Download Quantum Physics 2005 Notes-7 Operators, Observables, Understanding QM Notes 6

Quantum Physics 2005
Notes-7
Operators, Observables, Understanding QM
Notes 6
Quantum Physics F2005
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A summary of this section
• This section of notes is a brief overview of the ideas
in chapters 10-12 of Morrison.
• These chapters are the intellectual core of quantum
mechanics (as opposed to quantum physics).
• You will address the material in chapters 10-12 with
greater care in Intro to Quantum Mechanics.
• The purpose of these notes is only to introduce these
ideas, especially for those students who will not take
Intro to Quantum Mechanics.
• I will therefore skip lightly through these chapters,
picking only the most important concepts.
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Operators – Dirac notation
Dirac Notation - "bra"=
Q = !1 Qˆ !1 =
"ket"=
* ˆ
!
∫
1 Q!1dx =expectation value
allspace
!1 Qˆ ! 2 =
* ˆ
!
∫
1 Q! 2 dx
=matrix element
allspace
!1 !1 =
*
1
∫ ! ! 2 dx
=overlap integral
allspace
1) Take the complex conjugate of the function inside the
2) Act with the operator on the function to its right.
3) Integrate the integrand over all space.
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Hermitian Operators
Qˆ is Hermitian if for any two
physically admissible state functions
!1 Qˆ ! 2 = Qˆ !1 ! 2
(
)
*
Qˆ !1 ! 2 " ∫ Qˆ *!1 ! 2 dx
If an operator is Hermitian, the expectation value
of that operator will be real.
All operators for physical quantities are Hermitian.
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Is p Hermitian?
We will check explicitly whether !1 pˆ ! 2 = pˆ !1 ! 2 .
%
%x
Start by integrating !1 pˆ ! 2 by parts.
pˆ = $ih
!1 pˆ ! 2
%
 * # #


 % * 
= ∫ !  $ih ! 2  dx = $ih !1 ! 2 $ ∫ ! 2  !1  dx 
$#
%x
$#
$#


 %x  

#
*
1
*
#
%
 % *


= ih ∫ ! 2  !1  dx = ∫ ! 2  $ih !1  dx = pˆ !1 ! 2
%x 
$#
$#
 %x 

Therefore pˆ is Hermitian.
#
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Exercise
Show by writing out the real space integral
representation that the operator xˆ is Hermitian.
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Operators and Eigenvalues
In quantum physics, the form of the eigenvalue equation
for an observable Aˆ with eigenstate & a and eigenvalue a is:
Aˆ& = a& .
a
The collection of all eigenvalues is called the
spectrum of this operator.
The most familiar eigenstate is the stationary state.
! = & e$ iEt / h which satisfies: Hˆ ! = E !
E
E
In an eigenstate of an observable, the uncertainty in
that observable is zero.
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The eigenstate of one observable might not be an eigenstate
of another observable (but it can happen ).
ExampleAn energy eigenstate of the square well is:
E1 h( 2
 ( x  $ i'1t
with '1 =
.
! ( x, t ) = A cos 
=
e
2
h 2mL
 L 
% 

Is this an energy eigenstate of momentum  pˆ = $ih  ?
%x 

(
%
 ( x  $ i'1t
 ( x  $ i'1t
ˆp! = $ih A cos 
= ihA sin 
) p! NO!
e
e
L
%x
 L 
 L 
Another example:
An energy eigenstate of the free wave is:
pˆ ! = $ih
Notes 6
! = Ae (
i kx -'t )
.
%
i kx -'t
Ae ( ) = hk ! YES>
%x
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Hermiticity: Real eigenvalues and
orthonormal eigenfunctions
•
•
The eigenvalues of a Hermitian operator are real.
The eigenvalues of a Hermitian operator are the
only values that we can observe in a measurement
of that observable.
AND
The eigenfunctions of a Hermitian operator are
orthogonal, constitute a complete set, and satisfy
closure. (p 465, Morrison).
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Proof of orthogonality
Take two different eigenstates of the same observable:
1) Qˆ& = q& and 2) Qˆ& = q '&
q
q
q'
q'
We want to demonstrate that & q ' & q = 0.
Multiply both sides of 1) by complex conjugate of & q ' :
& q ' Qˆ& q = q & q ' & q = q
ˆ to simplfy the LHS:
Use Hermiticity of Q
& q ' Qˆ& q = Qˆ& q ' & q = q '& q ' & q = q '* & q ' & q = q ' & q ' & q
So we have ( q - q ') & q ' & q = 0, but we assumed q ) q '
so & q ' & q = 0
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Hermiticity: Completeness
Completeness of a set of eigenfunctions means that
any well-behaved function of the variables on which
the eigenfunctions depend can be expanded upon the set.
1)
f = ∑ cq& q
q
We can choose to represent any function in a given range of
variables by an expansion in a set of eigenfunctions of any
operator.
It's easy to find the coefficients cq . Multiply 1) by & q '
& q ' f = & q ' ∑ cq& q = ∑ cq & q ' ∑ cq& q = cq '
q
q
q
The coefficient cq is the "projection" of f onto the eigenfunction & q
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Summarizing Eigenfunction
Expansions
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The meaning and use of expansions
• The collection of coefficients in the expansion of a state
function in any complete set is merely an alternate way
to represent the state function.
• These coefficients and the eigenfunctions contain the
same information as the state function.
• Expressing a state function in terms of eigenfunctions
can make apparent some properties of the state
function.
• Expressing the state function in terms of eigenfunctions
can allow us to get information about other variables.
(Expressing the wavefunction (position variable) in
terms of momentum eigenfunctions allows us to
determine the momentum properties of the state
function.)
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Using expansions to calculate
expectation values
• If we have a wavefunction that is not an
eigenfunction of a given operator, the
expectation value may be hard to calculate in
the old way.
• Expanding the wavefunction in terms of the
eigenfunctions of that operator may make
things easier.
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Calculation of the mean
Q (t ) = ∫ ! * Qˆ !dx
*
= ∫ [ ∑ cn ' (t )& n ' ( x )] Qˆ ∑ cn (t )& n ( x)  dx
*
= ∫ [ ∑ cn ' (t )& n ' ( x )] [ ∑ cn (t )qn& n ( x) ] dx
2
*
= ∑ ∑ cn ' cn & n ' & n = ∑ cn qn
n' n
n
If we know the projection coefficients,
we know the expectation value.
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Example
Calculate the expected value of the energy for
the following non-stationary state of the SHO.
1
& 1e $ i'1t + 7& 2 e $ i'2t 
! ( x, t ) =
50
1 $ i'1t
7 $ i'2t
c1 (t ) =
e ;
c2 (t ) =
e
(all others=0)
50
50
#
1 3
49 5
62
2
E (t ) = ∑ cn En =
h'0 =
h' 0
h'0 +
n =0
50 2
50 2
50
Note that any individual measurement of E
will yield an eigenenergy.
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The Commutator
A simultaneous eigenstate is one whose state function
is an eigenstate of two operators:
and Rˆ& = r&
Qˆ& = q&
q ,r
q ,r
q ,r
q ,r
Because they are eigenstates:
Q = q *Q = 0 and
R = r *R = 0
ˆ ˆ - RQ
ˆ ˆ =0
Qˆ , Rˆ  = QR


1) Operators that commute define a complete set of
simultaneous eigenfunctions.
2) Two operators that share a complete set of
eigenfunctions commute.
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Commutators and uncertainty
principles
ˆ commute, then they share simultaneous eigenfunctions.
If operators Rˆ and Q
Measurement of each observable for an eigenfunction yields a precise value.
The only uncertainty relation that can apply is: *Q*R , 0.
A Generalized Uncertainty Principle: *Q*R ,
(For momentum and position:
1  ˆ ˆ
i Q , R 
2
ˆ
[ xˆ, pˆ ] = ih1)
If Qˆ , Rˆ  ) 0 then there exist no eigenstates of either observable.
(Think about + ( x) and e-i ( kx-'t ) .)
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Commutators of x and p
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An aside
• Charles Hermite (December 24, 1822 - January 14, 1901) was
a French mathematician who did research on number theory,
quadratic forms, invariant theory, orthogonal polynomials, elliptic
functions, and algebra. Hermite polynomials, Hermite normal
form, Hermitian operators, and cubic Hermite splines are named
in his honor.
• He was the first to prove that e, the base of natural logarithms, is
a transcendental number. His methods were later used by
Ferdinand von Lindemann for the proof of his celebrated
theorem that π is transcendental.
• Upon Weierstrass' discovery in 1861 of continuous curves that
are nowhere differentiable - they possess no tangent at any
point - Hermite famously remarked: “I turn aside with a shudder
of horror from this lamentable plague of functions which have no
derivatives.”
• http://en.wikipedia.org/wiki/Charles_Hermite
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