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

0:00 in early 1926 Irwin Schrodinger published

0:03 a series of papers that completely reshaped

0:05 physics over the previous three decades

0:08 it had become increasingly clear that existing

0:10 physics approaches simply didn't work at very small scales the equation

0:15 at the core of shinger papers effectively

0:17 replaced Newton's Second Law at the atomic

0:19 scale describing the behavior of particles

0:22 like electrons incredibly accurately scher's equation

0:25 is very similar to the Heat and wave equations from classical physics

0:30 with one exception the imaginary number I what is the I doing

0:35 here shorting your's equation critically and controversially

0:38 replaces the notion of a particle

0:40 with a wave and says that for a given point in Space

0:44 the value of this matter wave changes in time proportionally to the curvature

0:47 of the wave in space this proportionality makes a ton of sense

0:51 for the heat equation it tells us that for example in regions

0:54 that quickly change from cold to hot to cold the hot area will

0:58 become cooler as the heat spreads out but in Schrodinger's equation the time

1:02 derivative is multiplied by the imaginary number I how does multiplying by I

1:07 turn a heat equation into an incredibly accurate description of matter itself

1:12 imaginary numbers would go into to play a central role in quantum

1:16 physics what makes imaginary numbers so useful in one of our most fundamental

1:20 and successful theories of nature in 1925 Einstein published this paper where

1:26 he referenced a recent PhD thesis from an obscure Frenchman named new Le

1:30 de Bry 20 years before in 1905 Einstein and Max plank famously

1:34 showed that light comes in discret packets that we now call photons

1:38 and that the energy of each photon is related to its frequency

1:41 by Plank's constant in his thesis de Bry showed that if he treated matter

1:45 not as discret particles but instead as waves and extended the plank

1:49 Einstein relation to these matter waves he could accurately predict the behavior

1:53 of the hydrogen atom when Einstein's

1:56 paper reached the physicist Irwin Schrodinger he

1:58 quickly realized that de bry's work was a more elegant and general version

2:01 of his own investigations into guge Theory and became obsessed with the idea

2:06 that matter might actually be a wave after giving a talk under

2:10 Bry matter waves at his home University of Zurich in November sher's colleague

2:14 Peter Dubai remarked that this way of thinking was childish and that if

2:18 matter waves were real there would have to be a matter wave

2:22 equation this comment stuck with schinger and when he left for winter holiday

2:26 in the Swiss Alps a few weeks later he brought along his papers

2:29 and books to work on the problem in his room in the mountains

2:33 shinger sat down and tried to find the wave equation for matter

2:37 shinger began with the classical wave equation and worked to modify it

2:41 to be compatible with de bry's matter wave results in this one-dimensional

2:46 classical wave equation Y is a function of position and time that represents

2:50 the displacement of the wave for example the position of a point

2:54 on a vibrating string above or below its resting position and V

2:57 is the speed of the wave a common approach to solving the classical

3:01 wave equation is to break apart its spatial and time components resulting

3:05 in two new differential equations one that depends only on position and one

3:09 that depends only on time the position equation roughly says that the curvature

3:14 of the wave should be proportional to the negative displacement of the wave

3:18 this makes a lot of sense in our vibrating string example

3:21 a point with high positive displacement corresponds to a high negative curvature

3:25 and vice versa mathematically the position equation says that should be some

3:30 function of X that when differentiated twice is equal to itself times

3:34 some negative constant both s and cosine

3:37 have this property the second derivative

3:39 of s of K* X is equal to minus k^ 2 time the original function sin of KX exactly

3:46 satisfying our differential equation importantly for Schrodinger

3:50 when we fix the ends of our string setting F equal to Z

3:53 at xal 0 and xal L the length of our string only

3:57 very specific values for the constant k will work visually this just

4:01 means that we can fit half of a sine wave between the fixed

4:04 ends of our string or a whole sine wave or a sine

4:07 wave in a half and so on but nothing in between this behavior

4:11 is what gives vibrating strings a very pure tone musically the frequencies

4:16 of vibration are simple multiples

4:18 of the fundamental frequency this Behavior was critical

4:21 for sher's attack plan like the vibrating string the hydrogen atom produces

4:25 energy but only at very specific frequencies however for the the hydrogen atom

4:30 these frequencies are not at simple even spacings scher's Hope was

4:34 that if he modified the classical wave equation using dy's matter wave approach

4:39 the solutions to his new wave

4:40 equation would match the observed emission spectrum

4:43 for hydrogen first switching to the Greek letter s to represent the matter

4:47 wave and rewriting the wave number K in terms of wavelength shinger

4:52 then substituted into bry's formula that relates the wavelength of a matter wave

4:56 to its momentum expressed as mass time velocity the constant term

5:00 in the classical wave equation now depends on the mass of the matter wave

5:04 squared times the velocity of the matter wave squared from classical physics

5:08 kinetic energy is equal to 1 12 mass time velocity squared so we

5:12 can rewrite our numerator as 8 pi^ 2 m* the kinetic energy

5:16 of the wave finally taking the total energy e as the kinetic energy

5:20 plus the potential energy V Shoring your solve for the kinetic energy

5:24 and substitute it the hydrogen atom has one proton and one electron Shing

5:29 your assume that the proton was fixed

5:31 creating an electric potential for the electron

5:33 of the charge of the electron e^ s divided by the distance

5:36 R between the electron and proton atoms are of course three-dimensional so

5:41 we need to expand our spatial derivative to include X Y and Z

5:45 from here schinger needed to find the solution to his matter wave

5:48 equation just as we found earlier that s of KX was a solution

5:51 for the vibrating string the math is of course trickier here referencing

5:56 his mathematics books and with some

5:57 helpful correspondence from the mathematician Herman while

6:00 shinger was able to solve his wave equation for hydrogen like our solution

6:05 to the vibrating string problem where K could only take on very

6:08 specific values shinger showed that the energy term e in his equation

6:12 was also quantized and that the spacing of these energy values approximately

6:17 matched The observed emission spectrum for hydrogen

6:20 shinger submitted his results for publication

6:22 in this paper on January 27th 1926 the response from the scientific

6:27 Community was quick and positive Robert

6:30 Oppenheimer later called Schrodinger's result perhaps

6:32 the most perfect most accurate and most lovely theories that man has discovered

6:37 and the physicist Paul dur remarked

6:39 that Schrodinger's result contains much of physics

6:41 and in principle all of chemistry the orbital electron patterns that you

6:45 may have learned in chemistry class are the solutions to Schrodinger's equation

6:51 now up until this point none of Schrodinger's mathematics required the use

6:54 of imaginary numbers this would change in the summer of 1926 when shinger

6:59 expanded his approach to include systems

7:01 that change over time getting the details

7:05 right on complex topics like this requires a ton of research here's

7:09 all the books I reviewed when writing the script for this video spending

7:12 this amount of research time would not be possible without the support

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9:13 make all this research possible now back to exactly how imaginary

9:17 numbers snuck into Schrodinger's equation

9:20 in Schrodinger's initial approach he started

9:22 with the part of the classical wave equation that only depends

9:25 on position to completely solve our vibrating string example we have

9:29 to multiply our spatial Solutions F by our Solutions in time G

9:33 to compute the final position y of each point on the string

9:36 as a function of position and time in the classical wave

9:39 equation the spatial and time components are the same differential equation

9:44 just with different constants so the Solutions in time are also just

9:47 s and cosine waves a helpful mathematical trick used by physicists

9:51 including schinger is to express these Solutions using complex exponentials so

9:57 instead of writing the cosine of Omega* t is a solution

10:00 we instead write e to the power of I* Omega T

10:04 by oil's formula the real part of e to the power

10:07 of I* Omega T is exactly equal to our original cosine solution

10:12 differentiating complex exponentials is simple we just drop down the exponent

10:17 so the first time derivative of e to the I Omega T

10:20 is just I Omega times our original function and our second

10:24 derivative is just i^ 2* Omega 2* our original function this shows

10:28 that e to the I Omega T is a valid solution

10:31 to our differential equation importantly up until this point in physics

10:36 although complex numbers were used frequently like this in computation

10:40 the final answer was always just the real part of the result

10:43 and everything physical in the problem like the displacement of the string

10:47 corresponded to the real part of complex exponentials to expand his equation

10:52 into the time domain shinger started with a complex exponential representation

10:56 of the wave function as usual assuming he would be able

10:59 to take the real part once he was done calculating since

11:02 the energy of a matter wave is proportional to its frequency

11:05 by the plank Einstein relation we can rewrite this complex exponential

11:09 in terms of the total energy e of the wave differentiating we

11:13 can show that the energy of the wave times the wave function

11:16 is proportional to I times the derivative of the wave function now

11:20 returning to Schrodinger's time independent equation we can isolate e* the wave

11:25 function and substitute obtaining the final

11:28 modern version of the the Schrodinger

11:30 equation Schrodinger found this path early on but hesitated to publish

11:34 it writing to the physicist Hendrick Lorent what is unpleasant

11:38 here and indeed directly to be objected to is the use

11:41 of complex numbers Sai is surely fundamentally a real function the I

11:48 explicitly showing up next to the time derivative in shing's equation

11:51 means that purely real wave functions will not work the wave function

11:55 itself has to be made from complex numbers as we'll see

11:59 the complex wave function and multiplication of the time derivative by I

12:03 turned out to be a feature not a bug it allows

12:07 Shing your's equation to elegantly describe the behavior of matter let's

12:11 consider how shinger equation applies to a free particle in one

12:14 dimension such as an electron far away from any other particles

12:18 in this case our potential energy V is zero let's temporarily

12:22 combine our constants together into a single constant that we'll call C

12:26 and set it equal to 0.1 and assume a very simple

12:29 starting configuration for the wave function with a value of one

12:33 surrounded by zeros we can estimate the second spatial derivative

12:36 at our central location numerically by adding

12:39 together the adjacent wave function values

12:41 and subtracting two times the wave function at our central location

12:45 so 0- 2+ 0= -2 we can keep track of each part

12:50 of shing's equation over time using a table at time T equals

12:54 0 we set our initial wave function value to one and we

12:57 just estimated our second spatial derivative to be minus 2

13:01 from here all that's left to do is to compute DC DT

13:04 using shing's equation multiplying our estimate

13:08 for the spatial derivative by 0.1*

13:10 I we compute a complex number with zero for the real

13:14 part and minus 0.2 for the imaginary part now shorting your's

13:18 equation says that this value is equal to how much the wave

13:22 function will change in time again taking a numerical approximation approach

13:27 we can add DDT to to our current value of s

13:30 to get the value of s at our next time step

13:34 so our wave function now equals 1us 0.2 I on the complex

13:39 plane taking the second spatial derivative was equivalent to multiplying s

13:43 by minus 2 which flips it across the origin on the complex

13:46 plane we then multiplied by 0.1 scaling our result now

13:52 importantly before we add this result to update our current value

13:55 of s sher's equation tells us to multiply by I which

13:59 on the complex plane rotates our vector by 90° to the left

14:04 we then add this Vector to our current value of s

14:07 to compute the next value of s repeating this process we

14:12 estimate the second spatial derivative of our new wave function which

14:15 again flips s across the origin we again scale and rotate

14:19 by 90° and update sigh with our new value of D

14:23 SI DT after a few more steps a clear Trend emerges

14:28 our spatial d derivatives are pushing our wave function around the complex

14:32 plane on a curved or circular path in the classical wave function

14:36 we can think of the spatial curvature as pushing the wave

14:39 up and down in time while in Schrodinger's equation the spatial

14:43 curvature of the wave is pushing the complex wave function

14:45 in circles around the complex plane we can get a broader sense

14:50 for this Behavior over space and time by considering a simple

14:53 solution to sher's equation known as a plane wave this wave function

14:57 is a complex exponential in terms of both position and time visually

15:02 This Plane wave looks like a real cosine and imaginary sign traveling

15:05 to the right as time advances thinking of each point

15:09 in our one-dimensional space as a little complex plane the real cosine part

15:13 of our wave moves left and right and the imaginary sign

15:16 part of our wave moves up and down taken together these components

15:20 form a complex number that moves around the unit circle

15:23 As Time advances just as we saw numerically differentiating our plane wave

15:29 twice with respect to position gets us minus k^ 2times our original

15:33 plane wave sign so just as we saw numerically this spatial

15:37 derivative is directly across the origin from our wave function value

15:40 on the complex plane and its magnitude depends on the spatial

15:43 frequency K of our plane wave remember that sher's equation tells

15:48 us to multiply this value by I to get DC DT which

15:52 rotates our spatial derivative by 90° and it's this Vector

15:56 that effectively pushes our wave function around the complex plane larger values

16:01 of K mean our plane wave oscillates more rapidly in space increasing

16:05 the curvature of the wave function and increasing the second derivative which

16:09 following shinger equation pushes our wave function around the complex plane

16:13 faster in time plugging in the full plane wave into sher's equation

16:19 we can show that the plane wave is a solution and recover

16:21 the exact relationship between spatial frequency K and frequency and time

16:26 Omega this is the relationship that shinger started with from De

16:29 bry's matter waves so our plane wave looks like spirals

16:33 on a series of complex planes and shing's equation connects the behavior

16:37 of these spirals in space and time now how are these complex

16:42 valued matter waves at all a reasonable description of physical particles

16:46 like electrons aside from including the imaginary number I another important

16:51 feature of Shing your's equation is that the wave function and its

16:55 derivatives are not raised to any powers this makes sure equation

16:59 linear it means that if we have two wave functions let's

17:02 call them s 1 and S 2 if we add them together

17:06 s 1 plus S 2 is also a valid solution to shing's

17:10 equation this means that shing's equation will work for any combination

17:14 of plain waves for example if we had an identical second

17:18 spiral wave function but shifted in Space by half of a wavelength

17:22 the sum of these wave functions would exactly cancel out resulting

17:26 in an overall wave function that equals zero everywhere the ability

17:30 of wave functions to interfere like this is critical to the wav

17:33 likee properties of matter we see in situations like the double

17:36 slit experiment firing a stream of electrons through a single narrow

17:41 slit into a detector we see a smooth distribution of detections where

17:46 the most likely place for an electron to land is directly

17:48 behind the slit with this probability dropping off smoothly as we

17:52 move further away from the slit now if electrons were simply

17:56 particles when we opened a second slit we would expect the distributions

18:00 for the first and second slit to just add together resulting

18:03 in an overall detection pattern that looks very similar to the pattern

18:06 for a single slit however in practice this is not the behavior

18:11 we see as first demonstrated with electrons by Davidson and germer

18:15 in 1927 we instead see a wavy pattern where electrons almost

18:20 never arrive at certain locations on the detector we can make sense

18:24 of this strange behavior of matter by using Schrodinger's matter waves

18:28 we'll repres the electron as a little packet of waves this little

18:32 packet is also a valid solution to shing's equation and we'll

18:35 switch from thinking in one spatial Dimension to two in two

18:39 Dimensions it's more difficult to think of each point in space

18:42 as a little complex plane so let's switch to visualizing the amplitude

18:46 of our complex wave function as the height of our surface

18:49 and we'll represent the angle of our complex number using the color

18:52 of the surface if we close one of our slits

18:55 and pass our matter wave through we see the matter wave sprad

18:58 spr out evenly after passing through the slit running our experiment

19:02 again with the other slit we see similar results now before we

19:07 run our experiment with both slits open let's have a closer

19:10 look at how our two patterns line up notice that the amplitudes

19:14 of our wave functions are smooth curves as we expect from particle

19:18 like Behavior but the angles of the complex numbers that make

19:21 up our wave function change across the surface of our detector

19:25 and the colors from one experiment to the other do not

19:27 always line up meaning that our matter waves are out

19:30 of phase at certain locations this means that when we open up

19:34 both slits we expect destructive interference

19:36 at these locations running our experiment

19:40 with both slits open this is exactly the behavior we see

19:43 with the electrons matter wave canceling itself out at these locations

19:47 on the detector matching the behavior we see experimentally so the angles

19:51 of our complex numbers in our wave function also known

19:54 as the phase store important information about the matter wave of the electron

19:59 causing the wave to interfere with itself destructively at locations

20:02 in space consistent with experimental results a few days after shinger submitted

20:07 the final paper in his groundbreaking series The physicist Max Bourne

20:11 submitted this paper including what today we call the borne rule which

20:15 with some caveats states that the square of the amplitude

20:18 of the wave function is equal to the probability of finding the particle

20:21 at a certain location in space following the borne rule

20:25 the amplitude of the wave function allows us to figure out where

20:28 the particle is likely to be in space while the angle

20:31 of the complex wave function captures how matter waves interfere with themselves

20:35 and other matter waves there are other ways we can accomplish

20:39 this Behavior mathematically but complex numbers

20:41 are very convenient here and it's

20:43 interesting that imaginary numbers fall out of the classical wave equation

20:47 when we combine it with de bry's simple matter wave relationship

20:50 as Schrodinger did if we place our 2D wave packet

20:54 in a box the potential energy term in shing's equation will cause it

20:57 to reflect off the borders and interfere with itself resulting

21:01 in a set of discret fixed wave patterns this behavior is analogous

21:06 to the quantized energy levels Shing are found when applying his equation

21:09 to the hydrogen atom in three-dimensional space it's remarkable that the same

21:14 complex valued wave function can describe the behavior of a free

21:17 electron in the double slit experiment and the quantitized energy levels we

21:20 see for bound electrons in atoms years later in a lecture

21:25 in 1970 the great Quantum physicist Paul dur had this to say

21:29 about the wave function so if one asks what is

21:32 the main feature of quantum mechanics I feel inclined now to say

21:36 that it is the existence of probability amplitudes which underly all

21:40 Atomic processes now a probability amplitude is related to experiment but only

21:46 partially the square of its modulus is something we can observe

21:50 that is the probability which the experimental people get but besides

21:54 that there is a phase a number of modulus Unity which we

21:57 can modify without affecting the square of the modulus and this phase

22:01 is all important because it is the source of all interference

22:05 phenomena but its physical significance is obscure so the Real Genius

22:09 of Heisenberg and Schrodinger you might say was the discovery

22:13 of the existence of probability amplitudes

22:15 containing this phase quantity which is

22:18 very well hidden in nature and it is because it was

22:21 so well hidden that people hadn't thought of quantum mechanics much earlier

22:26 the phase durak is referring to here is the angle

22:29 of the complex numbers that make up the wave function imaginary and complex

22:33 numbers give us an elegant tool for representing and working

22:36 with this phase which is an integral part of how we understand

22:39 matter to work on these small scales the rise of quantum

22:43 mechanics is such an astounding chapter in the story of imaginary numbers

22:47 the physicist Freeman Dyson writes that imaginary numbers showing up

22:50 in wave mechanics is one of the most profound jokes of Nature

22:54 and that the square root of minus1 in Shing your's equation

22:57 means that nature works with complex numbers and not real numbers there's

23:01 some debate to be had about just how essential complex numbers

23:04 are here shinger himself seems to have never fully accepted a truly

23:08 complex valued wave function although he would go on to use

23:11 it in his work and Communications what's absolutely all inspiring to me

23:16 is that the numbers we rejected for so long as impossible

23:19 or imaginary ended up showing up so profoundly in one

23:23 of our deepest and most accurate theories of nature if you

23:28 liked this video I really think you'll enjoy the shringer equation chapter

23:32 in my imaginary numbers book the figures came out great

23:36 from the detailed numerical walkthrough to the plane wave analysis to the wave

23:39 packets and double slit experiment I wasn't sure how all

23:42 these detailed animations would translate to book format it took some trial

23:46 and error but I'm really happy with the results I love

23:49 video but I always struggle with how much detail to include

23:52 and how fast or slow to move no matter what I

23:56 know that my videos will be too fast and lose some people

23:58 and be too slow and bore others one thing I love

24:02 about reading books is being able to go at the exact right

24:04 pace for myself so whether you're an expert looking for a different

24:07 angle on a topic you know well or just picking up

24:10 this stuff for school work or fun I really think you'll

24:13 enjoy the book there's also exercises including a numerical walkthrough of Shing

24:18 your's equation that you can compute your way through yourself huge

24:21 thanks to everyone who pre-order the book it'll be shipping out

24:24 in just a few weeks over the past few months I've

24:26 landed on what I think is a really nice type setting

24:29 with extra wide margins for notes and figures and I found

24:32 a great book printer that prints on this really nice heavyweight paper

24:35 with great color and fine detail reproduction the book starts

24:39 with an introduction to imaginary numbers and works up to remon surfaces Oilers

24:42 formula and finally Schrodinger's equation

24:45 every chapter includes exercises with Solutions

24:47 in the back my initial print run is 70% sold out order

24:51 yours today to get it in time for the holidays it

24:54 also makes a great gift finally thanks to everyone who has reached

24:57 out about internet national shipping I'm still only able to offer

25:00 us shipping right now but I'm planning to expand internationally next year