Stern-Gerlach Experiment

The Stern-Gerlach experiment is a fundamental experiment in quantum mechanics that demonstrates the quantization of angular momentum. It's results are a direct contradiction to classical mechanics. This page is highly inspired by the section on the Stern-Gerlach experiment in Sakurai's textbook, as well as the Eigenchris video on Jones vectors.

Table of Contents

• Table of Contents
• The Setup and its Results
• Sequential Stern-Gerlach Experiments
  • Case 1: z-z
  • Case 2: z-x
  • Case 3: z-x-z
• Polarization of Light
  • Circular Polarization
• SU(2) States for Polarization
  • Summary of the Groups
• Jones Vectors are Spinors
• Summary and Next Steps
  • References
• Appendix: Magnetism Review
  • Magnetic Moment

The Setup and its Results

Imagine a beam of silver atoms traveling through a magnetic field. In classical mechanics, we would expect the magnetic field to exert a force on the atoms, causing them to deflect.

Stern-Gerlach Experiment

By Tatoute - Own work, CC BY-SA 4.0, Link

The Stern-Gerlach experiment was designed to test this prediction. In the setup, a furnace heats up silver atoms (1 in the image). They escape through a small hole and travel through a collimator (2), which ensures that the beam is narrow and well-defined. Then, the beam passes through an inhomogeneous magnetic field, which causes the atoms to deflect. (An inhomogeneous magnetic field is a magnetic field that varies in strength and direction over space.)

The electronic configuration of silver atoms is (in the ground state). Other than the electron, all other electrons are paired up, so they don't contribute to the angular momentum.

Below, I will provide two explanations for the classical prediction for the Stern-Gerlach experiment: one using some mathematical machinery and the other using an intuitive analogy.

Mathematical Explanation

Let the electron spin angular momentum be . The magnetic moment of the electron is proportional to the angular momentum, so we can write:

The energy of the interaction between the magnetic moment and the magnetic field is given by:

Therefore, the force on the atom is given by the gradient of the energy:

If we assume that the magnetic field is in the -direction, then the force on the atom is given by:

According to classical mechanics, since the atom can have any orientation of the magnetic moment, the -component of the magnetic moment can have any value from to . Therefore, the force on the atom can have any value from to . We would expect the beam to spread out in the -direction (4 in the image).

Intuitive Explanation

Magnetic forces appear when charge is moving. For example, when a current flows through a wire, a magnetic field is generated around the wire. This is how electromagnets work; you coil a wire around a piece of iron, and when you pass a current through the wire, the iron becomes magnetic.

In the Stern-Gerlach experiment, the south-pole of a magnet is placed at the top, above the north-pole of another magnet. The key point is that the magnetic field is inhomogeneous, meaning it varies in strength and direction over space. Specifically, the south-pole part is much stronger than the north-pole part. This means that the magnetic field is stronger at the top than at the bottom.

Suppose you throw a bar magnet into a SG apparatus setup. If it has its north pole pointing up, there would be a strong attraction towards the south pole at the top, and a weaker attraction towards the north pole at the bottom. Hence, the bar magnet would be pulled up.

Conversely, if the bar magnet has its south pole pointing up, there would be a strong repulsion from the south pole at the top, and a weaker repulsion from the north pole at the bottom. Hence, the bar magnet would be pushed down.

If the bar magnet is tilted slightly, it would experience a force that is a combination of the two. Overall, its path would be curved, and it would hit the screen at a point that corresponds to the combination of the two forces. Since you can continuously tilt the bar magnet, you would expect to see a continuous distribution of spots on the screen.

During the times of the SG experiment, the Bohr model of the atom was prevalent. In it, the electron orbits the nucleus in a circular path. Silver atoms have one unpaired electron in the 5th shell - this means that there is charge moving in a circular path. And as I said, a moving charge experiences magnetic forces. In fact, we can model the entire atom as a tiny bar magnet. Therefore, we would expect the beam to spread out in the -direction.

Instead, what happens is that the beam splits into two distinct beams, and only two spots are observed on the screen (5 in the image). This means that in reality, the magnetic moment can only have two values: and . In other words, the angular momentum of the electron can only have two values. Let's call these values the "up" state and the "down" state .

This is a direct contradiction to classical mechanics, which predicts that the magnetic moment can have any value between and .

Sequential Stern-Gerlach Experiments

Now, let's consider what happens when we pass the beam through another Stern-Gerlach apparatus. The goal is to understand how passing the beam through multiple Stern-Gerlach apparatuses affects the angular momentum of the atoms.

To remind ourselves, in the classical picture, the angular momentum is defined as . As long as we know the shape and mass distribution of the object, we can calculate the angular momentum in any direction.

Case 1: z-z

In the first case, we pass the beam through a SG apparatus with the magnetic field in the -direction (we will call this the -SG apparatus). This will cause the beam to split into two beams, one with up-spin and the other with down-spin . We block the down-spin beam and pass the up-spin beam through another -SG apparatus.

Since no external force changes the angular momentum, all the atoms in the up-spin beam will remain up-spin. Thus, only one beam will be observed on the screen.

The experimental result aligns with this prediction.

Sakurai uses a visual diagram to represent the state of the beam, which I replicated below.

Case 2: z-x

In the second case, we pass the beam through a -SG apparatus, block off the down-spin beam, and pass the up-spin beam through an -SG apparatus.

It turns out that after passing through the -SG apparatus, the beam splits into two again, one being and the other being .

What does this mean? One guess is that in the original beam, half of the atoms have both and , and the other half have both and .

The visual representation of this is shown below.

Case 3: z-x-z

Perhaps the most interesting case is when we do what we did in the second case and then pass the beams through another -SG apparatus. To remind ourselves, this is the order of the experiments:

1. Pass through -SG apparatus, splitting into and .
2. Block off .
3. Pass through -SG apparatus, splitting into and .
4. Block off .
5. Pass through -SG apparatus. What happens?

It turns out that the beam splits into two again, one being and the other being . This should be completely unexpected, as we blocked off the down-spin beam in the first step.

The visual representation of this is shown below.

What this tells us is the following: when the -SG apparatus selects for or , the atom loses its previous information. In other words, we cannot simultaneously measure and . This is completely different from classical mechanics, where we can measure the angular momentum in any direction.

Polarization of Light

The Stern-Gerlach experiment has a direct analogy in the polarization of light. Light is an electromagnetic wave, and the electric field of the wave can oscillate in any direction perpendicular to the direction of propagation. The direction of oscillation is called the polarization of the light.

Note that we are using light purely as an analogy to the Stern-Gerlach experiment, in hopes of creating a mathematical framework that can describe both phenomena. In this analogy, we treat light purely classically as an electromagnetic wave. There is a lot to say about the quantum nature of polarization - watch this video for more. In fact, watch it even if you don't care about the quantum nature of polarization.

If a light beam is -polarized, its equation can be written as:

It is often convenient to use complex notation to represent the electric field. Of course, the physical electric field is real, but the complex notation simplifies calculations.

We can also add a phase factor to the electric field:

A -polarized light beam can be written similarly as:

More generally, an electric field can be written as a linear combination of and polarizations:

We can separate the exponential and the polarization vector:

The vector is called the Jones vector.

In order to get -polarized light, we need to pass the light through a polarizer that only allows light oscillating in the -direction to pass through. We can call this an -filter.

In a similar manner to the Stern-Gerlach experiment, we can pass light through a series of filters and observe the polarization of the light:

1. Case 1: Pass through -filter, then -filter.

   In this case, the light will be -polarized after the first filter. Then, no light will pass through the -filter, as it only allows -polarized light to pass through. Thus, the light will be completely blocked.

2. Case 2: Pass through -filter, then a -filter, then -filter.

   In this case, there will actually be light passing through the -filter.

Let's examine the second case in more detail.

Denote the direction of the -filter as . Once the -filter is applied, the light will be -polarized regardless of the initial polarization. Doesn't this seem familiar to the Stern-Gerlach experiment?

SG Experiment	Light Experiment
After SG apparatus: Lose spin information	After filter: Lose polarization information
atoms	, -polarized light
atoms	, -polarized light

The basis vectors and are rotated versions of and , as shown below:

The key is that once light passes through an -filter, its polarization is somewhere on the -axis, meaning it has a component in the -direction. But this also means that it has a component in the -direction.

More precisely speaking, we can write down the coordinate transformation:

Another way of thinking about it is that mathematically, a filter simply projects the Jones vector onto the basis vector of the filter.

Since we can represent the polarization of light as a vector as we have seen, perhaps we can use the same vector representation for the spin of particles. Of course, this means that the particles have a property that can be represented as a vector in a two-dimensional space.

We can write the spin of a particle as a vector in a two-dimensional space. In the -direction, we have the and states, and in the -direction, we have the and states. Just like the polarization, we can rotate the spin states to get the and states:

Next, how do we represent spin in the -direction?

By symmetry, if we pass in the -direction through a -SG apparatus, it should yield the same result as passing in the -direction through an -SG apparatus. But it seems that we have already used up all possible linear combinations of to get . How do we represent ?

Circular Polarization

Another type of polarization is circular polarization, where the electric field rotates in a circle.

To understand this better, let's get a deeper understanding of the Jones vector.

Recall that the Jones vector can be represented by combining the and polarizations:

Circular polarization occurs when there is a phase difference between the and components. For example, consider the simple case where and are both and and . Then, the Jones vector is:

Options
Samples
Position slice	Show Hide
Overall wave	Show Hide

Jones vector:

With imaginary components, it is a bit harder to visualize the Jones vector. If we get the full expression for the electric field, we will see the following:

At the point , for example, if and , then the electric field is:

Using Euler's formula, we can expand it out and then take the real part:

In other words, the electric field rotates in a circle clockwise. Go to the above visualizer and set the following values: , , , , , . If you turn on the "Position Slice" option, you will see a slice of the wave at . I have also added vectors representing the horizontal and vertical components of this slice.

This wave is left-circularly polarized. If one points their left thumb in the direction of propagation, the electric field will rotate in the direction of the fingers.

Overall, the Jones vector for different types of polarization is summarized in the following table:

Polarization	Jones Vector
Linear in
Linear in
Linear at (diagonal)
Linear at (antidiagonal)
Linear at
Left-circular
Right-circular

Circular-polarized light can be created by passing linear-polarized light through a quarter-wave plate, which introduces a phase difference between the and components. Notice that if we pass circular-polarized light through a linear polarizer, we still get linear-polarized light, yet circular-polarized light is different from the -polarized light we used previously. This is perfectly analogous to the states we were looking for - ones which are different from the states but behave similarly. This calls for the following correspondence:

SG Experiment	Light Experiment
After SG apparatus: Lose spin information	After filter: Lose polarization information
atoms	, -polarized light
atoms	, -polarized light
atoms	Circular-polarized light

We can hence write down the states in the same way we wrote down the circular-polarized light:

SU(2) States for Polarization

In the physical world, we transform the polarization of light using wave plates and polarizers. Mathematically, each of these devices corresponds to a transformation on the Jones vector. These transformations can be represented by matrices called Jones matrices.

Wave plates are usually birefringent crystals - crystals that have different refractive indices for different polarizations. This means that light slows down differently depending on the direction, and hence have a different phase shift for different polarizations. This causes the phase difference between the and components of the Jones vector to change, hence creating circularly polarized light.

A quater wave plate - one that introduces a quarter-cycle phase shift in the -component of the Jones vector - can be represented by the matrix:

This changes a diagonal-polarized wave to a left-circular-polarized wave . This can be seen mathematically as follows:

The interesting part of this occurs when we keep applying these transformations:

In other words, the quarter wave plate matrix allows us to rotate between the different types of polarization in the order . In the matrix, we can also replace with a general phase factor to continuously rotate between the different types of polarization. A visual representation of this polarization space is shown below.

Shape of polarized light

Abstract polarization space

Now imagine that we have the same birefringent crystal, but this time we rotate it by . What kind of polarization will it create? The key is that instead of rotating the birefringent crystal, we can keep it stationary, rotate the light by , and then pass it through the crystal. Then, rotate the light back by and it will be our desired polarization.

Mathematically, this is equivalent to applying the following transformations: (1) rotate the light by , (2) apply the quarter wave plate, (3) rotate the light by :

An important thing to note is that the factor is a global phase factor - it just moves the entire wave by a constant phase. It does not apply any physical change to the polarization of the wave, so we can actually ignore it. Starting from horizontally polarized light (), we can apply the above transformation multiple times again to get the following:

In other words, the rotated quarter wave plate matrix allows us to rotate between the different types of polarization in the order . We can rotate continuously between the different types of polarization by using a generic angle in our original matrix multiplication:

Now we have two different matrices that allow us to rotate between the different types of polarization. If we put their circles together, we form a sphere as follows:

The transformations are represented by the matrices:

Blue transformation	Yellow transformation	Green transformation
		?

As you can see, by placing the different polarization states on a sphere, we can identify another transformation that we're missing. This sphere is called the Poincaré sphere. The third transformation is the following:

It turns out that these three matrices all belong to the Unitary Group of matrices, denoted as . When we rotate between the different types of polarization on the Poincaré sphere, the magnitude of the Jones vector remains constant. The squared magnitude of a real vector can be written as , where is the Hermitian adjoint of . If is a rotation vector, then applying to should not change the magnitude of . In other words, . Using the properties of the adjoint, we can expand this out to . Hence, , where is the identity matrix. This is the defining property of a unitary matrix.

If a matrix has only real entries, then its adjoint is the same as its transpose. Given that a, say, real matrix is unitary, it means that the columns of the matrix form an orthonormal basis. We can see this by taking the adjoint of the matrix and multiplying it by the original matrix. Below is an example of a unitary matrix:

These matrices are called orthogonal matrices, and the group of all orthogonal matrices is denoted as . Another type of matrix is a reflection matrix, which flips the sign of one of the basis vectors. It inverts the orientation of space, and hence has a determinant of . If we restrict ourselves to matrices with a determinant of , we get the Special Orthogonal Group of matrices, denoted as .

We can see that the matrices are similar to the matrices. The former are complex matrices, while the latter are real matrices with a determinant of . matrices do not necessarily have a determinant of , only its magnitude is preserved. This can be seen by starting from and taking the determinant of both sides:

Hence, the determinant can be any complex number with a magnitude of , or where is a real number.

We also know that because the global phase shift does not matter, there are multiple matrices that can represent the same transformation. For example, and represent the same transformation. We can isolate one specific matrix by enforcing another constraint - that the determinant of the matrix is exactly .

If the determinant of a matrix is , then we can rescale it by to get a matrix with a determinant of . This is called the special unitary group of matrices, denoted as .

Summary of the Groups

Below is a diagram that summarizes the different groups of matrices - , , , and . It is taken from the Eigenchris video.

: Unitary Group of Matrices, Preserves the magnitude of the vector;
	Special Unitary Group of Matrices,
: Orthogonal Group of Matrices, Preserves the magnitude of real vectors;
(Reflections)	(Rotations) Special Orthogonal Group of Matrices,

Jones Vectors are Spinors

The Jones vectors we have been using to represent the polarization of light are actually mathematical objects called spinors. To see why, consider the horizontal and vertical polarization vectors and .

In physical space, these are apart, yet on the Poincaré sphere, they are apart. In other words, starting from , a quarter-turn in physical space leads to , the same result as a half-turn on the Poincaré sphere. Then, taking another quarter-turn in physical space leads to , and on the Poincaré sphere, it leads back to . The signs do not matter since if you remember, the global phase shift does not matter.

This is the characteristic of a spinor - it requires a rotation to return to its original state. A rotation of on the Poincaré sphere corresponds to a rotation of in physical space.

Summary and Next Steps

This was quite a tough "introduction" to quantum mechanics. We started with the Stern-Gerlach experiment, which led us to the concept of spin states. Then, we postulated that the spin states are similar to the polarization states of light.

Here are the key points to remember:

• The Stern-Gerlach experiment is a thought experiment that separates particles based on their spin.
• From the SG experiment, it is clear that angular momentum is quantized. That is, particles can only have certain values of angular momentum.
• Performing sequential SG experiments in different directions leads to the conclusion that spin states can only exist in one direction. That is, when we measure or set the spin of a particle in one direction, the spin in the other directions is lost.
• The spin states are similar to the polarization states of light. The states are similar to the and polarizations, while the states are similar to the and polarizations.
• Mathematically speaking, the two spin states are members of a two-dimensional complex vector space and act as a basis for the space.
• The states are similar to circularly polarized light.
• The polarization of light can be represented by the Jones vector, which can be transformed by matrices called Jones matrices. They can be represented in an abstract space known as the Poincaré sphere.
• Jones matrices are elements of the group, the unitary group of matrices.
• The group is similar to the group, the special orthogonal group of matrices.
• Jones vectors rotate twice in abstract polarization space as they do in physical space, making them spinors.

While it may seem unnecessary to go through all of the matrix math and group theory, it gives us a huge headstart in understanding quantum mechanics. It will become very important when we study angular momentum in detail.

In the next chapter, we will apply the different matrix transformations back to the Stern-Gerlach experiment to see how they affect the spin states of particles.

References

• J.J. Sakurai, "Modern Quantum Mechanics", section 1.1 ("The Stern-Gerlach Experiment").
• Eigenchris "Spinors for Beginners", videos 2 and 3.

Appendix: Magnetism Review

Magnetic Moment

The magnetic moment of a particle is a measure of the strength and orientation of its magnetic field. It is a vector quantity, denoted by . The direction of the magnetic moment is the same as the direction of the magnetic field that the particle generates.

Mathematically, the magnetic moment is defined as:

where is the current flowing through the loop and is the area vector of the loop.

If one needs more detail, virtually any introductory electromagnetism textbook will have a section on magnetic moments.