Transcript MODULE 4

MODULE 2
Integrals over Operators
In quantum chemistry we seek to make contact between
calculations done using operators and the actual outcome of
experiments.
This usually requires us to evaluate certain integrals, all of which
have the form
I 
space
ˆ g d
f *
In Dirac notation the integral would be symbolized as
ˆ g
I f 
A closed bracket
or
ˆ
I 
fg
the RHS has the appearance of an element of a matrix placed at
the intersection of row f and column g. For this reason the
integral is referred to as a "matrix element".
MODULE 2
In many cases, the operator is simply multiplication by 1 and then
the matrix element has a special symbol
f 1 g  f g S
The parameter S is given the name "overlap integral" and is a
measure of the similarity of a pair of functions.
S can take values from 0 to 1.
We have seen the limiting values of S in the orthonormality
condition.
f g   f ,g
MODULE 4
MORE ABOUT POSTULATES
We need to establish a firm link between what we calculate and
what we measure in the laboratory.
operator
ˆ w w w

eigenket (function)
The operator represents some dynamical observable (energy,
momentum, dipole moment, ...) and has an eigenvalue w
It is a common practice to name the ket with its eigenvalue.
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Postulate 4
“The only possible values that can result from a series
of measurements of a dynamical variable are the
eigenvalues of the operator that corresponds to the
dynamical variable.”
Thus if a system is in the state defined by the ket w1
every measurement of the property represented by ̂
will yield w1 as the result (within the error of the
experiment).
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For example, the 1s, 2s, 2px,y,z, 3s, etc states of the hydrogen atom are the
designations for a series of eigenstates of the hamiltonian operator (see
later).
If we prepare a large number of H atoms in the 2s state and measure their
energies, every one will yield the same result, since the 2s state is an
eigenfunction (eigenket) of the hamiltonian operator.
Recalling our first equation, with the ket normalized
ˆ w w w

Multiplying from the left by the (normalized) bra w
ˆ w  w w w w w w
w 
w
closed bracket
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when the ket (and the bra) are eigenvectors (eigenfunctions) of
the operator, the closed bracket is the eigenvalue, a real
number.
However, the system might not be in an eigenstate of the operator
we are interested in
how to deal with such a situation?
To determine the result of experiments of the observable
represented by the operator ̂ on the state represented by the
ket  that is not an eigenket of ̂ , we must evaluate the
ˆ 
expression 
MODULE 4
a general function can be expressed as a linear combination of a complete set
of eigenfunctions (e.g., a line from sine functions) of an operator.
This is known as the Superposition Principle, and it is centrally important in
Quantum Theory. Thus:
  c1 w1  c2 w2  c3 w3  ...
When the eigenkets
   ci wi
i
wi
form a discrete series, then


ˆ
ˆ
      ci wi 
 i

Considering only two states
ˆ  
ˆ c w  c w

1
1
2
2

ˆ w c 
ˆ w
 c1
1
2
2
 c1w1 w1  c2w 2 w 2
applying the operator to a ket that is not one of its own eigenkets gives a linear
combination of its eigenkets multiplied by their eigenvalues.
MODULE 4
Now multiply the last equation from the left by the bra 
ˆ    (c w w  c w w )
 
1 1
1
2 2
2
and realizing that
   c1* w1  c2* w 2

Combining these and applying the condition of orthonormality
2
2
ˆ
    w1 c1  w 2 c2
The closed bracket on the LHS is a real number because the
square moduli are real
The wi are eigenvalues of a Hermitian operator (always real).
MODULE 4
the algebra for the general case is too boring, so we simply state
the result
2
ˆ
     ci wi
i
Thus where the system is not in an eigenstate of the operator of
the required observable the result of a large number of
measurements of the observable is seen to be a weighted sum
of the eigenvalues of the operator.
Each individual eigenvalue contributes to the sum according to the
square modulus of the coefficient that governs the contribution
of its corresponding eigenket in the LC.
New postulate
MODULE 4
POSTULATE 5
"If the system is in an eigenstate of the operator that corresponds
to the observable in question, determination of the observable
(on a large number of identical preparations) always yields a
single, unique result that is the eigenvalue of the operator.
If the system is not in an eigenstate of the operator in question, it
can be expressed as a superposition of the eigenstates of the
operator.
Then a single measurement of the observable yields a result that
corresponds to one of the eigenvalues of the set of
eigenfunctions of the operator.
Many determinations of the observable (on a large number of
identical preparations) will lead to a distribution of the
eigenvalues of the operator.
The probability
that a particular eigenvalue is measured is equal
2
to ci where ci is the coefficient of wi in the superposition."
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A single measurement can provide only one result.
A pointer can indicate only one value per determination.
Repeated measurements on a large number of identical systems
will generate a mean value.
This mean is either a unique number (the eigenvalue case).
Or is a distribution of numbers about an average value
(the not-an-eigenvalue case).
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2
ˆ
     ci wi
i
The RHS is an average value.
Thus the matrix element on the LHS is also an average value
It is called the expectation value of the operator (a QM average
value).
In real laboratory experiments it is very unusual for a large
number of identical systems to be in a single eigenstate.
Measurements will generate the expectation value of the operator.
MODULE 4
In the case of un-normalized wavefunctions the expectation value
is given by
* ˆ
ˆ

w  w  w  w d
ˆ
 

*
ww

 w w d
For particle in a one-dimensional well with infinite walls
 n x 
 n  N sin 

L


n2 h2
En 
8mL2
Suppose we make measurements of the kinetic energy (E) for a
series of identical wells in a particular eigenstate
because E is an eigenvalue of the KE operator (V = 0) every
measurement of E will be identical to every other one (within
experimental error.
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So much for the energy, now what about measurements of
momentum?
For the classical particle we can write p2 =2mE and since E is an
eigenvalue of T^ we might expect p2 and therefore p to be
eigenvalues of their corresponding operators
Operate on the wavefunction with the momentum operator
d n
d 
Nn
 n x  
 n x 
pˆ x n 

N sin 
cos 
 


i dx
i dx 
 L  i L
 L 
Thus n is not an eigenfunction of the momentum operator (sine
has become cosine on differentiation)
MODULE 4
according to our postulate, a series of measurements of
momentum will yield a distribution of results.
Let’s obtain the expectation value of the momentum operator for
one of the set of wavefunctions, the kth, for a normalized
wavefunction
L
 k x  k 
 k x 
2
pˆ x k  k pˆ k  N  sin 

 cos 
 dx  0
 L  i L 
 L 
0
Thus we find that the expectation value of the momentum
operator is zero
therefore the average value of a large number of measurements
of the momentum will be zero!!!!
MODULE 4
To resolve this apparent paradox we examine the square of the
momentum in the x direction.
The appropriate operator for this is
d
d
pˆ x 
.

i dx i dx
2

2
d2
 k x 
N sin 

2
dx
 L 
2
2
d
dx 2
2 2
k

 k x 
2
N sin 

2
L
L


Thus we see that k is an eigenfunction of the operator for the
square of the momentum
a set of measurements of px2 on identical systems will always
provide the same result, namely the eigenvalue.
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For the n = 1 energy level we have
px2
1

2
2
2
L
 2mE
px 1  (2mE1 )
1/ 2
Now we see the resolution of our paradox.
The quantity p2 (an eigenvalue) is the same for all determinations,
but p can either be +(2mE)1/2 or – (2mE)1/2.
Both of these quantities is an eigenvalue of the momentum
operator, but the wavefunction is not an eigenfunction thereof.
MODULE 4
A single measurement of p will always yield one of these values
if we make a large number of such measurements we shall find an
equal number of the positive and negative values, from which
the average value will be zero.
the momentum solutions appear with equal weights.
We never know in advance whether a single determination will
yield the positive or the negative value; but we know that we
shall find one or the other and that eventually we shall find an
equal number of each.
In terms of the motion of the particle we can imagine that the two
momentum values correspond to motions in the +x and -x
directions.