 So let's see what we have learned already from this most basic scenario, a particle moving at constant speed free from external forces. There is a wave equation, which means that the solutions will not have definite localization. These are waves. They describe a phenomenon that's not specifiable to any one location in space, and that means they're spread out. And as a result of that, and because they describe something that is oscillating, they're led to questions like, what is it that's oscillating? What are the implications for measuring things like position or momentum of particles when fundamentally they're waves, and they're not localizable to any one definite location in space at any one time. I may not be able to know everything that I thought I could know about particles from matter waves, which are really what matter is. Like waves are variations in the strength of electric and magnetic fields, and mechanical waves are variations in say the density of a medium, or the displacement of a medium. What's oscillating in a matter wave? We'll come back to that. Now to better understand these solutions, we need to confront the mathematics of these complex functions a bit more closely, and I don't want you to be daunted by the presence of either complex numbers or complex functions. They're basically just a representation of information that becomes necessary when a problem has too much information to be described by only one class of numbers, say real numbers. It's okay, it just means that matter waves contain more information than real numbers alone can capture, and there's nothing scary about that. All we have to do is become more comfortable with the language of complex numbers and how to get real values out of them, because after all real numbers are the only things that are realized in the physical world. It may be true that phenomena can be described by complex numbers and complex functions, but when we make measurements of the natural world, we don't get the answer I back from it. We get numbers like 5 or negative 52 or 73.771 back from measurements. Those are all real valued, and so regardless of the fact that the wave equation may be complex and its solutions may be complex, somehow we've got to get real numbers and only real numbers out of these things. And to really understand that, we need to take a look at complex numbers and a little bit of the algebra related to complex numbers. But basically, complex numbers just double the available amount of information you can store in a single number. That's all they do. So our working solution to the free particle wave equation is of this form. We've seen it a bunch of times now. And it looks weirdly similar to that representative complex number Z I showed you earlier, X plus I, Y. It's a complex structure with a real part, which we could represent by X and an imaginary part which you could represent by Y. It looks very similar to a simple complex number. But as I've said, observations of the natural world are conventionally described by real numbers, not imaginary ones. Now that's okay, I mean, we've already kind of hinted at the fact that complex numbers look a lot like vectors, and we're used to dealing with vectors with an X component and a Y component. And from those, we're comfortable summarizing the information content of a vector using the concept of its length or its magnitude, a single real number. For instance, you might have a three dimensional velocity with a vx, a vy, and a vz component. And that's all very complicated, but you're very comfortable going look. The speed of the particle is v, where v is the square root of vx squared plus vy squared plus vz squared, a single valued real number that summarizes the overall thrust of the velocity vector. So that's not scary at all. That's something you've been doing since beginning introductory physics. The question here is, how does one get a single real number out of a complex one? How do we get the measurable out of the complex function or number? Well, you might just try your old friend the square, right? Square the complex number and see if that gives you a real value. But unfortunately, it gives you a complex polynomial. You wind up with a real number x squared and a real number negative y squared, but a complex piece 2ixy, that's the sum of the cross terms of this square. If only we could get rid of that cross term, we'd be home free. We would actually recover something that looks weirdly like the Pythagorean theorem with an x squared and a y squared term. This is almost a hypotenuse squared, but it's not real valued. So this may be the hypotenuse in some space, but it's not the hypotenuse in the real number space of measurement. Okay, so that won't work. Now, instead, to get a real number, you need to do something like this. And this is part of what defines the algebra of complex numbers. You're gonna take z and you're gonna multiply it by a special version of itself, known as z star. But this is just x plus iy, the original complex number, times x minus iy. And you'll notice that when you do that distributed multiplication out and add all the terms together, you wind up with x squared plus y squared and no cross terms. Well, this looks weirdly like the Pythagorean theorem. You've got an x component squared, you've got a y component squared. And this is somehow related to a sort of square of the complex number, although this funny thing z star is required. So, while that yields something more consistent with, for instance, your experience with the Pythagorean theorem about the length of a vector, but it does it with a complex number with real and imaginary components, what is this thing z star that we've employed to get away with this? And the answer is that z star is what is known as the complex conjugate of the complex number. All you do to take the complex conjugate of a complex number is take all the numbers i inside the number and replace them with negative i. That's it. You're gonna send i to minus i, you're gonna flip the sign of all the i's and that is all z star represents. Now, to keep this kind of consistent with our instincts about vectors and lengths and magnitudes and things like that, we have a shorthand notation for z times z star, z times its own complex conjugate. To indicate that it is the square of the real length, the thing we would really measure as a consequence in nature if we described a problem using complex numbers. And that is denoted by the magnitude or absolute value bars of z all squared. So the magnitude of z squared is defined as z, z star. So if you see this notation, absolute value or magnitude of z squared in complex space, that denotes the product of z with its complex conjugate z star. That's how you get the real valued length of a complex number or a complex function. Now, another interesting thing about the free particle solutions is that one can simplify the notation that we've been using to carry around these free particle solutions. And that is the language of signs and cosines and exponential functions. So for instance, it's really clunky to have to keep writing out these signs and cosines in our free particle wave function solution to the wave equation. It would be nice if we could compactify this notation somehow. And mathematics does offer us a more compact representation of the same information and will also give us some practice with imaginary numbers like i. Ultimately, we will be able to summarize the free particle solutions as a single exponential function rather than a sum of signs and cosines. To get there, let's consider a Taylor expansion of the sign function, sine of x. So the sine of x, Taylor expanded into a series of terms, becomes x minus x cubed over three factorial plus x to the fifth over five factorial, etc. Similarly, the cosine function can be Taylor expanded into the following one minus x squared over two factorial plus x to the fourth over four factorial, etc. Notice that the sine involves only the odd powers of x and the cosine involves only the even powers of x, so x to the zero is one, x squared, x fourth, and so forth. And the sums all have alternating pluses and minuses that are used to combine the terms together. Now, recall that the Taylor expansion of the exponential function, e to the x, looks like the following. If you Taylor expand e to the x, you wind up with one plus x, plus x squared over two factorial, etc. So if you stare at these three things for a second, you're dangerously close to being able to find some combination of sine and cosine that when added together yields e to the x. But it's not going to be real valued because the sine and cosine expansions have alternating plus and minus signs in front of their terms. Whereas the e to the x expansion is all sums. And so we see a problem here. We would like to use e to the x to represent some combination of sine and cosine of x. But we can't do that because we have these stray minus signs on alternating terms that complicate our ability to use only real numbers to do this trick to make sine and cosine combine to get e to the x. Well, again, leaving that expansion of e to the x up here, let's go back and revisit a little bit the use of the imaginary number i and the implications it might have for combining sine and cosine. So note that while the expansion of e to the x involves the sum of a bunch of power of x and the sine and cosine expansions have alternating sums and subtractions, we might use this rule that when you see stray minus signs that they might be indicative of products of the imaginary number i, we can crack the puzzle. So let's think creatively for a moment and let's recall that i squared equals negative one. And that allows us to then rewrite terms like negative x squared, which appears in the expansion of the cosine function as i squared x squared. Or in other words, ix all squared. So it's as if we replace the argument of the cosine function with i times the argument that we started with. Now in the sine expansion, we have odd numbered powers of x, like negative x cubed, for instance. And that could be rewritten as i squared x cubed, but that's not very satisfying. We have different powers of i and x in this. But let's keep in mind that if we have a term that looks instead like negative i x cubed, that can be rewritten, and you can practice this for yourself as i cubed x cubed, which is just ix all cubed. So with those things in mind, let's recall our free particle solutions are of the form a times the cosine of an argument x plus i times the sine of an argument x. Well, if we stare at that for a second and we plug in the Taylor expansions of cosine and sine, we would get this, that we have a times, for instance, just keeping the first two terms in the Taylor expansion, 1 minus x squared over 2 factorial. And we're gonna add to that iA times this expansion of sine, keeping only the first two terms, x minus x cubed over 3 factorial. Now if we distribute the imaginary number i into the parentheses on the right hand side of this sum, we can start employing the identities and relationships that I wrote up here. So for instance, negative ix cubed is just ix all cubed. And negative x squared is just ix all squared. So for instance, I wind up with terms like this. I have ix here, which is fine, we can leave that alone. I have negative ix cubed, and that can be replaced with positive ix all cubed, and that's done here. Now for the cosine, I have 1 minus x squared over 2 factorial. Well, negative x squared can be replaced with ix all squared. And you'll notice what's happening. We're eating up the minus signs in algebra involving the number i. So we wind up with a positive sum of these terms. 1 plus ix plus ix all squared over 2 factorial, plus ix all cubed over 3 factorial, etc. If we were keeping more terms in the Taylor expansion. This thing here can simply be rewritten as a times e to the ix. The argument of the cosine and sine was x, but combining them in this way with a multiplicative i in front of the sine term, we get to rewrite that sum as a e to the i times x, the original argument of the sine and cosine function. So we've traded a real valued function for a complex function, but it's a much more compact notation than what we had before. And this allows us to rewrite the free particle solutions in this more compact form as a times e to the i times the quantity kx minus omega t. And this is a little bit easier to carry around on a piece of paper than the sums of sines and cosines with the imaginary number i and only one of the two terms. Now what is the magnitude of our free particle solution? And let's keep in mind that we don't know if the constant out in front of the function a is real or complex. So let's try to calculate the magnitude squared of the wave function of the free particle. Let's do that. So we're trying to calculate the absolute value of psi squared. And remember in a complex space of functions or numbers, that's defined as psi times its complex conjugate psi star. Well, what is that? Well, psi is just a times e to the i kx minus omega t. The complex conjugate of psi would involve changing i to negative i everywhere we see it. But we don't know if there's an i hiding inside of the pre-factor a that multiplies the exponential function. So to be very careful about this in case the a is also a complex number, we're going to replace a with a star and i with negative i up here. And that's about as far as we can go with this. If we now group terms together in the multiplication, we have a times a star. We have e i kx minus omega t. And grouping the exponents together, we have then negative i times kx omega t. These exponents completely cancel each other out to zero. And we're left with a term that's just e to the zero. E to the zero is one. So this then simplifies to a times a star. Or just the magnitude of a squared. So the measure of the wave particle function for a free particle is just a real number, the magnitude of a squared. But what is it that we've just evaluated? What is this function that solves the wave equation and what is the meaning of its length? These are the questions that really racked people's brains in the 1920s and 1930s. This was a real intellectual struggle in confronting the wave nature of matter. So one is forced to interpret these functions and their meaning. There is no easy answer from first principles in nature about what the wave function is because it's a complex function. You don't actually have any physical meaning to its real and imaginary parts. It's only the magnitude of the wave function squared that has any physical meaning. And so you have to lay an interpretation down as to what you think the underlying wave function is and what is waving. It's not energy because energy is a real thing. It's something else. And I have to tell you that in the history of physics, and you may have seen this in popular videos on quantum mechanics, which often are rife with misunderstandings of the underlying math and subject material. It's this contest of intellectual ideas that has caused the most hand rubbing and consternation and some of the most bitter disagreements and strong opinions in the history of science. And it's all been over a function whose direct value has no physical meaning because it's based in part on imaginary numbers, which themselves have no physical interpretation. It's only the real valued magnitude of the complex function or the complex numbers that have any physical meaning. It's not those numbers themselves. It's only the measure of their overall information content that has meaning in the physical world. Now the most practical interpretation, one which has also been met with the most experimental success since Erwin Schrodinger first published his wave equation, is that of a probabilistic meaning to the square of the wave function. That is to say, this thing, the magnitude of psi squared, this amplitude squared of the wave function is interpreted as representing a probability per unit distance per unit time in one dimension. In two dimensions, it's per unit area, and in three, it's per unit volume. Now to obtain raw probabilities, one has to specify the exact conditions under which the free particle has been prepared. For instance, where was it starting from? Exactly. And what was its momentum and things like that? And then you can answer questions such as, given that this is a matter wave and it's not localized once it's released to any one place in space, what's the probability of finding this particle between, say, one centimeter and two centimeters from the point of origin? Or what's the probability of finding the particle a distance of three centimeters from the point of origin one second after it starts its journey? These are questions you can try to answer in the framework of the matter wave equation, the Schrodinger wave equation, and all the math that goes along with it. And we don't have that framework available. We're going to develop that framework going forward and try to get answers to questions like this. So that's our goal. We're going to conclude our discussion of the implications of the wave nature of matter in this lecture, and later lectures we'll begin to think about specific problem statements and then how we use the Schrodinger wave equation to attack those problem statements and interrogate the solutions to get answers that can be measured in a laboratory experiment. The wave function itself is not directly accessible, but its amplitude squared in different situations has physical consequences for measurement. Now that said, because we're mathematical beings that can imagine things that are not physically realizable in the world around us, we can use some math and computer aids to try to visualize the wave function of our matter particle that's free from external forces. But to do this, we have to concoct a space of the imaginary value of the wave function and the real value of the wave function. Now these are not physical axes in space. They don't have physical extent. Remember that this is an oscillating probability. Probability itself is not physical, but the probabilities of outcomes are physical. And so you have to be very careful to separate your visualization of the wave function from physical meaning, which is only derivable from the square of the wave function, the complex conjugate times the original wave function. Nonetheless, because we are mathematical beings and we can think abstractly, let's attempt to visualize what the wave function of a free particle would look like without specifying how it was prepared. In that case then, it's the solution that we've written down already. And we can imagine thinking about the amplitude of the wave function along its imaginary axis and along its real axis. So along its imaginary axis, it's a sine function whose amplitude starts out at zero, goes to a maximum, plunges to a minimum and returns to zero after one cycle. And along the real value axis of the wave function, it starts off at maximum amplitude, eventually goes through zero to a minimum, back through zero to a maximum after one cycle of the matter wave. And note that the maximum of the matter wave in the real value part of the wave function is achieved at the same location as the zero point of the imaginary part of the wave function, which is what you would expect from a cosine and a sine function combined together. Now, of course, if we construct this in 3D space with our imaginary axis, our real axis, and then the spatial location and physical space of the particle, we wind up with a helical structure, a helical surface that winds through imaginary and real space, keeping in mind that we're talking about the imaginary and real components of the wave function. But at all points in space, as we've seen, the amplitude squared of this is a constant valued number that doesn't depend on space and time. And so whatever this wave function is doing, varying in its real and imaginary parts, in physical space, it represents a constant probability density everywhere in space in time. So there's nothing waving in physical space. In the space of the wave function, you have oscillation, and that oscillation is related to the probability of finding the particle at that point in space at that moment in time. But in physical space, all you have is the magnitude squared of the wave function. That's the only physical thing that manifests in the measurable world. Now, to close out this lecture, let's take a look at what it means to try to measure both the position and the momentum of a matter wave representation of a particle. So here's a real valued part of the wave function of a matter wave. It's the cosine. It starts at 1, goes to negative 1, returns to 1 after 1 cycle. And you see I've got two wavelengths represented in this picture. I've ignored the complex part, but it's also waving at the same time. We're looking now just at the physical position of the particle versus the value of the real component of the wave function. The imaginary component of this wave also has an important role in what happens with the physical reality of the particle, but it's not shown here. I just want to concentrate your energy now on thinking about what it means to measure momentum and position for a wave, or at least a particle described as a wave. Now, measuring the position of a free particle boils down to determining where it is along the x-axis. So for instance, I might do that by zooming in more and more on this wave and saying, OK, I'm localizing the particle more and more and more by spotting the little chunk of its wave function in the real valued component located at that point in space. But measuring the momentum of the same particle boils down to a different observation. Measuring the momentum of the particle is related to determining the second derivative of this wave with respect to space. That is determining the curvature of this wave. That's what the second derivative with respect to space tells you. It tells you about the spatial curvature of the real part, or the imaginary part, of the wave function. And it's that curvature, the degree to which the wave bends to move toward the next part of its cycle that determines momentum. Now, it's very easy to determine the momentum in this picture. We clearly have two wavelengths. We could sit down and easily determine from the information on this page what the wavelength of this wave is. But we might be a little less certain about where it is because there's a couple of cycles of its real valued part of its wave function here. So maybe it's here, or maybe it's here, or maybe it's here. All right, so knowing the momentum really well might preclude knowing the position really well. But what if we really localized this particle to one specific place in position space? So what we want to do is try to locate the particle more and more precisely by zooming in on the wave function to really localize the phenomenon to one narrow region of space. And this is equivalent to identifying where it is in a range x and x plus delta x. And then sending delta x more and more toward zero to zoom way, way, way, way in on a narrow slice of the wave. All right, but as we'll see, it's going to become harder and harder to establish the curvature of the wave as we do this. And thus, the momentum of the wave is going to slip from our grasp. Now, to help you with this exercise, what I want you to do is really stare at the wave in this region right here where I'm indicating with the mouse cursor. So really stare at the wave here. Right now, you can clearly see that there's well-defined curvature. You could easily and readily determine the wavelength of this phenomenon. How about now? Can you easily determine the wavelength of this phenomenon? I've zoomed in, localizing more in space where I want to see where the particle is. But in doing so, I've traded a lot of the curvature away in order to do that. It's getting harder to determine the wavelength of this wave, but you could still maybe do it. You've got a peak over here, and you can see how it's declining. There's lots of curvature to determine the momentum of this wave. But how about now? I've zoomed in even more. Stare at that. Are you confident you could determine the curvature of that wave? And you may be remembering the old wave. But as you continue to stare, can you determine the curvature of the wave? Well, I messed with you a little bit. While you were staring at the wave, while I was daring you to think about the curvature of that line, I did one more change to the wave. I'm still zoomed way in on it. But I changed the wavelength by 10%. Did you notice? Did you notice that the wavelength changed from the previous zoom in to the zoom in you're looking at now? An astute observer might have noticed while they were staring at it that the grid behind here changed when I did that. And that corresponds to a change in where I was zoomed in on the wave. But the starting value and the ending value of the wave in this picture didn't change. The heights of the waves where they enter the picture and exit the picture were concocted identically, giving you the impression that you were confident that the wavelength was the same as the wave from before. But it's not. I changed the wavelength by 10%. But presented you with a similar zoomed in region. And this is meant to confuse you on purpose to show you that the more you close in on the wave function, the harder and harder and harder it's going to be to determine the curvature of the wave. Is this line straight? Is it bending gently? How much is it bending? You don't have infinite resolution available to you in the universe. You're going to hit a limit at some point, and it's going to get extremely hard to determine if this is a straight line or not a straight line. And if it's not a straight line, you're going to struggle with determining exactly what its radius of curvature is. And that struggle is reflected in a loss of control over your knowledge of the momentum of the particle. Knowing the position too well comes at the cost of knowing the momentum. So let me repeat that statement one more time. When you're dealing with matter waves, knowing the position very well comes at the cost of knowing the momentum with any precision. Knowing the momentum very well comes at the cost of knowing the position with any precision. That I reflected in my earlier statement about being zoomed out looking at many cycles of the wave. You're very confident when you're zoomed out that you know the wavelength of this phenomenon. But because there are many places where the particle is likely to be and less likely to be represented by the changing amplitude of the wave in real space, you're getting kind of confused about where it might actually be. Is it more at one of the maxima or more at the other maximum or more at the third maximum or the fourth maximum or the fifth maximum. Gaining confidence in momentum comes at the cost of confidence and precision. And it was the physicist Werner Heisenberg who worked out the mathematics of this particular issue in 1927. Now the real way to do this of course is to take the wave equation and to work through the Fourier transform, which tells you something about the information content of the wave in position and frequency or momentum space. That's a little above the ability of a course like this to work out, although you are welcome to look into it on your own if you're comfortable with integrals and derivatives at a high level, at least at the level of say calc 2 and calc 3. Heisenberg codified the relationship between the certainty or uncertainty of our knowledge and momentum and the uncertainty of our knowledge and position in what is known as the Heisenberg uncertainty principle. And it's a very definitive statement, albeit an inequality. It says that the uncertainty in the knowledge of momentum, delta P, times the uncertainty in the knowledge of position, delta X, must always be greater than or equal to h bar, the reduced Planck's constant, divided by 2. Why is it that we don't worry about knowing how fast our car is moving while also knowing its position on the road? We don't freak out about that. Like, if we're going to stare at the speedometer for a moment, we're suddenly going to look up and realize we're in New York City, whereas we were in Dallas at the beginning of our glance down at the speedometer. That doesn't happen in the real world. You don't increase your confidence in your current velocity and thus your current momentum, and then suddenly look up and realize you're on the moon. I mean, this is essentially what we're talking about here with tiny matter waves, right? Is that once you become very confident you know where the particle is, you suddenly lose all confidence about its momentum and vice versa. Well, it's no wonder we didn't notice it. H bar over 2 is a number that is approximately 10 to the minus 35 joule seconds. That's an insanely small number. It's no wonder we didn't notice this before and that it would only manifest at the scale of things tiny, like atoms or electrons or the nucleus of the atom or things like that. But this statement holds for matter waves, no matter what situation you're in. You cannot know the position and the momentum at the same time with infinite precision. And you can see that if you did try to know one of them with infinite precision, that is delta x exactly equal to zero, so you want to know exactly the position of a matter wave. So you specify an experiment that lets you get infinite precision, no uncertainty on the position. You completely lose control of the momentum, the uncertainty on the momentum blows up to infinity in order to hold this as a constant. That's the only way to satisfy this inequality, is if delta p blows up to infinity as delta x goes to zero. This is a limit imposed by the wave nature of matter, it's unavoidable. You cannot know this pair of variables, x and p, with any simultaneously perfect precision. Now of course the y of this is buried deeply in things like the Fourier transform and in the algebra of matrices, that is collections of numbers in multiple dimensions, which is another form of language that can be used to derive quantum mechanics, which is where we are essentially at now. That's above the pay grade of this particular class, but I just want to say that because you are going to encounter quantum mechanics again in a dedicated higher level course than this one. And I want you to understand that I'm having to wave my hands quite a bit at this level in order to motivate this. Nonetheless, you will have a second crack at this where you'll begin to see the y's of all of this. Where is this coming from? Why h bar over two? Why this particular product of momentum and position? Are there other products of things that similarly in pairs are uncertain when you know one you don't know the other and vice versa? These are excellent questions and I don't expect you to be satisfied with this right now, but this is where we can get in a course at this level after two semesters of introductory physics. So let's review what we have learned in this lecture. We've learned about mechanical and electromagnetic wave equations. And from that, we've learned how to infer the nature of the wave equation for matter. And this has given us some ability to get at the meaning of the waves described by the matter wave equation, albeit by interpreting what's going on based on our experience with the natural world. The wave equation involves complex numbers and the solutions to the wave equations involve complex functions. We have to get real numbers out of this thing if we want to map it onto the real world. And the only way to do this is, for instance, to calculate the amplitude squared of the wave function. In doing that, however, we lose any ability to understand or map the physicality of the wave function itself onto the real world. It's only the amplitude of the wave function that has implications for the real world. So the wave function describes oscillating probabilities and it's the amplitude squared that tells us the probability per unit distance per unit time for something to be true in the Schrodinger wave equation, describing a matter wave involving either no forces or some forces. But the wave nature of matter ultimately imposes a limit of absolute knowledge on our ability to understand the world around us. What we learn from exploring the wave part of the wave function of the matter waves is that there's a limit to our knowledge. If we know the position of this wave very well, we lose control over its momentum. If we get control over its momentum at a high degree, we lose our confidence and information about the position of the particle any longer. These pair of variables are related to each other in their uncertainty by the Heisenberg uncertainty principle. And fundamentally, this imposes a limit of absolute knowledge on what we can know about a system of particles at any given moment in time by making measurements. These are the foundations of quantum mechanics that we will build on going forward. And we will spend the rest of this course essentially applying quantum mechanics and special relativity to problems involving the very small things in the universe like atoms and individual subatomic particles to make predictions about the natural world and understand phenomena like atoms and the behavior of particles trapped in systems like you would find, for instance, in semiconductors. These are all basic applications that are at our fingertips now that we have a foundational equation that we can solve in order to understand the outcomes of these particular situations.