In this video, we will discuss Stern-Gerlach experiment. We first want to introduce the electron spin, electron is known to have intrinsic angular momentum that is not associated with its orbital motion. And that is even when it is not rotating about any axis, it has angular momentum. This intrinsic angular momentum is called spin and just like orbital angular momentum, spin angular momentum produces magnetic moment. And therefore electrons interact with magnetic field through both orbital and spin angular momentum. So here in the figure down below, we depict these electron spin state corresponding to these counterclockwise rotation about these axis. In this case the electron spin will point upward and then spin pointing upward will produce a magnetic moment which is in the opposite direction. That is the N pole points down and then S pole is on the up side. And the other case, spin down case corresponds to the opposite rotation and producing the N pole pointing upward and S pole downward. It should be emphasized that electron is not actually spinning, electron produces angular momentum as if it were spinning. And we call that spin, but that does not mean that electron is actually spinning. It only produce, it only exhibits angular momentum corresponding to or similar to a spinning motion. So now we discuss Stern-Gerlach experiment. And the apparatus is shown here on the right and there is an electron source, typically a hot source. So you basically heat up a filament and it produces electrons. And it goes through some culminating optics and and go through a region of non-uniform magnetic field. And then the final electron position is recorded on this detector screen. Now, if the incident electron coming into this non-uniform magnetic field region, if that electrons have spin up state, then the N pole points down. And therefore electrons will be pushed downward in this apparatus. And if the incident electron has spin down state then it will be pushed upwards. If the incident electron has horizontally aligned spin state, then it will not experience any net force. And it will simply pass through this non-uniform magnetic field region without its trajectory modified. So if the electrons are produced from this oven thermal source with random spin state. Then we should expect a uniform more or less a uniform distribution on the screen as shown here in figure four here. However, in actual experiments we record only two dots as shown in five here, and that means that the electron spin is quantized. It's either spin up or down. It does not have any in between state. So, the Stern-Gerlach experiment shows that the angular momentum state, intrinsic angular momentum state of electron is quantized into either up or down. Even when we rotate the magnet so that the N and S poles of these magnet is aligned in some other axis than z, we still get the same results. That is we get these two dots on the screen. These two dots will simply be rotated together with the rotating magnetic field. Now here we want to take this experiment further and consider sequential Stern-Gerlach experiment. So here is the electron source producing randomly distributed spin states. And it goes through the first set of Stern-Gerlach experiment. And this apparatus here that we're using is aligned along the z-axis and therefore the resulting electrons will be either spin up or spin down. Now we block these spin down states and let only the spin up electrons pass through. And we have a second set of Stern-Gerlach apparatus also aligned along the z-axis. In this case all of the incident electrons have spin upstate already and therefore we should expect that we only get spin up state and there is no spin down state. And that is what the experiments show. Now, what we do is we repeat the same experiments except the last, the second Stern-Gerlach experiment apparatus is now aligned along x-axis. Now in this case, classically thinking you have a spin state aligned along z coming in, and it does not have any net angular momentum component along the x-axis. So what should you expect from these spin measurements through the Stern-Gerlach experiments along the x-axis. The result shows that we have 50 50 split between the spin up and spin down state along x-axis. To make matters more complicated, what we do is we use the random source, let it pass through the Stern-Gerlach experiment along z. Select only the spin up state along z let it pass through the Stern-Gerlach experiment along x. And then we block the spin down state along x, and only let spin up state along x pass through. And then let that go through another Stern-Gerlach experimental apparatus along z again. This time we observe that there is again 50 50 split between spin up and spin down along z. Despite the fact that we already have filtered the electron spin along z in the first experiment, and selected only the spin up state along z direction. So how do you explain all these very confusing and complicated experimental results? In quantum mechanics, when we make a measurement, we collapse the state of the system into one of the allowed states. So in the case of Stern-Gerlach experiment, these allowed states are either spin up or spin down state. The measurable quantity in quantum mechanics is represented by an operator. And the quantum state, allowed states are represented by the eigenstates of that operator. So for example, if you recall time-independent Schrodinger equation, it is an eigenvalue equation. And you can write down the Schrodinger equation as this H being the Hamiltonian operator and the Hamiltonian operator is spelled out here. And given Hamiltonian operator, we are to find both the wave function, psi, and eigenvalue E. So the eigenvalue represents the energy, allowed energy, and the eigenfunction corresponding to that eigenvalue represents the wavefunction associated with that state. Now let's suppose we measure the energy of a quantum system and we obtain a certain value E sub n. The operator for energy is Hamiltonian, and so quantum mechanical explanation is when you measure energy, you collapse the quantum system into one of the allowed eigenstates of Hamiltonian. When the result of the energy measurements is E sub n, what you have done is you collapsed the quantum system into the eigenstate of operator corresponding to the eigenvalue E sub n. The probability of measuring E sub n depends on the initial state. Obviously other values are possible, and we can only predict the probability of measuring certain values. We cannot deterministically predict exactly what will happen when we do these measurements. Now with these quantum mechanical interpretations on measurements, we will come back to the Stern-Gerlach experiment and give proper quantum mechanical interpretation of the results. Before we do that, we first need to establish some mathematical apparatus in order to be able to build our argument and analysis. So in the next few videos, we will provide some mathematical background. And then we will come back and discuss the Stern-Gerlach experiments using the quantum mechanical measurement theory.
