Transcript courses:lecture:wvlec:qmoperators_wiki.ppt

1
BASICS OF QUANTUM MECHANICS
Reading:
QM Course packet – Ch 5
We will state two things without proof, and you'll see why they
are reasonable, later.
1. In the "position representation" or "position basis", the
position operator is represented by the variable x:
xˆ =˙ x
1. In the "position representation" or "position basis", the
momentum operator is represented by the derivative with
respect to x:
d
pˆ =˙ -i
dx
1. This follows if you accept (2). The energy operator is:
2
2
2
ˆ
p
d
Hˆ =
+ Vˆ =˙ + V ( x)
2
2m
2m dx
Now let's think about eigenfunctions of these operators
(worksheet)
2
If the momentum operator operates on a wave function and
IF AND ONLY IF the result of that operation is a constant
multiplied by the wave function, then that wave function is
an eigenfunction or eigenstate of the momentum operator,
and its eigenvalue is the momentum of the particle.
p̂j ( x ) = Cj ( x )
eigenvalue
dj ( x )
iCx /
-i
= Cj ( x ) Þ j ( x ) = Ae
dx
operator
• not all states are eigenstates – and if the are not, they can be
usually be written as superpositions of eigenstates
• if a state is an eigenstate of one operator, (e.g. momentum),
that state is not necessarily an eigenstate of another operator,
though it may be.
3
Look more closely at the momentum eigenfunction (in the
position representation) or momentum eigenstate:
j p ( x ) = Ae
±ipx /
1. Why did we change C to p? And why the subscript?
2. What is the probability distribution for this state?
3. Is it normalized? Normalizable?
4. It is degenerate (new word, maybe?)
5. What sort of particle would be represented by this function?
6. Where is this particle "located"? Could we "improve" the
description of a particle by localizing it?
Position eigenstates:
This is a useful (but a bit pathological) representation of a
position eigenstate: j x' x = d x - x'
( )
1. Normalizable?
2. Otherwise reasonable?
(
)
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5
BASICS OF QUANTUM MECHANICS
REVIEW
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Review language of PH425
Kets and wave functions
Probability density
Operators – position, momentum, energy
Eigenfunctions
Mathematical representations of the above