Spin will turn out to have a major effect on how quantum particles behave. Therefore, quantum mechanics as discussed so far must be generalized to include spin. Just like there is a probability that a particle is at some position , there is the additional probability that it has spin angular momentum in an arbitrarily chosen -direction and this must be included in the wave function. This section discusses how.
The first question is how spin should be included in the wave function
of a single particle. If spin is ignored, a single particle has a
wave function , depending on position and on
time . Now, the spin is just some other scalar
variable that describes the particle, in that respect no different
from say the -position of the particle. The “every possible combination” idea of allowing every possible
combination of states to have its own probability indicates that
needs to be added to the list of variables. So the complete wave
function of the particle can be written out fully as:
(5.16) |
But note that there is a big difference between the spin
coordinate
and the position coordinates: while the
position variables can take on any value, the values of are
highly limited. In particular, for the electron, proton, and neutron,
can only be or , nothing
else. You do not really have a full axis
, just
two points.
As a result, there are other meaningful ways of writing the wave
function. The full wave function can be thought
of as consisting of two parts and that only depend
on position:
(5.17) |
The two-dimensional vector is called a “spinor” to indicate that its components do not change like
those of ordinary physical vectors when the coordinate system is
rotated. (How they do change is of no importance here, but will
eventually be described in derivation {D.69}.) The spinor
can also be written in terms of a magnitude times a unit vector:
This book will just use the scalar wave function
; not a vector one. But it is often convenient
to write the scalar wave function in a form equivalent to the vector
one:
(5.18) |
spin-upfunction and the
spin-downfunction are in some sense the equivalent of the unit vectors and in normal vector analysis; they have by definition the following values:
The function arguments will usually be left away for conciseness, so
that
Key Points
- Spin must be included as an independent variable in the wave function of a particle with spin.
- Usually, the wave function of a single particle with spin will be written as
where determines the probability of finding the particle near a given location with spin up, and the one for finding it spin down.
- The functions and have the values
and represent the pure spin-up, respectively spin-down states.
What is the normalization requirement of the wave function of a spin particle in terms of and ?
Inner products are important: they are needed for finding
normalization factors, expectation values, uncertainty, approximate
ground states, etcetera. The additional spin coordinates add a new
twist, since there is no way to integrate over the few discrete points
on the spin axis
. Instead, you must sum over these
points.
As an example, the inner product of two arbitrary electron wave
functions and is
In other words, the inner product with spin evaluates as
Another way of looking at this, or maybe remembering it, is to note
that the spin states are an orthonormal pair,
Key Points
- In inner products, you must sum over the spin states.
- For spin particles:
which is spin-up components together plus spin-down components together.
- The spin-up and spin-down states and are an orthonormal pair.
Show that the normalization requirement for the wave function of a spin particle in terms of and requires its norm to be one.
Assume that and are normalized spatial wave functions. Now show that a combination of the two like is a normalized wave function with spin.
There is no known internal physical mechanism
that
gives rise to spin like there is for orbital angular momentum.
Fortunately, this lack of detailed information about spin is to a
considerable amount made less of an issue by knowledge about its
commutators.
In particular, physicists have concluded that spin components satisfy
the same commutation relations as the components of orbital angular
momentum:
Further, spin operators commute with all functions of the spatial
coordinates and with all spatial operators, including position, linear
momentum, and orbital angular momentum. The reason why can be
understood from the given description of the wave function with spin.
First of all, the square spin operator just multiplies the
entire wave function by the constant , and
everything commutes with a constant. And the operator of spin
in an arbitrary -direction commutes with spatial functions and
operators in much the same way that an operator like
commutes with functions depending on and
with . The -component of spin
corresponds to an additional axis
separate from the
, , and ones, and only affects the
variation in this additional direction. For example, for a particle
with spin one half, multiplies the spin-up part of the wave
function by the constant and by
. Spatial functions and operators commute with
these constants for both and hence commute with
for the entire wave function. Since the -direction is
arbitrary, this commutation applies for any spin component.
Key Points
- While a detailed mechanism of spin is missing, commutators with spin can be evaluated.
- The components of spin satisfy the same mutual commutation relations as the components of orbital angular momentum.
- Spin commutes with spatial functions and operators.
Are not some commutators missing from the fundamental commutation relationship? For example, what is the commutator ?
The extension of the ideas of the previous subsections towards multiple
particles is straightforward. For two particles, such as the two
electrons of the hydrogen molecule, the full wave function follows from
the every possible combination
idea as
(5.22) |
Restricting the attention again to spin particles like
electrons, protons and neutrons, there are now four possible spin
states at any given point, with corresponding spatial wave functions
(5.23) |
The wave function can be written using purely spatial functions and purely spin functions as
The inner product now evaluates as
(5.24) |
Key Points
- The wave function of a single particle with spin generalizes in a straightforward way to multiple particles with spin.
- The wave function of two spin particles can be written in terms of spatial components multiplying pure spin states as
where the first arrow of each pair refers to particle 1 and the second to particle 2.
- In terms of spatial components, the inner product evaluates as inner products of matching spin components:
- The four spin basis states , , , and are an orthonormal quartet.
As an example of the orthonormality of the two-particle spin states, verify that is zero, so that and are indeed orthogonal. Do so by explicitly writing out the sums over and .
A more concise way of understanding the orthonormality of the two-particle spin states is to note that an inner product like equals , where the first inner product refers to the spin states of particle 1 and the second to those of particle 2. The first inner product is zero because of the orthogonality of and , making zero too.
To check this argument, write out the sums over and for and verify that it is indeed the same as the written out sum for given in the answer for the previous question.
The underlying mathematical principle is that sums of products can be factored into separate sums as in:
As an example, this section considers the ground state of the hydrogen
molecule. It was found in section 5.2 that the ground
state electron wave function must be of the approximate form
Including spin, the ground state wave function must be of the general
form
So the approximate ground state including spin must take the form
(5.25) |
Key Points
- The electron wave function for the hydrogen molecule derived previously ignored spin.
- In the full electron wave function, each spatial component must separately be proportional to .
Show that the normalization requirement for means that
In the case of two particles with spin , it is often
more convenient to use slightly different basis states to describe the
spin states than the four arrow combinations ,
, , and . The more
convenient basis states can be written in ket notation,
and they are:
The and states can be written as
Incidentally, note that components of angular momentum simply add up, as the Newtonian analogy suggests. For example, for , the spin angular momentum of the first electron adds to the of the second electron to produce zero. But Newtonian analysis does not allow square angular momenta to be added together, and neither does quantum mechanics. In fact, it is quite a messy exercise to actually prove that the triplet and singlet states have the net spin values claimed above. (See chapter 12 if you want to see how it is done.)
The spin states and that apply for a single spin- particle are often referred to as the “doublet” states, since there are two of them.
Key Points
- The set of spin states , , , and are often better replaced by the triplet and singlet states , , , and .
- The triplet and singlet states have definite values for the net square spin.
Like the states , , , and ; the triplet and singlet states are an orthonormal quartet. For example, check that the inner product of and is zero.