Chapter 5: Combining two spin-1/2 particles

Note

I think I can safely say that nobody really understands quantum mechanics – Richard Feynman

Summary

Attention

Up to now, we were only concerned with single-particle systems. We thoroughly characterized their angular momentum and developed a formal proof to determine their spectral properties (i.e., the eigenvalues of their angular momentum operators). We even looked at interactions between the single particles and an external field (e.g., magnetic or electric field). Here, we start looking at what happens when dealing with a collection of particles. We will start humbly: we will look at systems of two particles. Yet, this will already open our eyes on some of the most intriguing properties of quantum mechanics, including entanglement and so-called spooky action at a distance.

First, we introduce a new notation. Here, students are sometimes confused: we have introduced the ket vector for single particle in Chapter 1: Stern-Gerlach Experiments. We also remember that we used labels in the ket notations to express information we may have on the state of the particle. Here, we continue indexing the states, but using properties for the two particles. Note that in departure from the excellent textbook used in this course, I suggest mixing commas (to separate properties of a given particle) and semi-colons (to separate properties of different particles), for example:

$\ket{+\mathbf{z}; +\mathbf{z}}=\ket{s_{1}=\frac{1}{2}, m_{1}=+\frac{1}{2}; s_{2}=\frac{1}{2}, m_{2}=+\frac{1}{2}}$

It is worth examining this notation in details: this state represents a system of two particles. each with a total spin-1/2 and with a projection 1/2 along the z axis. The semi-colon is used to separate the information on the two particles. When we have two such particles, we can represent the state of the system using the following basis:

$\begin{array}{llll} |+\mathbf{z};+\mathbf{z}\rangle, & |+\mathbf{z};-\mathbf{z}\rangle, & |-\mathbf{z};+\mathbf{z}\rangle, & |-\mathbf{z};-\mathbf{z}\rangle \end{array}$

It is most useful to try to understand that the states built using that basis live in a space that is a combination of the two single-particle spaces.

$|+\mathbf{z},-\mathbf{z}\rangle=|+\mathbf{z}\rangle_{1} \otimes|-\mathbf{z}\rangle_{2}$

The direct product ( $\otimes$ ) sign clearly indicates that we are combining two single-state particles into one two-particle state. This may look innocuous but it is not as we shall see in this chapter.

An obvious question is to consider the total angular momentum operator that characterizes the two-particle states. The best way to avoid making silly mistakes, is to consider the total spin operator as:

$\hat{\mathbf{S}}=\hat{\mathbf{S}}_{1} \otimes 1+1 \otimes \hat{\mathbf{S}}_{2}$

You can read this equation as follows: The total spin angular momentum is equivalent to the sum of the following operations: (1) “do nothing on particle #2 and operate $\hat{S}_1$ on the state of particle 1” and (2) “do nothing on particle #1 and operate $\hat{S}_2$ on the state of particle 2”. Because this combined operator obeys the general properties of an angular momentum (notably, the commutation relationships), we have:

$\begin{array}{l} \hat{\mathbf{S}}^{2}|s, m\rangle=s(s+1) \hbar^{2}|s, m\rangle \\ \hat{S}_{z}|s, m\rangle=m \hbar|s, m\rangle \end{array}$

In the previous equations, note that $s$ and $m$ now correspond to the spin of the combined system! After some mathematical manipulation, we realize that combining two spin-1/2 particles can yield two types of states: one state with spin-0 and a triply degenerate state with spin-1 (we note that the dimensions of 4 makes sense since it is the dimension of the basis):

$\begin{aligned} |1,1\rangle &=|+\mathbf{z};+\mathbf{z}\rangle \\ |1,0\rangle &=\frac{1}{\sqrt{2}}|+\mathbf{z};-\mathbf{z}\rangle+\frac{1}{\sqrt{2}}|-\mathbf{z};+\mathbf{z}\rangle \\ |1,-1\rangle &=|-\mathbf{z};-\mathbf{z}\rangle \\ |0,0\rangle &=\frac{1}{\sqrt{2}}|+\mathbf{z};-\mathbf{z}\rangle-\frac{1}{\sqrt{2}}|-\mathbf{z};+\mathbf{z}\rangle \end{aligned}$

Once again, it is important to properly understand the notations (in fact, if you take time to understand notations, you are usually halfway there!). On the left-hand side of the equation above, we use the notation $\ket{\lambda,m}$ introduced in Chapter 3: Angular momentum, where the labels correspond to the total spin and its projection on the z axis, respectively.

These eigenstates allow us to introduce a very important definition: a pure state is a state that can be expressed as the direct product of single-particle states. In contrast, an entangled state is a state that cannot be expressed as the direct product of single-particle states.

The existence of entangled states and the probabilitic interpretation of quantum mechanics lead to the famous EPR paradox (Physical Review, 1935). In that paper, EPR were concerned by the fact that the properies of the indidual were (1) not determined before measurement (counterfactual definiteness) and (2) a measurement performed at a distance would determine the outcome of another property (locality). These arguments are rooted in our every-day experience of the real world and it was proposed that a hidden variable theory could resolve this paradox. Howerver, in 1964, this paradox was resolved using Bell’s theorem, which showed that quantum mechanics is not “counterfactual definite and local”. The so-called spooky action at a distance was experimetnally demonstrated for entangled photons in 1981 by Alain Aspect and coworkers. It has been demonstrated many times since then with other particles as well.

This result is mind-boggling! However, and this is where science has to be understood as a ’knowledge in motion” (that is: based on what we learn from new developments not available at a given time). There is little doubt that if Bell’s theorem and Aspect’s experiments had been parts of the early 20’s century “established knowledge”, no paradox would have ensued. This is also humbling (and scientists must always remain humble!): we may be ready to revise our understanding in light of new data. It is not because something is uncomfortable (i.e., not part of our everyday experience), that it is wrong!

Finally, the chapter concludes with the thought experiment of teleportation (side note: cloning is not possible in quantum mechanics); using the so-called Bell’s basis (which is a basis using only entangled basis vectors). Again, this prediction was confirmed experimentally (see, e.g., this website where information on historical demonstrations and additional background is provided).

Note

Quick recap on types of products we have seen thus far in this course

Direct product: $|+\mathbf{z};-\mathbf{z}\rangle=|+\mathbf{z}\rangle_{1} \otimes|-\mathbf{z}\rangle_{2} \equiv |+\mathbf{z}\rangle_{1}|+\mathbf{z}\rangle_{2}$ (combine two states into a new one)
Inner product: $\langle+z \mid+x\rangle$ (bra(c)ket, yields a scalar)
Outer product: $|+\mathbf{z}\rangle\langle+\mathbf{z}|$ (dyad, yields an operator)

Warning

In this course, we have talked about a 2-state system and a 2-particle state. Let’s make sure we understand that these are two difference concepts. The former corresponds to a single-particle system that can only be found in two distinct states. The latter is the state corresponding to the combination of two particles.

Learning Material

Copy of Slides

The slides for Chapter 5 are available in pdf format here: 📂.

Screencast

Key Learning Points

Spin-spin interaction: $\hat{H}=\frac{2 A}{\hbar^{2}} \hat{\mathbf{S}}_{1} \cdot \hat{\mathbf{S}}_{2}$ . This interaction comes from the fact a system that has an intrinsic magnetic moment interacts with the magnetic field associated with the magnetic moment of the other particle.
Eigen-states of the angular momentum of a two-particle system made of two spin-1/2 particles:

$\begin{aligned} |1,1\rangle &=|+\mathbf{z};+\mathbf{z}\rangle \\ |1,0\rangle &=\frac{1}{\sqrt{2}}|+\mathbf{z};-\mathbf{z}\rangle+\frac{1}{\sqrt{2}}|-\mathbf{z};+\mathbf{z}\rangle \\ |1,-1\rangle &=|-\mathbf{z};-\mathbf{z}\rangle \\ |0,0\rangle &=\frac{1}{\sqrt{2}}|+\mathbf{z};-\mathbf{z}\rangle-\frac{1}{\sqrt{2}}|-\mathbf{z};+\mathbf{z}\rangle \end{aligned}$
Raising and lowering operators: $\hat{S}_{+}=\hat{S}_{1+}+\hat{S}_{2+}$ and $\hat{S}_{-}=\hat{S}_{1-}+\hat{S}_{2-}$ .
We can use any basis one wishes to use to represent combined states, so long as it is complete. For example, this basis is a basis of pure states:

${\ket{+z;+z}, \ket{+z;-z}, \ket{-z;+z}, \ket{-z;-z}}$

This is the basis we used to find the eigenstates of the total angular momentum operator. However, we can choose also a basis with entangled states, the so-called Bell’s basis:

$\begin{array}{l} \left|\Psi_{12}^{(\pm)}\right\rangle=\frac{1}{\sqrt{2}}|+\mathbf{z}\rangle_{1}|-\mathbf{z}\rangle_{2} \pm \frac{1}{\sqrt{2}}|-\mathbf{z}\rangle_{1}|+\mathbf{z}\rangle_{2} \\ \left|\Phi_{12}^{(\pm)}\right\rangle=\frac{1}{\sqrt{2}}|+\mathbf{z}\rangle_{1}|+\mathbf{z}\rangle_{2} \pm \frac{1}{\sqrt{2}}|-\mathbf{z}\rangle_{1}|-\mathbf{z}\rangle_{2} \end{array}$
The teleportation experiment is based on a measurement corresponding to a projection on one of the Bell’s basis vector.

Test your knowledge

How would you write a three-particle state obtained from the combination of three spin-1/2 particles?
Write a basis to represent the three-particle states of question 1.
Verify the action of the raising and lowering operators on that the eigenstates of the total angular momentum for the two-particle (spin-1/2) states.
Consider the four eigenstates of the total angular momentum and express them in the ${\ket{+y},\ket{-y}}$ basis. Interpret your result, espcually the important of the $\pm$ sign found in

$\begin{aligned} |1,0\rangle &=\frac{1}{\sqrt{2}}|+\mathbf{z};-\mathbf{z}\rangle+\frac{1}{\sqrt{2}}|-\mathbf{z};+\mathbf{z}\rangle \\ |0,0\rangle &=\frac{1}{\sqrt{2}}|+\mathbf{z};-\mathbf{z}\rangle-\frac{1}{\sqrt{2}}|-\mathbf{z};+\mathbf{z}\rangle \end{aligned}$
Build the representation of the $\hat{S}_y$ operator in the basis of the eigenstates of $\hat{S}_z$ of the two-particle (spin-1/2) systems.
In the hydrogen atom, the hyperfine Hamiltonian represents:
1. Something that is of extremely high-quality.
2. The interaction between the magnetic moment of the proton and that of the electron.
3. The electrostatic interaction between the proton and the electron due to their electrical charges
Imagine 2 particles of spin- $\frac{1}{2}$ , the dimension of the corresponding space where their state vectors live is:
1. 1
2. 2
3. 3
4. 4
Imagine 1 particle of spin-1. How many possible projections of the spin angular momentum can you get?
1. 1
2. 2
3. 3
4. 4
Imagine 2 particles of spin-1. What is the dimension of the corresponding space where their combined state vectors live?
1. 1
2. 3
3. 6
4. 9

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: 5.1, 5.2, 5.5, 5.14

Recitation Assignment

Solve the following problems from the textbook: 5.4, 5.6, 5.7