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 that is also called
.
Moreover, the operator
possesses eigenvalues
lying in the continuous
range
(because 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,
![$\displaystyle \langle x' \vert x''\rangle = \delta(x'-x'').$](img347.png) |
(2.26) |
[See Equation (1.88).]
The eigenkets satisfy the extremely useful relation
![$\displaystyle \int_{-\infty}^{+\infty} d x' \, \vert x'\rangle\langle x'\vert= 1.$](img348.png) |
(2.27) |
[See Equation (1.94).]
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 (2.27). Thus,
![$\displaystyle \vert A\rangle = \int_{-\infty}^{+\infty} dx' \,\langle x'\vert A\rangle \vert x'\rangle$](img349.png) |
(2.28) |
The quantity
is a complex function of the position eigenvalue
. We can write
![$\displaystyle \langle x'\vert A\rangle = \psi_A(x').$](img351.png) |
(2.29) |
Here,
is the famous wavefunction of quantum mechanics [99,32].
Note that state
is completely specified by its wavefunction
[because the wavefunction can be used to reconstruct the state ket
using Equation (2.28)].
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 (a properly normalized) 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
![$\displaystyle P(x', dx') = \vert\psi_A(x')\vert^{\,2}\, dx'.$](img356.png) |
(2.30) |
This formula is only valid if the state ket
is properly normalized:
that is, if
. The corresponding normalization for
the wavefunction is
![$\displaystyle \int_{-\infty}^{+\infty} dx'\, \vert\psi_A(x')\vert^{\,2}= 1.$](img357.png) |
(2.31) |
Consider a second state
represented by a state ket
and
a wavefunction
. The inner product
can be written
![$\displaystyle \langle B\vert A\rangle = \int_{-\infty}^{+\infty} dx'\,\langle B...
...\vert A \rangle = \int_{-\infty}^{+\infty} dx'\,\psi_B^\ast (x') \,\psi_A'(x'),$](img359.png) |
(2.32) |
where use has been made of Equations (2.27) and (2.29). 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
giving
![$\displaystyle \psi_B(x') = f(x')\, \psi_A(x'),$](img365.png) |
(2.34) |
where use has been made of Equation (2.26). (See Exercise 3.) Here,
is the same function
of the position eigenvalue
that
is of the position operator
.
For instance, if
then
. It follows, from the previous result,
that a general state ket
can be written
![$\displaystyle \vert A\rangle = \psi_A(x) \rangle,$](img368.png) |
(2.35) |
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
![$\displaystyle \langle \psi_A^{\,\ast}(x) \stackrel{\rm DC}{\longleftrightarrow} \psi_A(x)\rangle.$](img373.png) |
(2.36) |
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
2016-01-22