 This is quantum mechanics 7, angular momentum. Welcome. In video 6, we delved into the solutions of Schrodinger's equation for the hydrogen atom. These orbitals describe the probability of finding the electron at some point in space. The different states are indexed by three integers, and specifies the energy of the orbital, while different values of L and M correspond to different orbital structure and orientation. We looked at so-called SPD and F orbitals, and we saw evidence that the shapes of these have physically observable consequences, hence they seem to correspond to something, quote, real. Here, we want to see what physical significance, if any, the quantum numbers L and M have. Let's step back and look at the big picture for a moment. The foundation of quantum theory is Planck's relation. Energy equals Planck's constant times frequency. The energy of a quantum particle associated with a state that varies sinusoidally in time with frequency nu is E equals H nu. Wave particle duality is expressed by De Broglie's relation. Momentum equals Planck's constant over wavelength. The momentum of a quantum particle associated with a state that varies sinusoidally in space with wavelength lambda is P equals H over lambda. Since there are three dimensions of space, there are three components of momentum, X, Y, and Z. Let's think of a state of definite energy and momentum. It varies sinusoidally in space, and in order to vary sinusoidally in time, the entire wave has to be moving. Looking at the left side of the image, it's clear that the wave amplitude at a given point oscillates in time. We saw in video 5 on the Schrodinger equation that this sinusoidal variation has to have two parts, a real part, shown in red, and an imaginary part, shown in green. So the wave function would have the form cosine of kx minus omega t plus the imaginary unit i times sine of kx minus omega t. Here we've used the shorthand omega for 2 pi times frequency nu and k for 2 pi over wavelength lambda. To quickly review material from video 5, the slope of the wave function in time, which we did out by a curly D over a curly Dt, slope in time or rate of change in time, is omega times the sine minus i omega times the cosine. Since omega is 2 pi nu, multiplying by h over 2 pi, which remember we call h bar, reduces to h nu equals e. So h bar omega equals e. If we multiply our slope through by i h bar, we end up with e times the cosine plus i e times the sine. And the final result is that i h bar times the slope of the wave function in time equals the energy times the wave function. As we've seen, this is a key term in the Schrodinger equation. Now, looking at the slope in the x space coordinate, we follow a very similar process except we get a minus k factor in place of omega. h bar times k equals h over lambda, which is the momentum p. So for a state of definite x momentum, minus i h bar slope in x of the wave function produces the x momentum times the wave function. And we can do the same thing for the y and z coordinates. People began to think of quantum mechanics in terms of these so-called operators. We put a little hat over a symbol to denote an operator. So the energy operator e hat is i h bar times the slope in t. The x momentum operator p hat sub x is minus i h bar times the slope in x and so on. These are called operators because they need to operate on or be applied to a wave function to give meaningful results. If the system is in a state of definite energy, then the energy operator e hat applied to the wave function gives the energy value e times the wave function. Likewise, if the system is in a state of definite x momentum, then the x momentum operator p hat sub x applied to the wave function gives the x momentum value p sub x times the wave function. And in general, if the system is in a state of definite o where o is any physically observable property, then the operator o hat applied to the wave function gives the value of o times the wave function. An important aspect of quantum weirdness is that the system need not be in a state of definite o. This happens when the operator gives you anything other than a scaled version of the wave function. A classic example that we'll treat in a future video is shooting your cat in the observables whether the cat is alive or dead. In the macroscopic world, a spinning object tends to keep spinning. We say it has angular momentum. Provided no external forces act on the object, angular momentum is conserved. Imagining an electron in a circular bore type orbit, if we curl our right fingers around the orbit, our extended thumb will point in the direction of the angular momentum vector, which we denote by the letter L. The orientation of this arrow is perpendicular to the plane of the orbit. Its length represents the amount of angular momentum. Specifically, if a particle is on the x-axis, a distance x from the origin and is traveling in the y-direction with a linear momentum P y, then the angular momentum vector is in the z-direction with magnitude x P y. A particle on the y-axis, a distance y from the origin traveling in the minus x-direction with linear momentum minus P x will contribute minus y P x to this component of the angular momentum. The combination of these gives the z-component of angular momentum for a particle at any position with any linear momentum. We can likewise work out the angular momentum components along the x and y-axes. Now, applying the operator concept, we replace each momentum value by the corresponding momentum operator to obtain three angular momentum operators. These expressions are a bit involved, but they're simply step-by-step instructions for how to turn a mathematical crank and extract information from the wave function. The operator for the squared magnitude of angular momentum is the sum of the squares of the x, y, and z operators. Setting up these operators and turning the math crank, we find the following results. The hydrogen orbitals are states of definite angular momentum magnitude. L hat squared operating on an orbital wave function produces a number L squared times the wave function, where L squared equals quantum number lowercase L times L plus 1 times h bar squared. Taking the square root, we find that the magnitude of angular momentum is the square root of L times L plus 1 times h bar. The orbitals are also states of definite z-component of angular momentum. L hat z operating on an orbital wave function produces a number Lz times the wave function, where Lz equals the quantum number m times h bar. We now see the physical significance of the three quantum numbers n, l, and m. n determines the energy, l determines the magnitude of angular momentum, and m determines the z-component of angular momentum. Quantum mechanics doesn't appear to allow us to know the full angular momentum vector, only its length and one component. Let's try to visualize this. Imagine an electron following a classical orbit shown here in green. It seems that we're able to know the length of the red angular momentum vector in terms of the quantum number l and its projection on the z-axis in terms of the quantum number m. This could describe any angular momentum vector falling on the brown circle. Apparently there is some uncertainty about the orientation of the orbit. Say the quantum number l equals 2, this would be a so-called d orbital. The quantum number m can take on the values minus 2, minus 1, 0, 1, and 2. We know that the length of the angular momentum vector is the square root of 2 times 3, which is square root of 6, times h bar. The z-component is one of the five values shown. Say m is minus 2. Then the angular momentum vector could be anywhere on the brown circle. If m is minus 1, it could be anywhere on this next brown circle, and so on for all five values of m. Now, Schrödinger's equation doesn't describe the electron as a particle following a classical orbit, but instead gives us the probability for finding the electron throughout space. Despite the somewhat fuzzy picture of the electron, it seems that we're able to say that not only does it have a definite energy, but it has a definite angular momentum, too. We can also know one component of the angular momentum vector, but it seems that we can't know all three components. This is a surprising result, so let's take a look at this in some detail by considering how we might actually measure a component of angular momentum. Moving charges produce a magnetic field, which is the basis of the electromagnet. Suppose we have a loop of wire with a bunch of electrons moving around it. This will act more or less like a little bar magnet with north and south poles. Here we have a suspended permanent magnet aligned with Earth's magnetic field. Running current through a coil of wire generates a new magnetic field, which allows us to pull on the magnet. Does this process work at the atomic level with our fuzzy wave particle picture of the electron? Let's guess that it does work out the consequences and see if there's something to observe experimentally. Thinking of a classical orbit for visualization purposes and realizing that the electron is moving so fast, about 1% the speed of light, that it effectively blurs into a loop of current, it'll have an angular momentum vector. The orbiting electron will also essentially appear to form a tiny magnet parallel to the angular momentum vector. Suppose we generate a magnetic field in the laboratory. We put our atomic magnet inside the field. We'll call a little north-south pole magnet like this a magnetic dipole. Opposite poles attract so the dipole will align with the magnetic field just like the needle of a compass. If we twist the dipole, those attractive forces will be constantly trying to twist it back into alignment with the field. So we'll have to put energy into the system to change the orientation of the dipole. It seems plausible, therefore, that a magnetic field might cause orbitals with different dipole orientations that take on different energy levels. Even before the development of quantum mechanics, Peter Zeeman had won one of the first Nobel Prizes in physics for discovering the effect later named after him. Zeeman found that in the presence of a magnetic field, a single line of an atomic spectrum may split into an odd number of lines. Here we see an example. With no magnetic field, we observe three single spectral lines. In the presence of a magnetic field, the right line stays single, the middle line splits into three and the left line splits into five. This is precisely what we would expect to observe if an atomic dipole had five possible orientations with respect to the magnetic field, as would be the case for a d orbital with l equals two. For a p orbital with l equals one, there would be three orientations, hence three split energy levels. For an s orbital with l equals zero, no angular momentum, hence no magnetic dipole, there should be no splitting. Therefore, transitions from p to s orbitals should split into three distinct lines. The Zeeman effect shows how we can observe the distinct z components of angular momentum predicted by quantum mechanics. Now, why can't we determine all three components of angular momentum? Our theory tells us that we can know the magnitude of the angular momentum vector and its component along the z axis. Now, imagine we introduce a magnetic field in the z direction in order to use the Zeeman effect to measure the quantum number m and the z component of angular momentum. The magnetic field will try to twist the magnetic dipole to make it align with the field. But the electron orbit has both a magnetic dipole and an angular momentum. When you try to twist an object which has angular momentum, you observe the strange behavior of gyroscopic procession. The angular momentum vector moves around a circle perpendicular to the direction you're trying to twist it. This circle is precisely the circle of constant z component of angular momentum. Imagine our electron orbit is a spinning bicycle wheel. The magnetic field wants to make the axle point downward. But instead of doing this, the axle will undergo procession. Turning off the field gets rid of the twist and stops the procession. So, if gyroscopic procession occurs at the atomic level, we can see the dilemma it creates for us. Even in a classical world, the very act of using a magnetic field to measure the z component of L causes the x and y components of L to change. If you then measure the x component of L with a magnetic field along the x axis, the y and z components will change, and you'll have just negated your first measurement. You can only know one of the three components, Lx, Ly, or Lz.