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 one that depends only on 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 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 of the matter wave squared 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 between your machine and the Pia server and hides your IP address this layer of privacy has a broad range of uses from keeping your data more secure when using public Wi-Fi networks to not allowing your internet service provider at home to track and potentially sell your browsing data being able to use the internet from any of Pia's servers in 911 countries and all 50 US states also allows you to get around Regional restrictions I find this especially useful when traveling internationally I can hop on a Pia server back home and quickly access the right version of Amazon and other sites and get around region blocking to access the same streaming content than I normally could my wife and I are planning a trip to Italy next year we love catching up on shows when we're vacationing don't judge us but we still haven't made it all the way through Game of Thrones however it's not available in Italy with Pia I know I can just hop on a US server and watch this also works for other streaming services like Netflix using the URL pn. cwelch laabs you can get 83% off of Pia that comes out to Just Around $2 a month plus an additional 4 months free all with a 30-day money back back guarantee if you're looking to either start using a VPN or to switch VPN providers this is a great deal and also helps me continue making great content you'll also find the url in the description below huge thank you to Pia for sponsoring this video and helping make all this research possible now back to exactly how imaginary numbers snuck into Schrodinger's equation in Schrodinger's initial approach he started with the part of the classical wave equation that only depends on position to completely solve our vibrating string example we have to multiply our spatial Solutions F by our Solutions in time G to compute the final position y of each point on the string as a function of position and time in the classical wave equation the spatial and time components are the same differential equation just with different 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 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 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.