What is the i really doing in Schrödinger's equation?

in early 1926 Irwin Schrodinger published a series of papers that completely reshaped physics over the previous three decades it had become increasingly clear that existing physics approaches simply didn't work at very small scales the equation at the core of shinger papers effectively replaced Newton's Second Law at the atomic scale describing the behavior of particles like electrons incredibly accurately scher's equation is very similar to the Heat and wave equations from classical physics with one exception the imaginary number I what is the I doing here shorting your's equation critically and controversially replaces the notion of a particle with a wave and says that for a given point in Space the value of this matter wave changes in time proportionally to the curvature of the wave in space this proportionality makes a ton of sense for the heat equation it tells us that for example in regions that quickly change from cold to hot to cold the hot area will become cooler as the heat spreads out but in Schrodinger's equation the time derivative is multiplied by the imaginary number I how does multiplying by I turn a heat equation into an incredibly accurate description of matter itself imaginary numbers would go into to play a central role in quantum physics what makes imaginary numbers so useful in one of our most fundamental and successful theories of nature in 1925 Einstein published this paper where he referenced a recent PhD thesis from an obscure Frenchman named new Le de Bry 20 years before in 1905 Einstein and Max plank famously showed that light comes in discret packets that we now call photons and that the energy of each photon is related to its frequency by Plank's constant in his thesis de Bry showed that if he treated matter not as discret particles but instead as waves and extended the plank Einstein relation to these matter waves he could accurately predict the behavior of the hydrogen atom when Einstein's paper reached the physicist Irwin Schrodinger he quickly realized that de bry's work was a more elegant and general version of his own investigations into guge Theory and became obsessed with the idea that matter might actually be a wave after giving a talk under Bry matter waves at his home University of Zurich in November sher's colleague Peter Dubai remarked that this way of thinking was childish and that if matter waves were real there would have to be a matter wave equation this comment stuck with schinger and when he left for winter holiday in the Swiss Alps a few weeks later he brought along his papers and books to work on the problem in his room in the mountains shinger sat down and tried to find the wave equation for matter shinger began with the classical wave equation and worked to modify it to be compatible with de bry's matter wave results in this one-dimensional classical wave equation Y is a function of position and time that represents the displacement of the wave for example the position of a point on a vibrating string above or below its resting position and V is the speed of the wave a common approach to solving the classical wave equation is to break apart its spatial and time components resulting in two new differential equations one that depends only on position and time the position equation roughly says that the curvature of the wave should be proportional to the negative displacement of the wave this makes a lot of sense in our vibrating string example a point with high positive displacement corresponds to a high negative curvature and vice versa mathematically the position equation says that should be some function of X that when differentiated twice is equal to itself times some negative constant both s and cosine have this property the second derivative of s of K * X is equal to minus k^ 2 time the original function sin of KX exactly satisfying our differential equation importantly for Schrodinger when we fix the ends of our string setting F equal to Z at xal 0 and xal L the length of our string only very specific values for the constant k will work visually this just means that we can fit half of a sine wave between the fixed ends of our string or a whole sine wave or a sine wave in a half and so on but nothing in between this behavior is what gives vibrating strings a very pure tone musically the frequencies of vibration are simple multiples of the fundamental frequency this Behavior was critical for sher's attack plan like the vibrating string the hydrogen atom produces energy but only at very specific frequencies however for the hydrogen atom these frequencies are not at simple even spacings scher's Hope was that if he modified the classical wave equation using dy's matter wave approach the solutions to his new wave equation would match the observed emission spectrum for hydrogen first switching to the Greek letter s to represent the matter wave and rewriting the wave number K in terms of wavelength shinger then substituted into bry's formula that relates the wavelength of a matter wave to its momentum expressed as mass time velocity the constant term in the

classical wave equation now depends on the mass of the matter wave squared times the velocity from classical physics kinetic energy is equal to 1 12 mass time velocity squared so we can rewrite our numerator as 8 pi^ 2 m * the kinetic energy of the wave finally taking the total energy e as the kinetic energy plus the potential energy V Shoring your solve for the kinetic energy and substitute it the hydrogen atom has one proton and one electron Shing your assume that the proton was fixed creating an electric potential for the electron of the charge of the electron e^ s divided by the distance R between the electron and proton atoms are of course three-dimensional so we need to expand our spatial derivative to include X Y and Z from here schinger needed to find the solution to his matter wave equation just as we found earlier that s of KX was a solution for the vibrating string the math is of course trickier here referencing his mathematics books and with some helpful correspondence from the mathematician Herman while shinger was able to solve his wave equation for hydrogen like our solution to the vibrating string problem where K could only take on very specific values shinger showed that the energy term e in his equation was also quantized and that the spacing of these energy values approximately matched The observed emission spectrum for hydrogen shinger submitted his results for publication in this paper on January 27th 1926 the response from the scientific Community was quick and positive Robert Oppenheimer later called Schrodinger's result perhaps the most perfect most accurate and most lovely theories that man has discovered and the physicist Paul dur remarked that Schrodinger's result contains much of physics and in principle all of chemistry the orbital electron patterns that you may have learned in chemistry class are the solutions to Schrodinger's equation now up until this point none of Schrodinger's mathematics required the use of imaginary numbers this would change in the summer of 1926 when shinger expanded his approach to include systems that change over time getting the details right on complex topics like this requires a ton of research here's all the books I reviewed when writing the script for this video spending this amount of research time would not be possible without the support of this video sponsor private internet access please take a minute to consider if Pia might be a good fit for you it really helps me out there are many good reasons to use a VPN and Pia is the VPN that I use personally Pia takes your privacy really seriously they don't keep any logs of their users activity and this no log policy has held up in court and been independently audited I was impressed to learn from the audits that this even includes your typical error and debug logs that could in some cases contain user data when you connect to a VPN like Pia you're effectively using the internet from a Pia server of your choice Pia encrypts your traffic 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constants so the Solutions in time are also just s and cosine waves a helpful mathematical trick used by physicists including schinger is to express these Solutions using complex exponentials so instead of writing the cosine of Omega * t is a solution we

instead write e to the power of I * Omega T by oil's formula the real part of e to the power of I * Omega T is exactly equal to our original cosine solution differentiating complex exponentials is simple we just drop down the exponent so the first time derivative of e to the I Omega T is just I Omega times our original function and our second derivative is just i^ 2 * Omega 2 * our original function this shows that e to the I Omega T is a valid solution to our differential equation importantly up until this point in physics although complex numbers were used frequently like this in computation the final answer was always just the real part of the result and everything physical in the problem like the displacement of the string corresponded to the real part of complex exponentials to expand his equation into the time domain shinger started with a complex exponential representation of the wave function as usual assuming he would be able to take the real part once he was done calculating since the energy of a matter wave is proportional to its frequency by the plank Einstein relation we can rewrite this complex exponential in terms of the total energy e of the wave differentiating we can show that the energy of the wave times the wave function is proportional to I times the derivative of the wave function now returning to Schrodinger's time independent equation we can isolate e * the wave function and substitute obtaining the final modern version of the Schrodinger equation Schrodinger found this path early on but hesitated to publish it writing to the physicist Hendrick Lorent what is unpleasant here and indeed directly to be objected to is the use of complex numbers Sai is surely fundamentally a real function the I explicitly showing up next to the time derivative in shing's equation means that purely real wave functions will not work the wave function itself has to be made from complex numbers as we'll see the complex wave function and multiplication of the time derivative by I turned out to be a feature not a bug it allows Shing your's equation to elegantly describe the behavior of matter let's consider how shinger equation applies to a free particle in one dimension such as an electron far away from any other particles in this case our potential energy V is zero let's temporarily combine our constants together into a single constant that we'll call C and set it equal to 0. 1 and assume a very simple starting configuration for the wave function with a value of one surrounded by zeros we can estimate the second spatial derivative at our central location numerically by adding together the adjacent wave function values and subtracting two times the wave function at our central location so 0 - 2 + 0 = -2 we can keep track of each part of shing's equation over time using a table at time T equals 0 we set our initial wave function value to one and we just estimated our second spatial derivative to be minus 2 from here all that's left to do is to compute DC DT using shing's equation multiplying our estimate for the spatial derivative by 0. 1 * I we compute a complex number with zero for the real part and minus 0. 2 for the imaginary part now shorting your's equation says that this value is equal to how much the wave function will change in time again taking a numerical approximation approach we can add DDT to our current value of s to get the value of s at our next time step so our wave function now equals 1us 0. 2 I on the complex plane taking the second spatial derivative was equivalent to multiplying s by minus 2 which flips it across the origin on the complex plane we then multiplied by 0. 1 scaling our result now importantly before we add this result to update our current value of s sher's equation tells us to multiply by I which on the complex plane rotates our vector by 90° to the left we then add this Vector to our current value of s to compute the next s repeating this process we estimate the second spatial derivative of our new wave function which again flips s across the origin we again scale and rotate by 90° and update sigh with our new value of D SI DT after a few more steps a clear Trend emerges our spatial d derivatives are pushing our wave function around the complex plane on a curved or circular path in the classical wave function we can think of the spatial curvature as pushing the wave up and down in time while in Schrodinger's equation the spatial curvature of the wave is pushing the complex wave function in circles around the complex plane we can get a broader sense for this Behavior over space and time by considering a simple solution to sher's equation known as a plane wave this wave function is a complex exponential in terms of both position and time visually

This Plane wave looks like a real cosine and imaginary sign traveling to the right as time advances thinking of each point in our one-dimensional space as a little complex plane the real cosine part of our wave moves left and right and the imaginary sign part of our wave moves up and down taken together these components form a complex number that moves around the unit circle As Time advances just as we saw numerically differentiating our plane wave twice with respect to position gets us minus k^ 2times our original plane wave sign so just as we saw numerically this spatial derivative is directly across the origin from our wave function value on the complex plane and its magnitude depends on the spatial frequency K of our plane wave remember that sher's equation tells us to multiply this value by I to get DC DT which rotates our spatial derivative by 90° and it's this Vector that effectively pushes our wave function around the complex plane larger values of K mean our plane wave oscillates more rapidly in space increasing the curvature of the wave function and increasing the second derivative which following shinger equation pushes our wave function around the complex plane faster in time plugging in the full plane wave into sher's equation we can show that the plane wave is a solution and recover the exact relationship between spatial frequency K and frequency and time Omega this is the relationship that shinger started with from De bry's matter waves so our plane wave looks like spirals on a series of complex planes and shing's equation connects the behavior of these spirals in space and time now how are these complex valued matter waves at all a reasonable description of physical particles like electrons aside from including the imaginary number I another important feature of Shing your's equation is that the wave function and its derivatives are not raised to any powers this makes sure equation linear it means that if we have two wave functions let's call them s 1 and S 2 if we add them together s 1 plus S 2 is also a valid solution to equation this means that shing's equation will work for any combination of plain waves for example if we had an identical second spiral wave function but shifted in Space by half of a wavelength the sum of these wave functions would exactly cancel out resulting in an overall wave function that equals zero everywhere the ability of wave functions to interfere like this is critical to the wav likee properties of matter we see in situations like the double slit experiment firing a stream of electrons through a single narrow slit into a detector we see a smooth distribution of detections where the most likely place for an electron to land is directly behind the slit with this probability dropping off smoothly as we move further away from the slit now if electrons were simply particles when we opened a second slit we would expect the distributions for the first and second slit to just add together resulting in an overall detection pattern that looks very similar to the pattern for a single slit however in practice this is not the behavior we see as first demonstrated with electrons by Davidson and germer in 1927 we instead see a wavy pattern where electrons almost never arrive at certain locations on the detector we can make sense of this strange behavior of matter by using Schrodinger's matter waves we'll repres the electron as a little packet of waves this little packet is also a valid solution to shing's equation and we'll switch from thinking in one spatial Dimension to two in two Dimensions it's more difficult to think of each point in space as a little complex plane so let's switch to visualizing the amplitude of our complex wave function as the height of our surface and we'll represent the angle of our complex number using the color of the surface if we close one of our slits and pass our matter wave through we see the matter wave sprad spr out evenly after passing through the slit running our experiment again with the other slit we see similar results now before we run our experiment with both slits open let's have a closer look at how our two patterns line up notice that the amplitudes of our wave functions are smooth curves as we expect from particle like Behavior but the angles of the complex numbers that make up our wave function change across the surface of our detector and the colors from one experiment to the other do not always line up meaning that our matter waves are out of phase at certain locations this means that when we open up both slits we expect destructive interference at these locations running our experiment with both slits open this is exactly the behavior we see with the electrons matter wave canceling itself out at these locations on the detector matching the behavior we see experimentally so the angles of our complex numbers in our wave function also known as the phase store important information about the matter wave of the electron causing the wave to interfere with

itself destructively at locations in space consistent with experimental results a few days after shinger submitted the final paper in his groundbreaking series The physicist Max Bourne submitted this paper including what today we call the borne rule which with some caveats states that the square of the amplitude of the wave function is equal to the probability of finding the particle at a certain location in space following the borne rule the amplitude of the wave function allows us to figure out where the particle is likely to be in space while the angle of the complex wave function captures how matter waves interfere with themselves and other matter waves there are other ways we can accomplish this Behavior mathematically but complex numbers are very convenient here and it's interesting that imaginary numbers fall out of the classical wave equation when we combine it with de bry's simple matter wave relationship as Schrodinger did if we place our 2D wave packet in a box the potential energy term in shing's equation will cause it to reflect off the borders and interfere with itself resulting in a set of discret fixed wave patterns this behavior is analogous to the quantized energy levels Shing are found when applying his equation to the hydrogen atom in three-dimensional space it's remarkable that the same complex valued wave function can describe the behavior of a free electron in the double slit experiment and the quantitized energy levels we see for bound electrons in atoms years later in a lecture in 1970 the great Quantum physicist Paul dur had this to say about the wave function so if one asks what is the main feature of quantum mechanics I feel inclined now to say that it is the existence of probability amplitudes which underly all Atomic processes now a probability amplitude is related to experiment but only partially the square of its modulus is something we can observe that is the probability which the experimental people get but besides that there is a phase a number of modulus Unity which we can modify without affecting the square of the modulus and this phase is all important because it is the source of all interference phenomena but its physical significance is obscure so the Real Genius of Heisenberg and Schrodinger you might say was the discovery of the existence of probability amplitudes containing this phase quantity which is very well hidden in nature and it is because it was so well hidden that people hadn't thought of quantum mechanics much earlier the phase durak is referring to here is the angle of the complex numbers that make up the wave function imaginary and complex numbers give us an elegant tool for representing and working with this phase which is an integral part of how we understand matter to work on these small scales the rise of quantum mechanics is such an astounding chapter in the story of imaginary numbers the physicist Freeman Dyson writes that imaginary numbers showing up in wave mechanics is one of the most profound jokes of Nature and that the square root of minus1 in Shing your's equation means that nature works with complex numbers and not real numbers there's some debate to be had about just how essential complex numbers are here shinger himself seems to have never fully accepted a truly complex valued wave function although he would go on to use it in his work and Communications what's absolutely all inspiring to me is that the numbers we rejected for so long as impossible or imaginary ended up showing up so profoundly in one of our deepest and most accurate theories of nature if you liked this video I really think you'll enjoy the shringer equation chapter in my imaginary numbers book the figures came out great from the detailed numerical walkthrough to the plane wave analysis to the wave packets and double slit experiment I wasn't sure how all these detailed animations would translate to book format it took some trial and error but I'm really happy with the results I love video but I always struggle with how much detail to include and how fast or slow to move no matter what I know that my videos will be too fast and lose some people and be too slow and bore others one thing I love about reading books is being able to go at the exact right pace for myself so whether you're an expert looking for a different angle on a topic you know well or just picking up this stuff for school work or fun I really think you'll enjoy the book there's also exercises including a numerical walkthrough of Shing your's equation that you can compute your way through yourself huge thanks to everyone who pre-order the book it'll be shipping out in just a few weeks over the past few months I've landed on what I think is a really nice type setting with extra wide margins for notes and figures and I found a great book printer that prints on this really nice heavyweight paper with great color and fine detail reproduction the book starts with an introduction to imaginary numbers and works up to remon surfaces Oilers formula and finally Schrodinger's equation every chapter includes exercises with Solutions in the back my initial print run is 70% sold out order yours today to get it in time for the holidays it also makes a great gift finally thanks to everyone who has reached out about internet national shipping I'm still only able to offer us

shipping right now but I'm planning to expand internationally next year