 This is Quantum Mechanics 5, the Schrödinger equation. Welcome. The Schrödinger equation, published in 1926 by Owen Schrödinger, is practically synonymous with quantum mechanics. It is the primary key which unlocks the secrets of the atom and of chemistry and which describes what Schrödinger called an undulatory theory of the mechanics of atoms and molecules. Now, since this is a video about an equation, it will necessarily contain some math. But, hopefully, you'll find the emphasis is on motivation and illustration of physical concepts. Schrödinger started with De Broglie's hypothesis that electrons are associated with waves having wavelength equal to Planck's constant over electron momentum. Recall from the previous video that this explained the discrete energy levels of the Bohr model of hydrogen as simply the requirement at the length of an orbit equal an integer number of wavelengths. Schrödinger sought the general equation governing these waves, a wave equation of the electron. To understand how he arrived at this, we first need to talk about wave equations in general. Wave equations describe the behavior of fields. A field which, following Schrödinger, will denote by Greek letter psi is simply some quantity which exists at all points of a space. An example is the field of Earth's surface temperature. At any point on Earth, this field's value is simply the temperature at that point. Atmospheric pressure defines another field, as does ground elevation. For any field, physics seeks an equation to describe its behavior. In the great majority of cases, nature, quote, wants a field to be as uniform as possible. Usually this means that in equilibrium, the field is uniformly constant everywhere. For example, if we dip a wire rectangle in soapy water, the resulting soap bubble will be, neglecting gravity, uniformly flat. The field of soap bubble elevation will be constant. But what if the field can't be constant? What then would, as uniform as possible, mean? Suppose we bend our wire rectangle into a curve. Rather than trying to stay flat, the soap film will relax into some smoothly curved shape. What determines this equilibrium shape? The answer is that the field value at each point equals the average of the neighboring values. This is what, as uniform as possible, means in general. Imagine the field psi at some point in three-dimensional space and a small sphere of radius A surrounding that point. Call the average field value on the sphere psi average. Then equilibrium simply requires that psi equals psi average. This is so common in physics that we have a special symbol in operation called the Laplacian that measures the difference between psi average and psi. The Laplacian is denoted by what looks like a triangle to the power two, applied to the field of interest. Specifically, for a very small radius A, the Laplacian is basically 6 over A squared times the difference of psi average and psi. In equilibrium, these values are equal, and we have Laplace's equation, which simply tells us that there is no difference between a field value and the average of its neighbors. But what if we disturb the equilibrium of a field? Say we use our finger to make an indentation in an initially flat, stretched membrane. We expect that the indebted part will feel a force pulling it upward. Neighborhood points will feel a corresponding force pulling them downward. As those points move in an attempt to re-establish equilibrium, they will disturb the equilibrium of yet other points, which in turn will move in response. The result is a wave which propagates away from the initial disturbance out to infinity, assuming our membrane is of infinite extent. If instead the membrane has a boundary, the wave will be reflected there and travel until it's reflected a second time, and a third time, and on and on. Unless energy is removed from the membrane, these fluctuations will continue indefinitely. Now, if we look at the displacement of a single point on the membrane over time, we find a fairly complicated curve. The equation that tells us how to predict this curve is called the wave equation. To develop the wave equation, we'll consider one-dimensional waves of balls connected by springs. In the situation shown here, the middle ball is pulled by two springs, and the net force would be zero if the ball were at the center of the line connecting its neighbors. This is simply the condition that its position is the average of its neighbors. The average position is higher than the actual position, so in this case the Laplacian is positive. We write that the acceleration of the ball is proportional to the Laplacian, that is, proportional to the difference of the equilibrium position and the actual position. We denote acceleration with two dots, and the constant of proportionality turns out to be the square of the wave velocity, the speed of light, or of sound, or of whatever it is we're describing. If we plot the ball's position, which is our field value, through time we get some curve. At a particular time we're at some point on the curve, and we can draw a line through the curve at this point. This is the velocity or the slope of the field in time, and we denote this with a single dot. We can also draw a circle through the curve at this point, and this is related to the acceleration, or the curvature of the field in time, that we denote with two dots. This curvature is also the slope of the slope of the field. So the wave equation tells us that when the field value at a point is not in equilibrium with its neighbors, the plot of that field value versus time will have a corresponding curvature. The greater the disequilibrium, the greater the curvature. If the field value is less than the equilibrium value, the curvature will be upward, as shown in this figure. Conversely, if the field value were greater than the equilibrium value, then the curvature would be downward. Starting with some initial condition, knowing this curvature at every point allows us to calculate the field at all future times. Now, in general, this process is pretty complicated. But it's possible to find special waves that vary in a relatively simple manner, such that the displacement at every point is a uniform sine wave, defining a single definite frequency. In quantum mechanics, frequency is related to energy, so we expect that such waves will turn out to be very important. If the field varies as sine of t, and at every point we plot the field's slope, we get another curve, which is the cosine of t. Plotting the slope of the cosine produces minus the sine of t. So for our purposes, if the field or position is sine of t, the slope, or velocity, is cosine of t, and the curvature, or acceleration, is minus the sine of t. Now imagine we make our sine wave oscillate faster. We write the field as sine to pi nu t, where nu is the frequency of oscillation. Since this compresses our curve horizontally, the slopes will increase, and our slope function will have a larger amplitude. It's 2 pi nu times the cosine of 2 pi nu t. Likewise, the curvature has an even larger amplitude. It's minus quantity 2 pi nu squared times sine of 2 pi nu t. This is minus quantity 2 pi nu squared times the field itself. For waves with this special single frequency behavior, the wave equation has a special form. Our general wave equation is time curvature of the field equals wave speed squared times the Laplacian of the field. For the single frequency case, the curvature is just minus quantity 2 pi nu squared times the field. If we divide by c squared, and use the relation between wave speed, wavelength, and frequency, the wave equation becomes the so-called Helmholtz equation. This is the wave equation for a single frequency, and it was well known in Schrodinger's time, having been used to describe light, radio, sound, and water waves. Now, finally, we're ready to develop the Schrodinger equation. Recall de Broglie's hypothesis for the relation between electron momentum and wavelength, the same relation we have for photons, p equals Planck's constant over wavelength. From this, 1 over lambda equals p over h, and plugging that into the Helmholtz equation gives us the result shown here. Now momentum, p, squared equals mv squared. And we can write this as 2m times 1 half m v squared. This second quantity is just the kinetic energy of the electron, k. So we have p squared equals 2m, k. Since the total energy of an electron is its kinetic energy plus its potential energy, we can write the kinetic energy as the total energy e minus the potential energy v. Plugging 2m times e minus v in for p squared, we obtain the Schrodinger equation for a single frequency, in essentially the form Schrodinger presented it in 1926. Incredibly, that's basically all there is to it. Since that time, people have found it useful to multiply through by h bar squared over 2m, where, recall, h bar is h over 2 pi, and rearrange to get the following form. Minus h bar squared over 2m times the Laplacian in the field plus the potential energy times the field equals the total energy times the field. What's attractive about this form is that it corresponds to the classical equation kinetic energy plus potential energy equals total energy. We see that in quantum mechanics, the Laplacian, the measure of the disequilibrium of the field associated with an electron, corresponds to the electron's kinetic energy. In the nearly 90 years since it was first presented, this form of Schrodinger's equation has proven to be one of the most consequential equations in all of physics. Nearly any scientific inquiry or technological application at the atomic scale must apply it at some level. Still, this form of the Schrodinger equation, as the Helmholtz equation from which we obtained it, is limited to single-frequency fields corresponding to Bohr's stationary states. We want to find the general form of the equation. To start, for a frequency new, Planck's relation gives the energy as h nu. So let's write e psi equals h nu psi. Now we've seen that if the field varies as sin 2 pi nu t, the slope in time is 2 pi nu times cosine 2 pi nu t. That factor of nu means that if we multiply by h bar, the 2 pi cancels and leaves us with h nu. So h bar times psi dot equals e times the cosine. This is almost what we're looking for, namely e times psi, but we have a cosine instead of a sine. We know that the curvature gives us a sine, but it also gives us a nu squared, which would produce a factor of e squared, and that's not what we're looking for. Instead, it seems that in general, the e psi term should be replaced by something involving the field's slope in time, but the nuts and bolts don't quite mesh yet. The way the nuts and bolts were put together struck many physicists, including Schrodinger, is rather bizarre and a bit unsettling. It was necessary to make use of the so-called imaginary unit i. Mathematicians invented i in order to be able to write i squared equals minus 1, or i equals the square root of minus 1. The square of any real number cannot be negative, so i is an imaginary number. Now that's fine for abstract math, but it would be shocking if a theory of physical reality employed an imaginary number. Yet that is what the general form of Schrodinger's equation does. Assume the field has a real part, sine 2 pi nu t, plus an imaginary part, i cosine 2 pi nu t. The field's slope in time, psi dot, is then 2 pi nu times the cosine, minus i 2 pi nu times the sine. Multiply this by i times h bar. The 2 pi's cancel, h times nu is energy e, and minus i squared is minus minus 1 equals 1. The resulting real part is e times the sine, and the imaginary part is i times e times the cosine. So i h bar psi dot equals e psi. Now we can replace e psi in the single frequency Schrodinger equation by i h bar psi dot to get the general Schrodinger equation shown here. Our mathematical description of physical reality employs unreal imaginary numbers. Incredible. Now the steps we just took might seem like they would only work for a field oscillating at a single frequency nu. So how is this a more general equation, you might ask? The key is that for the first form of the Schrodinger equation you need to specify the energy before you solve the equation. This necessarily limits you to a solution at the single frequency corresponding to the energy e. If indeed you're able to find a solution at all. In the second form the frequency is extracted by the slope operation, so you don't need to put it into the equation at the start. Moreover, suppose you have two fields oscillating at different frequencies and you add them together to get a new field. The slope term i h bar psi dot operates on the two components separately, and when you plug all of this into the general Schrodinger equation it takes up into two independent single frequency equations. This idea can be extended to the sum of any number of components. And using the theory of Fourier discussed in a previous video, this means that the second form of Schrodinger's equation is truly general, applying to any conceivable field. So we have two forms of the Schrodinger equation. The form for the general case is used when we need to describe dynamic situations, such as an electron traveling through the double slit experiment. The single frequency form is used to calculate the energy levels of atoms and the chemical bonds and molecules. In the next part we'll look at animations of Schrodinger equation solutions to get some idea of what all this math actually says about how the world works at atomic scales.