SOLVING the SCHRODINGER EQUATION | Quantum Physics by Parth G

hey what's up your lot path here and in today's video we are looking at how to solve the schrodinger equation as well as what it actually means to solve the schrodinger equation if you enjoyed this video then please do hit the thumbs up button as well as the subscribe and bell buttons and please do check out my patreon page if you'd like to support me on there let's get into it now the schrodinger equation can often be seen written in many different forms each of which corresponds to a different physical or mathematical scenario i want to take you through the process of solving the simplest version of the equation but i want to try and do it in an intuitive and hopefully visual way the logic discussed in this video is exactly the logic used to solve the schrodinger equation in more complicated scenarios as well like the hydrogen atom for example but first what are we dealing with when we're talking about the schrodinger equation well if you've seen my old video on this topic then you might recall that the schrodinger equation deals with a quantity known as the wave function the wave function is often denoted by the greek letter psi and it's used to describe everything that we know about a particular system that we happen to be studying for example if we happen to be studying an electron say just moving around in space then the wave function of the electron is basically a mathematical function that gives us all the information we have about that electron using our wave function we can find out things like the probability of us finding our electron at a particular point in space at a particular point in time more of this discussion in my wavefunction video all of these other symbols are basically specific constraints on how our wavefunction can look and behave as time progresses and say we're still studying our electron then the first term describes the kinetic energy of this electron and this second term describes the potential energy largely influenced by the surrounding environment for example in free space there's nothing influencing the electron at all we label this potential as zero but for example if we were to place a charged plate say here then this would clearly affect our electron because negative charges repel and so the potential would no longer be zero in this particular region of space and finally the quantity e can be thought of as the total energy of the electron although this is a bit hand wavy but essentially what we're looking at is kinetic energy plus potential energy is equal to the total energy ish also it's worth noting that we're only working with the time independent schrodinger equation so basically the total energy doesn't change over time it's constant but this is kind of the point like i said we want to be working with the simplest version of this equation and then looking at how to solve it so basically the terms in the schrodinger equation tell us something about the behavior of the wave function and solving the schrodinger equation just means finding what psi can be now in my opinion the simplest way to learn how to solve the schrodinger equation is to first consider a particle that can only move in one dimension that is it can move say left or right but it cannot move up or down or in and out of the screen this isn't exactly easy to recreate in real life but remember real life at this point doesn't matter we aren't necessarily dealing with actually physical things yet we just want to make sure that we can do the maths basically okay so our particle is allowed to move left and right but now we're going to place two barriers in its path specifically we're going to place a barrier at x is equal to zero and x is equal to a we're going to say that these are massive heavy unmovable walls impenetrable even the particle cannot get through these walls and cannot be found anywhere other than between the walls for those of you familiar with the idea of quantum tunneling yeah we're saying this doesn't happen here and the way to ensure that is that we say these walls are infinitely thick they go on forever again another non-physical assumption but it will make the mats easier really though these infinitely thick impenetrable solid ass walls are just one way of getting what we're actually aiming for a region of space where the potential is infinite which in simple terms means that the potential energy of a particle would have to be infinite in order for a particle to be found in those regions so again the same idea a particle cannot be found within or beyond these walls it can only be found between the walls and this setup is known as the one-dimensional particle in a box in this case we're saying that the potential in between the walls is zero because there's nothing influencing the particle in any way between the walls we can show this on an energy diagram let's say that the vertical axis of our graph is going to be the energy whether that's the energy of the particle or the external potential energy and the horizontal axis of course refers to the particle's position and we've got the walls at x is equal to zero and x is equal to a the potential is zero between the walls as we've said already and as soon as we hit the walls the potential becomes infinite which is represented like this now what we can do is to take our value of v in the nice friendly v is equal to zero region and substitute it into our schrodinger equation by saying v is equal to zero if we try to do this for the unfriendly regions then we'd have infinities to deal with but at least the infinities are sort of balanced out by this idea that our wave function the thing that we're trying to find psi must be zero in these regions that's how we've manufactured it because we don't ever want to find our particle in the regions where the walls exist and remember the wave function directly corresponds to our probability of finding the particle in a particular place so if the wave function in the walls is zero then the probability of finding a particle in the walls is zero as well but the main question really is what does the wave function look like in the middle region when v is equal to zero well at this point we've now got a differential equation that we can solve for those of you that are not familiar with this weird looking thing d two psi by dx squared this basically refers to what's known as the second derivative of our wave function psi for example if our wave function psi ends up looking like this then we can find its gradient at every single point or its slope if you prefer at every single point and that becomes d psi by dx and we can find the gradient or slope of that function and that gives us d2 psi by dx squared this is a very brief description of derivatives again more resources in the description below but what we've just seen is how to find d2 psi by dx squared if we know what psi is but that's not the case here we're trying to go the other way which is much trickier and we've got this e psi term to deal with as well luckily however all of these terms are constant and we can rearrange our equation so that we're left with this we can then combine all of the constants into one nice big constant which we'll call k squared for now you'll see y k squared and not just k later and at this point what we've got is that the second derivative of whatever our wave function psi is is equal to minus some constant k squared multiplied by our wave function now we're trying to find a function psi that obeys this equation and one type of function that does this really nicely is a sinusoid here is a diagram of y is equal to sine x here is its derivative dy by dx and here is that function's derivative d2y by dx squared as you can see what we've got is the original function in this case multiplied by a factor of negative one and so what we're saying is that when you start with a sine and differentiate it twice you still end up with a sinusoidal term and so if we carefully account for the constants in our equation our solution is going to look like a sinusoid feel free to pause the video here and have a go at differentiating this function twice to see if it really does obey the schrodinger equation at this point we're not quite there yet with the full solution though the next thing that we need to consider is that the wave function has to behave a particular way at the walls remember how we said the wave function must be zero within the walls well the wave function must also be zero at the walls there are lots of different reasons for this but here's one particular one let's say we decided that psi could be anything as soon as we got to the walls in other words the value of psi didn't need to be zero at the walls but in that case let's say we found that the value of psi was something else entirely by extension then the whole wave function of our system would look something like this and notice that at the walls the value jumps from 0 to some other value that's non-zero but then let's say we try to find the gradient of this function now deep side by dx well the gradient at the walls is infinite and so if we try to find d2 psi by dx squared again we've got infinities to be dealing with and so at the walls our equation breaks completely it's even worse than within the walls at least within the walls we had size equal to zero to offset the infinity that we had for the value of v so to recap the value of the wave function at both walls must be zero we can encode this mathematically we can first start by saying that when x is equal to zero psi must be equal to zero as well this works out quite nicely actually because on both sides of the equation we've got zero and zero is equal to zero it's the other wall that gets interesting when we substitute in psi is equal to zero for x is equal to a we get some really cool results we essentially find a restriction on the kind of sine wave that we can have as a solution for example this is one possible solution it's half a sine wave but the value of psi at each wall is zero as we need it to be and looking very carefully at this position we've gone through half a sine wave which means that this quantity within our brackets must be equal to 180 degrees because that's when we go through half a sine wave right sine of 180 degrees is equal to zero and if we use radians instead of degrees which is the other unit of measuring angles the much more natural unit of measuring angles then 180 degrees is actually equal to pi radians resources in the description below about radians if you're unfamiliar with them but so we find that this equation must hold true if our wave function is half a sine wave and when we rearrange it we now have something that tells us the value of the energy e in other words if our wave function looks like this then the energy of our particle is this another possible solution is a full sine wave fitting into this region so this time at the wall on the right the value in our brackets must now be equal to 360 degrees right because we've gone through an entire sine wave and 360 degrees is equal to 2 pi radians therefore we found that if the wave function looks like a whole sine wave then the energy of the particle is this and we can continue doing this for various lots of half sine waves so we could have three half sine waves in our region or four half sine waves in our region and so on and so forth and in each case we can calculate the energy of the particle when its wave function looks like those sine waves what we're seeing here is a quantum phenomenon known as quantization what we're finding is that because we can only have specific wave functions and those specific wave functions correspond to specific energies a particle therefore can only have specific energies it cannot be anything in between and it cannot be less than this minimum energy because we can fit a half sine wave in there but we can't fit anything less than that we can't fit like a quarter sine wave otherwise the wave function at the walls wouldn't be zero and some of you may have noticed that that's also why we didn't consider cosine solutions to our equation cosine works in the schrodinger equation it just doesn't work for this particular setup now as it turns out there's one more thing that we need to consider when finding the solution to this schrodinger equation which is known as normalization i want to discuss this in a separate video because it's a really interesting concept that we can go into a lot of detail about but the crux of the matter is that in this particular case it adds a factor of square root of 2 over a to our solution here's what this means physically though rather than mathematically let's say our particle is in the lowest energy level basically it has an energy e1 well in that case the wave function looks like this and remember the wave function corresponds directly to the probability of us finding that particle at a particular point in space and this relationship is that if we square our wave function technically take the square modulus then we get the probability of finding our particle at a particular point in space so for our particle in the lowest energy level it's highly likely we'll find it in the middle of the box and less likely we'll find it towards the edges seems pretty reasonable but let's now consider the second energy level where this particle has slightly more energy than the lowest energy level well in that case the wave function looks like this when we square it technically take it square modulus it looks like this and that tells us that we're highly likely to find the particle either here or here but absolutely no chance of finding it in the very center strange right super interesting now it goes without saying that everything we've discussed in this video is just the first step of understanding the schrodinger equation and how to solve it more complicated systems such as three-dimensional systems or spherically symmetric systems like atoms essentially use the same logic and mathematics just becomes more complicated to match those scenarios so with all of that being said i hope you enjoyed this discussion if you did then please do hit the thumbs up button and feel free to subscribe to my channel for more fun physics content and hit that bell button if you want to be notified when i upload please do consider checking out my patreon page if you'd like to support me on there i'd really appreciate it thank you all so much for your wonderful support as always and i will see you really soon thanks for watching

Channel: Parth G

Views: 62,625

Rating: 4.9818745 out of 5

Keywords: how to solve the schrodinger equation, solving schrodinger equation, schrodinger equation., schrodinger equation solved, parth g, quantum mechanics, quantum physics, physics, time independent schrodinger equation, particle in a box, differential equations

Id: sPZWtZ8vt1w

Length: 13min 4sec (784 seconds)

Published: Tue Jan 26 2021