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[15]
1.
Definitions and terms
— Define each of the following terms or concepts, as used in this course,
in one or two sentences written in your test booklet. Give the significance or importance as well, and
provide equations if immediately relevant.
[Note:
my answers here are more full than I expected from you.]
a)
stationary state, in quantum mechanics
ANS: A
stationary state
, in QM as in oscillations and waves, is a waveform for which every part changes with the same
single frequency. It can be written as a product of spatial and temporal terms, which is the same as saying the variables
x
and
t
are separable:
Ψ
(
x
,
t
)
=
ψ
(
x
)
φ
(
t
)
=
ψ
(
x
)exp
i
E
t
In QM, this is significant because it means that the probability density distribution is independent of time. This means, for
instance, that there is no change in the ‘electron cloud’ of a state in an atom, and for this reason we avoid the problem that
a classical electron orbiting a nucleus must be accelerating and radiating away energy. Thus, a stationary state can be
stable, everlasting — which makes sense because with a single frequency it follows that
Δ
E
= 0, and we actually require
therefore that
Δ
t
=
. (These stationary states also always provide for us a
vector-space basis
to express any general
wavefunction solution.)
b)
well-defined observable
ANS: A
well-defined observable
is an property or measurable quantity of a given wavefunction for which the uncertainty
is zero. In QM this is exactly associated with an eigenproblem relationship between the observable and wavefunction:
ˆ
O
Ψ
(
x
,
t
)
=
O
Ψ
(
x
,
t
)
where
ˆ
O
is an operator and
O
is the unique contant (
eigenvalue
) which it returns, when it operates on the wavefunction
(
eigenfunction
). It’s important, for example, in stationary states, for which the
energy
is well-defined, and therefore we can
speak of precise energy-levels.
c)
expectation value
ANS: An
expectation value
is a number <
O
> determined by an operator
ˆ
O
and wavefunction
Ψ
(
x
,
t
)
through the
following formula:
O
=
Ψ
*
all space
ˆ
O
Ψ
dx
It is the mathematical description of how observables are ‘extracted’ from the wavefunction (which contains all available
information). Therefore it is our theoretical prediction of what result an experimental measurement corresponding to
operator
ˆ
O
will produce. [Note: experimentally, we will average together many repeated measurements to give an
experimental value with measurement error as well as the intrinsic uncertainty;
however, the expectation value is not
exactly a weighted average of the operator by the probability density distribution — unless
ˆ
O
and
Ψ
commute
, i.e.,
ˆ
O
Ψ
=
Ψ
ˆ
O
, which is true if the operator multiplies as does
x
, and not true for an operator like
ˆ
p
=
i
x
.
[20]
2.
Multiple Choice
— In each part below, select the
best
single choice, and write your answer in
your test booklet.
No explanations are required; part-marks may be awarded for partially correct
answers.
All parts (i)–(iv) below have equal weight.
[Note:
no incorrect answers were given part-marks in this test;
correct answers scored 5 marks.]
i)
Which single statement below is
most true of the spatial wavefunction
ψ
(x)
shown at right?
C. it could be associated with a
rectangular well of potential 0
within two sides
U
0
, with
U
0
less than the energy
E
of the
wavefunction.


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