Next: Schrödinger Representation
Up: Position and Momentum
Previous: Poisson Brackets
Consider a simple system with one classical degree of freedom, which corresponds to
the Cartesian coordinate
. Suppose that
is free to take any value (e.g.,
could be the position of a free particle). The classical dynamical variable
is represented in quantum
mechanics as a linear Hermitian operator which is also called
.
Moreover, the operator
possesses eigenvalues
lying in the continuous
range
(since the eigenvalues
correspond to all the possible results of a measurement of
). We can
span ket space using the suitably normalized eigenkets of
.
An eigenket corresponding to the eigenvalue
is denoted
.
Moreover,
[see Equation (85)]

(117) 
The eigenkets satisfy the extremely useful relation [see Equation (87)]

(118) 
This formula expresses the fact that the eigenkets are complete, mutually
orthogonal, and suitably normalized.
A state ket
(which represents a general state
of the system)
can be expressed as a linear superposition of the eigenkets of the position
operator using Equation (118). Thus,

(119) 
The quantity
is a complex function of the position eigenvalue
. We can write

(120) 
Here,
is the famous wavefunction of quantum mechanics.
Note that state
is completely specified by its wavefunction
[because the wavefunction can be used to reconstruct the state ket
using Equation (119)].
It is clear that the wavefunction of state
is simply the collection
of the weights of the corresponding state ket
,
when it is expanded in terms of the eigenkets of the
position operator. Recall, from Section 1.10, that the probability of
a measurement of a dynamical variable
yielding the result
when the system is in state
is given by
, assuming that
the
eigenvalues of
are discrete. This result is easily generalized to dynamical
variables possessing continuous eigenvalues. In fact, the probability of
a measurement of
yielding a result lying in the range
to
when the system is in a state
is
.
In other words, the probability of a measurement of position yielding a
result in the range
to
when the wavefunction of the system is
is

(121) 
This formula is only valid if the state ket
is properly normalized:
i.e., if
. The corresponding normalization for
the wavefunction is

(122) 
Consider a second state
represented by a state ket
and
a wavefunction
. The inner product
can be written

(123) 
where use has been made of Equations (118) and (120). Thus, the inner product of two states is
related to the overlap integral of their wavefunctions.
Consider a general function
of the observable
[e.g.,
].
If
then it follows that

(124) 
giving

(125) 
where use has been made of Equation (117). Here,
is the same function
of the position eigenvalue
that
is of the position operator
:
i.e., if
then
. It follows, from the above result,
that a general state ket
can be written

(126) 
where
is the same function of the operator
that the wavefunction
is of the position eigenvalue
, and the ket
has the
wavefunction
. The ket
is termed the standard ket.
The dual of the standard ket is termed the standard bra, and is
denoted
. It is
easily seen that

(127) 
Note, finally, that
is often shortened to
, leaving
the dependence on the position operator
tacitly understood.
Next: Schrödinger Representation
Up: Position and Momentum
Previous: Poisson Brackets
Richard Fitzpatrick
20130408