenvectors of an infinite 1D square well
between 0 and L can be written
ψ n ( x) 
2
n x
sin
L
L
and any function that vanishes at x = 0 and x = L can be written
as a linear sum of these ψn(x) (Fourier analysis on an interval).
Hˆ , pˆ , Pˆ are not only linear operators, they are also physical
operators (“observables”) that represent physical quantities.
Therefore their eigenvalues must be real.
But how do we know which operators represent physical
quantities? How do we know whether an operator has only
real eigenvalues?
A partial answer: any observable  must be “Hermitian”:
φ Aˆ ψ  Aˆ φ ψ ,
where φ(x) and ψ(x) are any two wave functions.
Why Hermitian? We can prove that if  is Hermitian then
1. the eigenvalues ai of  are real, and
2. the eigenvectors a j of  can be chosen orthogonal.
Proof:
ˆ a a  a Aˆ a  a , hence every ai is real;
1. ai*  A
i i
i
i
i
ˆ a  a Aˆ a  a a a , hence
2. ai ai a j  A
i j
i
j
j i j
either ai = aj or ai a j  0 , i.e. ai and a j are orthogonal.
(If ai = aj then we can choose linear combinations of ai and
a j that are eigenvectors of  and orthogonal.)
The observable Ĥ has a special name: it is the “Hamiltonian”.
Let us show that the Hamiltonian is a Hermitian operator, i.e.
that φ Hˆ ψ  Hˆ φ ψ . We assume that all wave functions
φ(x) and ψ(x) vanish for |x| → ∞. Now
φ Hˆ ψ 


φ * Hˆ ψ dx
 2 d 2


φ * 
 V ( x) ψ dx
2

 2m dx



Integrating by parts twice, we get
d 2ψ
dφ * dψ
d 2φ *
φ* 2  

ψ
2
dx dx
dx
dx
 2 d 2

 2 d 2


φ * 
 V ( x) ψ dx 
ψ 
 V ( x) φ * dx
2
2

  2m dx
 2m dx






*
  d


ψ 
 V ( x) φ * dx
2
  2m dx


 Hˆ φ ψ .


2
2
Thus φ Hˆ ψ  Hˆ φ ψ and Ĥ is Hermitian. The fact that Ĥ
is Hermitian leads to two important conclusions:
1. Ei*  Hˆ ψi ψi  ψi Hˆ ψi  Ei , hence every Ei is real;
2. Ei ψi ψ j  Hˆ ψi ψ j  ψi Hˆ ψ j  E j ψi ψ j , hence
either Ei = Ej or ψi ψ j  0 , i.e. ψi and ψ j are orthogonal.
In fact, we can always form a complete orthonormal basis for
the vector space of ψi out of eigenstates of Ĥ. For any φ
we can write
φ 
c j ψ j
.
j
“Ortho” means that ψi ψ j  0 if i ≠ j, and “normal” means
that ψi ψi  1 for every i. Assuming that φ is normalized,
too, we find

1 φ φ  



i, j

i

ψi ci*  
 

ci*c j ψi ψ j 


i
j

cj ψj 


ci*ci
.
And now we can generalize our probability rule: Not only is
|φ(x)|2 the probability density to find a particle in the state φ(x)
at the point x, but also
φ ψj
2
 cj
2
is the probability to find a particle in the state φ to be in the
state ψ j . If the energy Ej is non-degenerate, then |cj|2 is the
probability to find the particle with energy Ej.
From here it is an easy step to define an expectation value:
φ Hˆ φ 
E j cj
j
2
is the average energy in the state φ .
Expectation values
Whatever we just proved about the Hamiltonian applies also to
every other observable, because every physical operator is
Hermitian. For example, the expectation value of momentum
in the state φ is
φ pˆ φ 

d 
φ(x)   i  φ(x) dx .

dx 


*
The expectation value (average value) of the position in the
state φ is
φ xˆ φ 


φ(x) * x φ(x) dx
,
which is just an ordinary common-sense average. So x̂ = x.
Eigenfunction expansions
We saw that every state φ has an expansion in eigenstates of
the Hamiltonian. The same is true for any other observable Â:
the state φ has an expansion in eigenstates of  :
φ 
c j a j
,
j
where Aˆ a j  a j a j and the coefficients cj depend on φ .
Dirac’s δ-function
What about x? We declared x̂ = x to be an operator, but what is
its eigenfunction? Apparently the eigenfunction of x̂ should be
a wave function that is zero everywhere except at some point a.
We can approximate such a function in many ways, e.g. as a
very thin gaussian function (ε → 0):
ψ a ( x) 
e
( x  a ) 2 / 2
4 
,
Dirac instead defined a “δ-function” δ(x−a) with the following
properties: for any function f(x),
x2
x
1
f (a) for x1  a  x2 ,
f ( x) ( x  a) dx  
0 otherwise .

Dirac’s δ-function
Dirac’s δ-function is an eigenfunction of x̂ with “δ-function
normalization”:

  ( x  a) ( x  b) dx   (a  b)
Using Dirac’s δ-function, we can write an eigenfunction
expansion for x̂ as follows:
φ( x ) 

 φ(a) ( x  a) da
,
where we regard φ(a) as the “coefficient” of the eigenfunction
δ(x−a).
Dirac’s δ-function
Dirac’s “δ-function normalization” applies also to momentum
ik1x
ik2 x
e
e
eigenfunctions
and
because their scalar
2
2
product is


k1 k 2 




e ik1x eik2 x
2
2
ei ( k2  k1 ) x
2
dx
dx
  (k1  k 2 ) .
(The integral is infinitely bigger for k1 = k2 than for k1 ≠ k2 .)
Dirac’s δ-function
One way to “tame” the integral


e
ikxx 2
2
dx 

eikx
 2
dx is to treat it as
ik 2 2
 ( x  )  k / 4
2
 e

2
dx 
e
 k 2 / 4
2 
in the limit ε→0. Multiplying this by f(k) and integrating over
k, we get

 f (k )
e
k 2 / 4
2 
dk  f (0)


e
 k 2 / 4
2 
dk  f (0) .
Commutators
Operators are different from numbers, and in particular, they
don’t obey the rule xy = yx that real and complex numbers x,y
obey. If we apply two operators Aˆ1 Aˆ 2 to a state ψ , we may
get a different result than if we apply them in the opposite
order:
Aˆ1 Aˆ 2 ψ  Aˆ 2 Aˆ1 ψ
 
 Aˆ1, Aˆ2  Aˆ1Aˆ2  Aˆ2 Aˆ1
.
The commutator Aˆ1 , Aˆ 2 of Aˆ1 , Aˆ 2 is defined as
operating on a state.
Commutators


ˆ , Hˆ  0 ,
If a Hermitian operator  commutes with Ĥ, i.e. A
then we can find eigenstates of  that are also eigenstates of Ĥ.
Example: If Ĥ commutes with the parity operator P̂, then for
any state ψ , Hˆ Pˆ ψ  Pˆ Hˆ ψ . So if ψ is an eigenvector of
Ĥ with eigenvalue E, i.e. Hˆ ψ  E ψ , then so is P̂ ψ :
 
 
Hˆ Pˆ ψ  Hˆ Pˆ ψ  Pˆ Hˆ ψ  Pˆ E ψ  E Pˆ ψ
.
If E is nondegenerate, then Pˆ ψ  α ψ for some number α, so
ψ is an eigenstate of P̂. If E is degenerate, then (1  Pˆ ) ψ / 2
2
are eigenstates of P̂ , since Pˆ  1.
Let’s calculate the commutator  xˆ , pˆ  by applying it to a wave
function ψ(x):
 xˆ , pˆ  ψ( x)   xˆ pˆ  pˆ xˆ  ψ( x)
 
d  
d  
  x  i     i  x  ψ( x)
dx  
dx  
 
d
d


 ix ψ    i [ xψ] 
dx
dx


d
d 

 ix ψ    iψ  ix ψ 
dx
dx 

 i ψ ;
hence
 xˆ , pˆ   i .
Generalized uncertainty principle
Definition:  Aˆ 
Aˆ 2  Aˆ
2
, where Aˆ  ψ Aˆ ψ etc.
ˆ ψ  Aˆ ψ  ( Aˆ ) ψ , where ψ is a
Theorem: A


vector orthogonal to ψ .
Proof: We have Aˆ ψ  c1 ψ  c2 ψ 
for some ψ  , and
ˆ . If we
if we apply the bra ψ to each side we obtain c1  A
take the scalar product of each side with itself, we obtain
(c2 )  Aˆ 2  (c1) 2  Aˆ 2  Aˆ
2
2
.
Generalized uncertainty principle
Suppose
Aˆ ψ  Aˆ ψ  ( Aˆ ) ψ 
Bˆ ψ  Bˆ ψ  ( Bˆ ) ψ  ' ,
where ψ  and ψ  ' are different, in general. Now if we
 Aˆ , Bˆ  we have
 Aˆ , Bˆ    Aˆ  Bˆ  ψ ψ '' ψ ψ
calculate
and therefore
  
1
ˆ
ˆ
A B 
2
 Aˆ , Bˆ 
  
 2  Aˆ  Bˆ
.
Application to position and momentum uncertainty
(Heisenberg’s uncertainty relation for Δx and Δp):
  
1
ˆ
ˆ
A B 
2
1
ˆ
ˆ
 x  p  
2

ˆ
ˆ
 x  p  
2
 Aˆ , Bˆ 
 xˆ, pˆ 


2