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Two-Electron System
Consider a system consisting of two electrons. Let
and
represent the position and spin operators of the first electron, respectively,
and let
and
represent the corresponding operators of the second electron. Furthermore,
let
represent the total spin operator of the system. Suppose that the Hamiltonian commutes with
, as is often the case.
It follows that the state of the system is specified by the position eigenvalues
and
, as well as the total spin
quantum numbers
and
. As usual, the eigenvalue of
is
, and the eigenvalue of
is
. (See Chapter 5.)
Moreover,
can only take the values
or
or
or
, respectively. (See Chapter 6.)
The overall wavefunction of the system can be written
|
(9.44) |
where
|
|
(9.45) |
|
|
(9.46) |
|
|
(9.47) |
|
|
(9.48) |
(See Section 6.4.)
Here, the spinor
denotes a state in which
and
, et cetera, where
and
are the eigenvalues
of
and
, respectively. The three
spinors are usually referred to as triplet spinors, whereas the single
spinor is called the singlet spinor. Note that the triplet spinors are all symmetric with respect to exchange of particles, whereas the
singlet spinor is antisymmetric.
Fermi-Dirac statistics requires that the overall wavefunction be antisymmetric with respect to exchange of particles. Now, according to Equation (9.44),
the overall wavefunction can be written as a product of a spatial wavefunction and a spinor. Moreover, when the system is in the spin-triplet
state (i.e.,
), the spinor is symmetric with respect to exchange of particles. On the other hand, when the system is in the spin-singlet
state (i.e.,
), the spinor is antisymmetric. It follows that, to maintain the overall antisymmetry
of the wavefunction, the triplet spatial wavefunction must be antisymmetric with respect to exchange of particles, whereas the singlet
spatial wavefunction must be symmetric. In other words, in the spin-triplet state the spatial wavefunction takes the form
|
(9.49) |
whereas in the spin-singlet state the spatial wavefunction is written
|
(9.50) |
The probability of observing one electron in the volume element
around position
, and the
other in the volume element
around position
, is proportional to
, or
Here, the plus sign corresponds to the spin-singlet state, whereas the minus sign corresponds to the spin-triplet state. We can immediately see that, in the
spin-triplet state, the probability of finding the two electrons at the same point in space is zero. In other words, the two electrons have
a tendency to avoid one another in this state. On the other hand, in the spin-singlet state, there is an enhanced probability of finding the
two electrons at the same point in space (because of the final term in the previous expression). In other words, the two electrons are attracted to
one another in this state.
Note, however, that the spatial probability distributions
associated with the singlet and triplet states differ substantially only when the two single-particle spatial wavefunctions,
and
,
overlap: that is, when there exists a region of space in which the two wavefunctions are simultaneously non-negligible.
Next: Helium Atom
Up: Identical Particles
Previous: Spin Statistics Theorem
Richard Fitzpatrick
2016-01-22