Subsections

5.5 Multiple-Particle Systems Including Spin

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.

5.5.1 Wave function for a single particle with spin

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)

The value of gives the probability of finding the particle within a vicinity of and with spin angular momentum in the -​direction .

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)

These two parts can in turn be thought of as being the components of a two-di­men­sion­al vector that only depends on position:

Remarkably, Dirac found that the wave function for particles like electrons has to be a vector, if it is assumed that the relativistic equations take a guessed simple and beautiful form, like the Schrö­din­ger and all other basic equations of physics are simple and beautiful. Just like relativity reveals that particles should have build-in energy, it also reveals that particles like electrons have build-in angular momentum. A description of the Dirac equation is in chapter 12.12 if you are curious.

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)

The square magnitude of function gives the probability of finding the particle near a position with spin-up. That of gives the probability of finding it with spin-down. The spin-up function and the spin-down function 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

is the way the wave function of, say, an electron will normally be written out.

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.

5.5.1 Review Questions
1.

What is the normalization requirement of the wave function of a spin particle in terms of and ?

5.5.2 Inner products including spin

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

or writing out the two-term sum,

The individual factors in the integrals are by definition the spin-up components and and the spin down components and of the wave functions, so:

In other words, the inner product with spin evaluates as

 (5.19)

It is spin-up components together and spin-down components together.

Another way of looking at this, or maybe remembering it, is to note that the spin states are an orthonormal pair,

 (5.20)

as can be verified directly from the definitions of those functions as given in the previous subsection. Then you can think of an inner product with spin as multiplying out as:

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.

5.5.2 Review Questions
1.

Show that the normalization requirement for the wave function of a spin particle in terms of and requires its norm to be one.

2.

Assume that and are normalized spatial wave functions. Now show that a combination of the two like is a normalized wave function with spin.

5.5.3 Commutators including 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:

 (5.21)

These equations are called the “fundamental commutation relations.” As will be shown in chapter 12, a large amount of information about spin can be teased from them.

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.

5.5.3 Review Questions
1.

Are not some commutators missing from the fundamental commutation relationship? For example, what is the commutator ?

5.5.4 Wave function for multiple particles with spin

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)

The value of gives the probability of simultaneously finding particle 1 within a vicinity of with spin angular momentum in the -​direction , and particle 2 within a vicinity of with spin angular momentum in the -​direction .

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)

For example, gives the probability of finding particle 1 within a vicinity of with spin up, and particle 2 within a vicinity of with spin down.

The wave function can be written using purely spatial functions and purely spin functions as

As you might guess from this multi-line display, usually this will be written more concisely as

by leaving out the arguments of the spatial and spin functions. The understanding is that the first of each pair of arrows refers to particle 1 and the second to particle 2.

The inner product now evaluates as

This can be written in terms of the purely spatial components as
 (5.24)

It reflects the fact that the four spin basis states , , , and are an orthonormal quartet.

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.

5.5.4 Review Questions
1.

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 .

2.

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:

This is similar to the observation in calculus that integrals of products can be factored into separate integrals:

5.5.5 Example: the hydrogen molecule

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

where was the electron ground state of the left hydrogen atom, and the one of the right one; was just a normalization constant. This solution excluded all consideration of spin.

Including spin, the ground state wave function must be of the general form

As you might guess, in the ground state, each of the four spatial functions , , , and must be proportional to the no-spin solution above. Anything else would have more than the lowest possible energy, {D.24}.

So the approximate ground state including spin must take the form

 (5.25)

where , , , and are constants.

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 .

5.5.5 Review Questions
1.

Show that the normalization requirement for means that

5.5.6 Triplet and singlet states

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:

 (5.26)

A state has net spin , giving a net square angular momentum , and has net angular momentum in the -​direction . For example, if the two particles are in the state , the net square angular momentum is , and their net angular momentum in the -​direction is .

The and states can be written as

This shows that while they have zero angular momentum in the -​direction; they do not have a value for the net spin: they have a 50/50 probability of net spin 1 and net spin 0. A consequence is that and cannot be written in ket notation; there is no value for . (Related to that, these states also do not have a definite value for the dot product of the two spins, {A.10}.

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.

5.5.6 Review Questions
1.

Like the states , , , and ; the triplet and singlet states are an orthonormal quartet. For example, check that the inner product of and is zero.