Applying Classical Geometry Intuition to Quantum Spin

Using concepts of geometric orthogonality and linear independence, we logically deduce the form of the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found by forcing expectation values of the different basis states to have the properties we expect of a classical, geometric coordinate system. The process highlights the correspondence of quantum angular momentum with classical notions of geometric orthogonality, even for the inherently non-classical spin-1/2 system. In the process, differences in and connections between geometrical space and Hilbert space are illustrated.


I. INTRODUCTION
Developing intuition for the quantum description of angular momentum can be very difficult for students, resulting in serious misconceptions. 1 Semi-classical descriptions, such as the vector model of angular momentum, 2 can leverage students' classical intuition. But these models are often misinterpreted or taken too literally.
To help students overcome misconceptions and gain insight, we describe an exercise to supplement traditional methods of teaching quantum angular momentum. Rather than creating a semi-classical model, the exercise emphasizes the correspondence between quantum and classical descriptions of angular momentum. It does not explain why quantum mechanics behaves as it does, but instead focus on working with quantum angular momentum states and on developing intuition.
In this exercise we deduce the relationships between |↑ x , |↓ x , |↑ y , |↓ y , |↑ z , and |↓ z , the components of the x, y, and z basis sets of the spin-1/2 system. Although there are several ways to derive these well-known relationships, including the use of rotation operations, 3,4 ladder operators, 5 or direct diagonalization, 6 in this exercise one deduces, rather than derives, the relationships by considering the properties that the basis states should have.
The exercise makes a connection between the arbitrary choices made when defining a Cartesian coordinate system and a set of arbitrary constants selected in the definition of orthogonal basis sets in Hilbert space. We limit our discussion to a spin-1/2 system for simplicity and clarity, and to avoid dealing with two different types of unbiased states, representing both spatially orthogonal angular momentum states and "angle" states. 7 After reviewing this exercise, several senior-level undergraduates and graduates stated that it had made them aware of misconceptions they didn't know they had -like confusing geometrical orthogonality and linear independence, thinking that the z basis states alone were not a complete basis, or not realizing that basis states made of angular momentum eigenstates along different axes are just as fundamental as the z basis.
Students commented that the reasoning in this exercise was easier to follow than more rigorous derivations, and said that reading through this paper helped solidify their understanding of quantum angular momentum, basis states, and working with Dirac notation. The exercise clarifies the physical significance of quantum phases, the meaning of orthogonality and linear independence, and the connection between "classical" geometric space and Hilbert space.

II. QUANTUM SPIN-1/2 FORMALISM
It is common to describe spin-1/2 particles using a basis consisting of the two eigenstates of the z-component of angular momentum, |↑ z and |↓ z . If a spin-1/2 particle, such as an electron or a proton, is in one of these states, the z component of angular momentum is precisely defined with no uncertainty. We assume that this basis exists and that every possible state of our particle can be expressed as a sum of these two states in the form where a and b are scalar, potentially complex, constants. If we measure the z-component of the particle's angular momentum we will always get either plus or minus /2, with probabilities of a * a and b * b if the state is normalized. To predict the outcome of a measurement along a different axis, we can write our quantum state in terms of the eigenstates of the component of angular momentum we plan to measure. Since the z axis is chosen arbitrarily, we know that a pair of eigenstates of any component of angular momentum must exist and have similar properties to the z basis states.
Two special basis sets are made up of the eigenstates of angular momentum along the x and y axes, respectively. Because the z basis forms a complete set, we should be able to write these basis states in terms of the z states. 8 We'll start by writing the x basis states in the general form where A, B, C, and D are constants. We can deduce what these constants must be simply by considering what properties these states should have and enforcing them using inner products. In the process we will see connections between physical space and Hilbert space.

III. ORTHOGONALITY
To find A and B in Eq. 2 we first note that a classical particle with its angular momentum in the x direction will have no component of angular momentum in the z direction. Of course, in a quantum spin-1/2 system we will always measure the z component to be plus or minus /2, never zero, regardless of the particle's state. So rather than mapping our intuition of geometrical orthogonality onto possible measurement outcomes, to force our x basis states to be geometrically orthogonal to z we will demand that the expectation value for the z component of angular momentum be zero.
For this to be true, we intuitively expect that |↑ x should be an equal superposition of the two z eigenstates so that we are exactly as likely to measure /2 as we are − /2. To show this more rigorously, we will use the operator to find the expectation value of the z component of angular momentum. Remember that, because our two z basis states are normalized and linearly independent, So the effect of operating with S z on a quantum state |↑ ψ is just to multiply the |↑ z and |↓ z components of the state by /2 and − /2. Using this operator to find the expectation value of the z component of angular momentum for a particle in the |↑ x state, we get We require that this expectation value be zero, such that As we expected, A and B must have the same magnitude, such that we have an equal superposition of |↑ z and |↓ z . But the complex phases of A and B are unrestricted. So the most general form that |↑ x can have, subject only to the limitations that it be normalized and geometrically orthogonal to the z basis states, is where φ 1 and φ 2 are arbitrary real constants. It makes sense that we get one arbitrary phase angle, since quantum mechanics always allows us an arbitrary overall phase factor. But why two? There's a classical analog to this freedom. After defining the z axis of a Cartesian coordinate system, the x axis can point in any one of an infinite number of directions which are orthogonal to z. These choices can be parametrized by an angle relative to some reference direction, as shown in Fig. 1. Because we are free to choose φ 1 and φ 2 , we will set them to zero -not just because this is simple, but because it gives us the conventional form of the basis state,

IV. LINEAR INDEPENDENCE
To find |↓ x we note that it, too, must be geometrically orthogonal to the z basis. So it must have the form We find an additional constraint when we consider that the x basis states could have been the z basis states had we simply chosen a different direction for our z axis. As such, since our z basis states are orthogonal to each other, we expect the two states in the x basis to be orthogonal to each other as well.
As is often done, we've unfortunately used the word "orthogonal" to mean two different things. When we say that the x states must be orthogonal to the z states, we are referring to geometric orthogonality. But when we say that the x states must be orthogonal to each other, we refer to orthogonality in Hilbert space or linear independence. Just as a dot product of zero assures geometric orthogonality, an inner product ↑ x | ↓ x = 0 guarantees that |↑ x and |↓ x are linearly independent, such that one can't be written in terms of the other.
The inner product of the x basis states is For this to be zero, the two phases must differ by π radians. If we choose to let φ 3 = 0, both for simplicity and by convention, we find that e iφ4 = −1 and V. THE y BASIS Just as we saw for the x basis states, for the y basis states to be normalized and geometrically orthogonal to the z axis, they must have the form Since we have the freedom to multiply each state by an arbitrary overall phase factor, for simplicity (and to arrive at the canonical form of the states), we can set φ 5 and φ 7 to zero. The other phase angles are constrained by that fact that, in addition to being geometrically orthogonal to z, these states must also be geometrically orthogonal to the x axis we've defined.
The operator which gives us information about the x component of spin can be found by noting that, when written in the x basis, this operator should look similar to S z represented in the z basis: To write this in the z basis we plug in Eqs. (8) and (11) to get We can use S x to find the expectation value of the x component of angular momentum for a particle in the state |↑ y : If we want |↑ y to be geometrically orthogonal to the x states then this must be zero, implying that φ 6 = ±π/2 and giving us only two unique possibilities: As we discuss in the next section, the choice of whether to use the upper or lower sign is not arbitrary, so we'll not select one over the other just yet.
Applying the same condition to |↓ y and forcing the inner product ↑ y | ↓ y to be zero gives us

VI. COORDINATE SYSTEM HANDEDNESS
There is a classical analog to the two possible choices in Eqs. (18) and (19). In a Cartesian coordinate system, once the z and x axes have been selected, there are still two possible directions for y. As illustrated in Fig. 2, one direction results in a right-handed and the other in a left-handed coordinate system.
We can find the handedness of a coordinate system with cross products. The link between the Pauli matrices and quaternions, 9 and between cross products and commutators in quaternion algebra 10 hints that there is a connection between cross products in geometric space and commutators in Hilbert space. So commutators of spin operators could tell us which choice of sign in our y basis states will result in a right-handed coordinate system.
Because there is no spatial representation of spin, we'll make an analogy with the orbital angular momentum operator: Here r is the position and p the momentum operator. Assuming a right-handed coordinate system, the components of L are We can use these components to calculate the commutator of L x with L y : Noting that position and linear momentum operators commute with operators for orthogonal spatial dimen-sions and that [z, p z ] = i , Eq. (24) can be written as If we had chosen a left-handed coordinate system (but still used a right-handed cross product), we would have gotten the same result but with a minus sign, The "anti-commutative" relationship of angular momentum operators reminds us of the cross products of unit vectors in geometric space. For example, ifn j is a unit vector along the j th axis, then for a right-handed coordinate system using a right-handed cross product, n x ×n y =n z . For a left-handed coordinate system we get a minus sign:n x ×n y = −n z .
By analogy, we may suppose that a right-handed coordinate system for a spin-1/2 particle is the one that results in the commutation relation while a left-handed coordinate system would result in a minus sign in the commutation relation. We get the commutation relation in Eq. 27 if we choose the upper sign in Eqs. (18) and (19). This gives us the last piece of the puzzle, and we have now "deduced" the relationships between the x, y, and z basis states for the spin-1/2 system.

VII. CONCLUSION
We deduced the relationships for |↑ x , |↓ x , |↑ y and |↓ y in terms of |↑ z and |↓ z using the concepts of or-thogonality and linear independence. By insisting that the two states in the x basis be normalized, "geometrically" orthogonal to the z states, and orthogonal to each other in Hilbert space, we arrived at expressions which were completely specified except for three arbitrary phase angles -two related to arbitrary overall phase factors, and a third analogous to choosing the direction for the x axis in a Cartesian coordinate system.
With the y basis we had less freedom because the states had to be geometrically orthogonal to both the z and the x basis states. We again had an arbitrary overall phase factor for each basis state. But we only had two possible choices for the remaining phase factors, analogous to the choice of handedness in a Cartesian coordinate system. We determined handedness by making a connection between cross products and commutators.
Many students fail to understand the connection between Hilbert space and classical three-dimensional physical space. This exercise, intentionally not rigorous for the sake of building intuition, can help students overcome misconceptions and develop a more intuitive feel for Hilbert space and the quantum description of angular momentum. We thankfully acknowledge Jean-François Van Huele for very helpful discussions, and we are grateful to Christopher J. Erickson and to the undergraduates and former undergraduate students of our department who gave feedback on this manuscript.