 This is how the time-independent Schrodinger equation looks like. And this is how the time-dependent Schrodinger equation looks like. We can already state that the Schrodinger equation is, mathematically speaking, a partial differential equation of second order. Differential equation means that the searched quantity is not a variable, but the function and an equation are derivatives of this function. The function we are looking for in the Schrodinger equation is the so-called wave function. Partial means that the equation contains derivatives with respect to multiple variables, such as derivative with respect to location x and with respect to time t. And second order means that the highest derivative that occurs in the differential equation is of second order. Most phenomena of our everyday life can be described by classical mechanics. The goal of classical mechanics is to determine how a body of mass m moves over time t. We want to determine the trajectory that is the path r of t of this body. In classical mechanics, the trajectory allows us to predict where this body will be at any given time. For example, we are able to describe the movement of our Earth around the Sun, the movement of a satellite around the Earth, the launch of a rocket, the movement of a swinging pendulum or a thrown stone. These are all classical motions that can be calculated with the help of Newton's second law of motion. So with the equation f equals m times a, or for the experts among you, with the differential equation m times the second time derivative of the trajectory is equal to the negative gradient of the potential energy function. By solving this differential equation, you can find the trajectory you are looking for for a specific problem. To solve this differential equation at all, the potential energy function must of course be given. The initial conditions characterizing the problem that you want to solve must also be known. For example, if you describe the motion of a particle, then an initial condition could be the position and velocity of the particle at time zero. Once you have determined the trajectory r of t by solving the differential equation, you can then use it to find out all other relevant quantities such as the particle's velocity, its momentum or its kinetic energy. In the atomic world, however, classical mechanics does not work. The tiny particles here, like electrons, do not behave like classical point-like particles under all conditions, but they can also behave like waves. Because of this wave character, the location of an electron cannot be determined precisely because a wave is not concentrated at a single location. And if we try to squeeze it to a fixed location, the momentum can no longer be determined exactly. This behavior is described by the uncertainty principle, a fundamental principle of quantum mechanics that cannot be bypassed. So we cannot determine a trajectory r of t of the electron as in classical mechanics and derive all other motion quantities from this trajectory. Instead, we have to find another way to describe the quantum world. And this other way is the development of quantum mechanics and the Schrodinger equation. In quantum mechanics, you do not calculate a trajectory r of t, but a so-called wave function psi. This is a function that generally depends on the location r and the time t, where now the location r is a space coordinate and not an unknown trajectory. The tool with which we can find the wave function is the Schrodinger equation. It is only through this novel approach to nature using the Schrodinger equation that humans have succeeded in making part of the microcosm controllable. As a result, humans are now able to build lasers that are indispensable in medicine and research today. Or scanning tunneling microscopes which significantly exceed the resolution of conventional light microscopes. It was only through the Schrodinger equation that we were able to fully understand the periodic table and nuclear fusion in our sun. This is only a small fraction of the application that the Schrodinger equation has given us. So let us first find out where this powerful equation comes from. Unfortunately, it is not possible to derive the Schrodinger equation from classical mechanics alone. But it can be derived, for example, by including the wave particle duality which does not occur in classical mechanics. However, experiments in modern technical societies show that the Schrodinger equation works perfectly and is applicable to most quantum mechanical problems. Let us try to understand the fundamental principles of the Schrodinger equation and how it can be derived from a simple special case. We make our lives easier by first looking at a one-dimensional movement. In one dimension, a particle can only move along a straight line, for example, along the spatial axis x. Now consider a particle of mass m flying with velocity v in x direction. Because the particle moves, it has a kinetic energy. Let us call it w-kin. It is also located in a conservative field, for example in a gravitational field or in the electric field of a plate capacitor. Conservative means when the particle moves through the field, the total energy w of the particle does not change over time. Consequently, the energy conservation law applies and the potential energy, let us call it w-pot, can be assigned to the particle. The total energy w of the particle is then the sum of the kinetic and potential energy. This is nothing new. You already know this from classical mechanics. The energy conservation law is a fundamental principle of physics, which is also fulfilled in quantum mechanics in modified form. The weirdness of quantum mechanics is added by the wave particle duality. This allows us to regard the particle as a matter wave. A matter wave characterized by the de Broglie wavelength lambda. Lambda equals h over p. p is the momentum of the particle, which is the product of the velocity v and the mass m of the particle. And h is the Planck constant, a natural constant that appears in many quantum mechanical equations. By the way, because of its tiny value of only 6.626 times 10 to the power of minus 34 joule seconds, it is understandable why we do not observe quantum mechanical effects in our macroscopic everyday life. According to the wave particle duality, we can regard the particle as a wave and assign physical quantities to this particle that are actually only intended for waves, such as the wavelength in this case. In quantum mechanics, it is common practice to express the momentum p, not with a de Broglie wavelength, but with a wave number k. k is 2 pi over lambda. Thus the momentum becomes h times k over 2 pi. h over 2 pi is defined as a reduced Planck constant, h bar. So we can write the momentum more compact with h bar k. We will need this equation later. The de Broglie wavelength is also a measure of whether the object behaves more like a particle or a wave. Particle-like behavior can be described by classical mechanics. More exciting is the case when the particle behaves like a wave. To distinguish it from classical point-like particles, such an object is called a quantum mechanical particle. A particle has a larger de Broglie wavelength if it has a smaller momentum p. In other words, smaller mass and velocity. Perfect candidates for such quantum mechanical particles are electrons. They have a tiny mass and their velocity can be greatly reduced by means of electric voltage or cooling in liquid hydrogen. Thus the classical particle behaves more like an extended matter wave, which can be described mathematically with a plane wave. We call it by the capital Greek letter psi. A plane matter wave generally depends on the location x and the time t. You can describe a plane wave, which has the wave number k, frequency omega and amplitude a by a cosine function. A times cosine kx minus omega t. It doesn't matter whether you express the plane wave with sine or cosine function. You might as well have used sine. When time t advances, the wave moves in the positive x direction, just like our considered particle. In order to do math with such waves without using any addition theorems, we transform the plane wave into a complex exponential function. First add to the cosine function the imaginary sine function. You have thus transformed a real function into a complex function. Where i is the imaginary unit, a times cosine kx minus omega t is the real part, and a times sine kx minus omega t is the imaginary part of the complex function psi. The good thing is that we can take advantage of the enormous benefits of complex notation and then declare that in the experiment we are only interested in the real part, that is the cosine function. In our case, we can simply label the imaginary part as non-physical and just ignore it. However, keep in mind that the complex plane wave can also be a possible solution to the Schrodinger equation. But we will deal with this later. In the next step we use the Euler relationship from mathematics. It connects the complex exponential function with cosine and sine. So that's exactly what you need right now. Because with it you can convert the complex plane wave to an exponential function. And you have already represented a plane wave in a complex exponential notation. Whenever you see a term like that, you know it's a plane wave. Remember, our original plane wave as a cosine function is contained in the complex function as information, namely as the real part of this function. You can easily illustrate the complex exponential function. It is a vector in the complex plane. The amplitude A corresponds to the magnitude of the vector, that is its length, and the argument kx minus omega t corresponds to the angle phi, also called phase, enclosed between the real axis and the psi vector. When the time t passes, the angle changes and the vector rotates in the complex plane, in our case clockwise. This rotation represents the propagation of the plane wave in the positive x direction. The complex exponential function is a function that describes a plane wave. Therefore, it is also called wave function, especially in connection with quantum mechanics. Often the wave function psi is also called the state of the particle. We say the particle is in the state psi. So always remember, when we talk about state and quantum mechanics, we mean the wave function. Of course, there are different states that different particles can take under different conditions. The plane wave with specific boundary conditions is just one simple example of a possible state. Next, multiply the equation for the total energy by the wave function. In this way, you combine the law of conservation of energy and the wave particle duality inherent in the wave function. But this equation does not help you much yet. You still have to find a way to convert it into a differential equation. A plane wave is a typical wave that appears in optics and electrodynamics when describing electromagnetic waves. And from there, we know that a plane wave is a possible solution of the wave equation. A one-dimensional plane wave, as in our case, solves the one-dimensional wave equation. C is the phase velocity of the wave. In the case of electromagnetic waves, it is the speed of light. In the case of matter waves, it is the phase velocity omega over k. But for us, this is not important for the time being. I quickly want to show you the wave equation to motivate our next step. On the left-hand side of the wave equation is the second derivative of the wave function with respect to x. So let's calculate the second derivative of our plane wave. The second derivative adds k squared and a minus sign. The minus sign because i squared is equal to minus 1. The wave function as an exponential function remains unchanged with derivation, as you hopefully know. Next, we again make steps which at first sight appear to be arbitrary, but in the end they will lead us to the Schroding equation. We will somehow try to connect the second derivative of the wave function with the conserved total energy. First, use the rewritten de Broglie relation for the momentum p equals h bar k and replace k squared. Now to bring the kinetic energy into play, replace the momentum p squared with the help of the relation kinetic energy is equal to p squared over 2 m. You can easily obtain this form from one-half mv squared by rearranging the momentum p is equal to m times v, squaring and inserting it into velocity v. If you now look at the law of conservation of energy multiplied by the wave function, you will see the w-kin times psi occurs there. So solve the equation for w-kin times psi. Now if you just insert that into the law of conservation of energy, you get the Schroding equation. This Schroding equation is one-dimensional and time-independent. You can recognize the one-dimensionality immediately by the fact that only the derivative would respect to a single space coordinate occurs, in this case with respect to x. And you can recognize the time-independence of the Schroding equation by the fact that a constant total energy occurs. The wave function, however, may be time-dependent. Let's recap for a moment. To derive the Schroding equation, we have combined the law of conservation of energy and the wave-particle duality, introducing the wave-particle duality by assuming a plane-matter wave. So you could say that the time-independent Schroding equation is the energy conservation law of quantum mechanics. This term stands for total energy, this one for kinetic energy, and this one for potential energy. Let's assume that you have solved the Schroding equation and found a specific wave function. It doesn't matter how exactly you did it. Of course, depending on the problem, you will generally not get a plane wave. The found wave function can also be a complex function. So you cannot just neglect the imaginary part of it as we agreed on in the beginning with our plane wave. By omitting the imaginary part, the result of the Schroding equation would no longer agree with the results of experiments. For an experimentalist, however, such complex wave functions are quite bad because they cannot be measured. But how can you check your calculation in an experiment if the complex wave function cannot be measured at all? What does the wave function actually mean? Here the predominant statistical interpretation of quantum mechanics comes into play, the so-called Copenhagen interpretation. It does not say what the wave function psi means, but it interprets its square of the magnitude. By forming the square of the magnitude, you get a real valued function, that is a function measurable for the experimentalist. In addition, the square of the magnitude is always positive, so there is no reason why it should not be interpreted as probability density because, as you know, probabilities are always positive, never negative. In the one-dimensional case, the square of magnitude would then be a probability per length and in three-dimensional case, a probability per volume. Let's stay with a one-dimensional case. If you integrate the probability density, that is the squared magnitude of the wave function over the location x within the length between a and b, then you get a probability p of t. The integral of the squared magnitude indicates the probability p of t that the particle is in the region between a and b at the time t. In general, the probability to find the particle at a certain location can change over time. If you plot the squared magnitude against x, you can read out two pieces of information from it. First, probability p is the area under the squared magnitude curve. Second, the most likely way to find the particle is to find it at the maxima, most unlikely at the minima. Note, however, that it is not possible to specify the probability of the particle being at a certain location, for example at x equals a, but only for a spaced region here between a and b. If you try to determine the probability at a single point in space, the integral becomes zero, because the area under the curve at a single point in space is zero. Therefore, we always calculate the probability of the particle being in a certain space region. In order for the statistical interpretation to be compatible with the Schrodinger equation, the solution of the Schrodinger equation, that is the wave function psi, must satisfy the so-called normalization condition. This means that the particle must exist somewhere in space. In one-dimensional case, it must therefore be found 100% somewhere between minus infinity and plus infinity. In other words, the integral for the probability integrated over the entire space must be one. The normalization condition is a necessary condition that every physically possible wave function must fulfill. After solving the Schrodinger equation, the found wave function psi must be normalized using the normalization condition. Normalizing means that you must calculate the integral and then determine the amplitude of the wave function so that the normalization condition is satisfied. The normalized wave function then remains normalized for all times t. If this were not the case, the Schrodinger equation and the statistical interpretation would be incompatible. There are of course solutions to the Schrodinger equation, for example, psi equals 0, which cannot be normalized. Such solutions are unphysical. They do not exist in reality. By the way, wave functions that can be normalized are called square integrable functions in mathematics. If you know with 100% that the particle is located between A and B, then you must reduce the normalization condition accordingly to the region between A and B. Let's take a specific example of how a wave function is normalized. Let us consider a simple one-dimensional case. An electron moves straight from the negative electrode to the positive electrode of a plate capacitor. The two electrodes have the distance d to each other. You have determined the wave function by solving the Schrodinger equation. It's just a plane wave. The amplitude A is unknown. Therefore, you use the normalization condition to normalize the wave function and determine A at the same time. You know with 100% probability the electron must be between the two electrodes. If the negative electrode is at x equals 0 and the positive electrode at x equals d, then the electron is somewhere between these two points. So the integration limits are 0 and d. First, you have to determine the squared magnitude. The magnitude of the wave function is formed in the same way as the magnitude of a vector. This is the first time the usefulness of the complex exponential function comes into play. It is always true that the magnitude of e to the i k x minus omega t is equal to 1. Perfect. You don't have to do complicated math. So the squared magnitude of the wave function is A squared. Insert the squared magnitude into the normalization condition. The amplitude A is independent of x, so it is a constant and you can put it before the integral. Integrate. Insert the integration limits. Rearrange for amplitude and you get 1 over square root of d as amplitude. In this way, you normalize the wave function and determine the amplitude for a given problem. In this example of the normalization condition, you can see from the amplitude that it has the unit 1 over square root of meter. Because the exponential function is dimensionless, the wave function has the same unit as the amplitude. In three dimensions, the unit of the wave function is 1 over square root of cubic meter. Once you have determined the wave function by solving the Schrodinger equation, you can use it to find out not only the probability of the particle's location, but also the mean value of the location and that of all other physical quantities. For example, the mean value of the momentum, the velocity or kinetic energy. In quantum mechanics, the mean value is written in angle brackets. Why only a mean value and not an exact value and how this can be determined, you will learn in detail in another video. Important for you is to know that you can describe a quantum mechanical particle with a wave function as well as you can describe a classical particle with a trajectory. We can learn something about the behavior of the wave function from the Schrodinger equation without having solved it already. In the Schrodinger equation, bring the term with a potential energy to the left-hand side and bracket the wave function. The potential energy generally depends on the location x. One could also call it potential energy function or ambiguously but briefly potential. It indicates the potential energy of a particle at the location x. This function could be, for example, quadratic in x, called harmonic potential. But the potential energy function also have a completely different behavior. Here we'll look at an example of a quadratic potential energy function. If a particle is in this potential, then it has greater potential energy when it is further away from the origin. The potential energy function should be given before you can solve the Schrodinger equation. According to the law of conservation of energy, the total energy w is a certain constant value regardless of where this particle is in this potential. In a diagram, it is a horizontal line that intersects our one-dimensional potential energy function in two points, x1 and x2. A classical particle can under no circumstances exceed this total energy. Consequently, it can only move between the reversal points, x1 and x2. Between these two points, it can completely convert its kinetic energy into potential energy and vice versa without moving outside of x1 and x2. The particle is trapped in this region. If we find the particle outside of x1 and x2, its potential energy would be greater than its total energy. So the kinetic energy, w minus w-pot, would be negative. A negative kinetic energy could have the particle only with an imaginary velocity. But imaginary velocity is not measurable, not physical. Therefore, a negative kinetic energy is also not physical. This is exactly why we can expect that the classical particle can never be outside of x1 and x2. We call the region outside x1 and x2 the classically forbidden region, and the region within x1 and x2 as the classically allowed region. In quantum mechanics, however, you often have the case that the wave function in the classically forbidden region is not zero. But if the wave function is not zero, the probability of finding the particle in the classically forbidden region is not zero either. This property of the wave function allows the particle to pass through regions that are classically forbidden. This behavior of the wave function is the basis for the quantum tunneling. At first glance, this seems to be a serious contradiction, because if the wave function enters the forbidden region, the quantum mechanical particle can be found there with a certain probability. But there, its potential energy is greater than its total energy. Consequently, the quantum mechanical particle would have to have a negative kinetic energy. But this contradiction is resolved by the Heisenberg's uncertainty principle. According to this principle, the potential and kinetic energy of a particle cannot be determined simultaneously with arbitrary accuracy. If the particle had a potential energy greater than its total energy, it can be calculated that the uncertainty in the measurement of kinetic energy is always at least as large as the energy difference w-w-pot. This energy difference is the kinetic energy of a classical particle, but not of a quantum mechanical particle. In quantum mechanics, you have to get rid of the idea that a quantum mechanical particle has an exact potential and exact kinetic energy simultaneously. Because of the uncertainty principle, you cannot claim that the kinetic energy in the forbidden region becomes negative because w-w-pot is not a kinetic energy. Therefore, a quantum mechanical particle can with a low probability be in the classically forbidden region without violating the principles of physics. From the Schrodinger equation, you can extract interesting information about the behavior of the wavefunction. This can be seen when you look at the signs of the energy difference and the wavefunction. There are two signs together with a minus sign on the other side of the equation determine the sign of the second spatial derivative of the wavefunction. The second spatial derivative is called curvature. You can visualize the curvature as follows. Imagine the wavefunction is a road that you want to ride along with a bicycle. A negative curvature means that the wavefunction bends to the right. You would have to steer your bicycle to the right. A positive curvature on the other hand means that the wavefunction curves to the left. You would therefore have to steer your bicycle to the left. Let us first look at the two cases where the energy difference is positive. Here the total energy is greater than the potential energy. So we are in the classically allowed region. The first case is the wavefunction at location x is positive. Then the right hand side with a minus sign in front of it must also be positive to satisfy the equation. Consequently, the curvature at this location x must be negative. The curvature causes the positive wavefunction to always bend towards the x-axis. Even if it rises a little initially, this rise will become smaller and smaller until the wavefunction inevitably falls. The second case is the wavefunction at the location x is negative. Then the curvature at this location x must be positive. Again, the curvature causes the negative wavefunction to bend from below towards the x-axis. If you compare the sign of the curvature with the sign of the wavefunction in these two classically allowed cases, you will see that they always have an opposite sign. In summary, this behavior results in an oscillation of the wavefunction around the x-axis. In the classically allowed region between the locations x1 and x2, we can say about each wavefunction that it oscillates. The higher the total energy w of the particle, the more the wavefunction oscillates. If the total energy is lower, the wavefunction oscillates less. Now let's look at the forbidden region and see how the wavefunction must behave there. In the forbidden region, the potential energy is greater than the total energy. Its energy difference is therefore always negative. In the classically forbidden region, the wavefunction and the curvature do not always have the opposite sign, but the same sign. Therefore, the wavefunction is no longer forced to bend towards the x-axis. Instead, it can show two other behaviors. On the one hand, it could grow into positive or negative infinity, but this behavior is not physical because it violates the normalization condition. On the other hand, it can drop exponentially. This behavior is compatible with the normalization condition and therefore physically possible. If you take a closer look, you will notice that this behavior is only achieved for certain values of the total energy. This is called quantization, which means the fact that the allowed total energies can only take discrete values. If you trap a quantum mechanical particle somewhere, as in our case between x1 and x2, the total energy of this particle is always quantized. We can then accept values w0, w1, w2, w3 and so on, but no energy values in between. For each of these allowed energies, there is a corresponding wavefunction, psi0, psi1, psi2, psi3 and so on. The different possible wavefunctions and the corresponding allowed energies are numbered with an integer n. n is a so-called quantum number. We say a particle with the smallest possible energy, w0, is in the ground state, psi0, and a particle that has an energy greater than w0 is in the excited state. Overall, we can summarize in the classically allowed region the wavefunction oscillates and in the classically forbidden region the wavefunction drops exponentially and the total energy of the trapped particle described by this wavefunction is quantized. You can generalize the one-dimensional Schroding equation to your three-dimensional Schroding equation. We assume that the wavefunction psi depends not only on one spatial coordinate x, but on three spatial coordinates x, y, z. You can also combine the three space coordinates more compactly to a vector r. Vectors are shown in bold here. In the one-dimensional Schroding equation you have to add the second derivative with respect to y and z to the second derivative with respect to x so that all three spatial coordinates occur in the Schroding equation. This is what multi-dimensional analysis tells you to do if you want to convert the one-dimensional Schroding equation into the three-dimensional Schroding equation. This is our time-independent three-dimensional Schroding equation. You can write it more compactly. Bracket the wavefunction. The sum of the spatial derivatives in the brackets form a so-called Laplace operator, nubla squared, sometimes also noted as delta. An operator like the Laplace operator only has an impact when applied to a function because an isolated spatial derivative makes no sense. Here you apply the Laplace operator to the wavefunction psi. The result nubla squared psi gives the second spatial derivative of the wavefunction that is exactly what we had before. Do you see another possible operator on the right-hand side? Bracket the wavefunction. The operator in the brackets on the right-hand side is called Hamilton operator H, or just Hamiltonian. You can use it to write the Schroding equation very compactly. Using the Hamilton operator, you formulated the Schroding equation as an eigenvalue equation, which you probably know from linear algebra. You apply the Hamilton operator, imagine it as a matrix, to the eigenfunction psi. Imagine it as an eigenvector. Then you get the eigenvector psi again unchanged, scaled with the corresponding energy eigenvalue w. With this eigenvalue problem, you can mathematically see why the energy w can be quantized in quantum mechanics. The energy eigenvalues depend on the Hamilton operator. These eigenvalues are discrete for most Hamilton operators that you will encounter. What if the total energy of the quantum mechanical particle is not constant in time? This can happen, for example, if the particle interacts with its environment, and thus its total energy changes. For such problems, the time-independent Schroding equation is not applicable. For this, you need a more general form of the Schroding equation, the time-dependent Schroding equation. Now we assume a time-dependent total energy w of t. For simplicity, we assume that the particle is not in an external field, and therefore has no potential energy. The total energy w of the particle is therefore only the time-dependent kinetic energy. Multiply the equation by the wave function psi. Does the expression w-kinsi look familiar to you? You've already seen it in the derivation of the time-independent Schroding equation when we were looking at the second spatial derivative of the plane wave. Let's use this expression. Now we do some magic. Take the time derivative of the plane wave. The total energy in our case corresponds to the kinetic energy, and this can be written using the frequency omega. Because of the wave particle duality, analogous to the de Broglie wavelength, w is equal to h bar omega. Use this equation to express the frequency omega with the total energy. Rearrange the equation for w-psi. To beautify the equation a little, extend the fraction h bar over i with i. So i squared becomes minus 1. This eliminates the fraction and the minus sign. Now our modified equation is ready for insertion into w-psi. And you have already obtained the time-dependent Schroding equation for a special case for a particle without potential energy. Evan Schrodinger now assumed that the equation is also fulfilled if the time-dependent potential energy multiplied by the wave function is added to the kinetic term. This is the one-dimensional time-dependent Schroding equation. It has a similar form as the time-independent Schroding equation with the only difference that the term for the total energy has changed. Analogous to the one-dimensional time-independent equation, the time-dependent Schroding equation can be extended to three dimensions. Just replace the second spatial derivative with the Laplace operator, nubla squared. And you're done. Solving the time-dependent Schroding equation is not that easy. But you can simplify the solving of this partial differential equation considerably if you convert it into two ordinary differential equations. One differential equation then depends only on time and the other only on space. The trick is called separation of variables. Because you separate the space and time dependencies from each other, this is a very important approach in physics to simplify and solve differential equations. The only requirement for variable separation is that the potential energy does not depend on time t, but it may well depend on location x. The wave function itself, of course, can still depend on both location and time. The following variable separation. Divide the wave function psi into two parts. Into a part that depends only on the location x, let's denote this function as the small Greek letter psi. And into a part that depends only on time t. Let's denote this function as the small Greek letter phi. Next, write the original total wave function psi as a product of the two separated wave functions. Not all wave functions can be separated in this way, but since the Schrodinger equation is linear, you can form a linear combination of such solutions and thus obtain all wave functions, even those that cannot be separated. As you can see from the time-dependent Schrodinger equation, the time derivative and the second spatial derivative occur there. Let's execute these two derivatives independently from each other. First, take the derivative of the separated wave function with respect to time t. Then, take the second derivative of the separated wave function with respect to x. Now you have two equations. You can now insert the time derivative and the space derivative into the Schrodinger equation. Additionally, insert the separated wave function in the term with the potential energy. You can make a little plastic surgery here. We know that phi of t only depends on time and that the psi of x only depends on space. You can imply this information by replacing the partial derivatives noted with curved del with the so-called total derivatives, noted with the regular d. This way you don't have to write phi of t or psi of x all the time, but can simply write phi and psi. From the notation of the derivative it is then clear that the function depends only on one variable, the one that's in the derivative. It is not important if you do not know what a total or partial derivative is. Just replace the del symbols with regular d symbols. Now you have to reformulate this differential equation so that its left-hand side depends only on time t and its right-hand side only on location x. This is achieved by dividing the equation by the product psi times phi. What does it do for you? If you vary the time t which only occurs on the left-hand side only the left-hand side of the equation will change while the right-hand side remains unchanged. But if the right-hand side does not change with time it is constant. It corresponds to the total energy w which is constant in time. Denote the right-hand side as a constant w. Bring i h bar to the other side. 1 over i becomes minus i. The same applies for the space coordinate. If you vary x on the right-hand side the left-hand side remains constant because it is independent of x. Because of the equality the left-hand side must correspond to the same constant w. Denote the left-hand side with a constant w. If you now multiply the differential equation by psi you get the time independent stationary Schrodinger equation. In this version the wave function psi depends only on x and not on time because of the variable separation. What did you achieve overall? As already mentioned with this you have obtained two less complicated ordinary differential equations from a more complicated partial differential equation. You can even immediately specify the solution for the temporal differential equation. It is easily solved with pencil and paper. Multiply the whole equation by dt. Now you just have to integrate both sides. On the left-hand side the integration of 1 over phi yields the natural logarithm and on the right-hand side the integration yields t. We can omit the integration constant and include it in small psi. Rearranging for the searched function phi gives the solution of the differential equation. You cannot solve the second differential equation that is the time independent stationary Schrodinger equation without a given potential energy function. But it's okay with the separation of variables we have simplified the solving process a lot. The great thing is now instead of solving a more complicated time dependent Schrodinger equation, you can solve this stationary Schrodinger equation. This way you only get the space dependent part psi of the total wave function. But if you look at the separation ansatz, you just have to multiply the space dependent part psi with the time dependent part phi to get the total wave function. And for phi you have found that it is an exponential function and this time dependent part is the same for all separable wave functions you will encounter. A wave function which can be separated into a space and time dependent functions describes a stationary state. By stationary we mean that the wave function itself is time dependent but its squared magnitude is not. All other physical quantities describing the particle are also time independent. For example a particle whose wave function is a stationary state has a constant mean value of energy, constant mean value of momentum and so on. So that's it. I hope that after watching this video you have gained a solid basic knowledge of the Schrodinger equation. But remember the Schrodinger equation is not generally applicable. It is a non-relativistic equation. This means that it fails for quantum mechanical particles that move almost at the speed of light. Furthermore, it doesn't naturally take into account the spin of a particle. All these problems are only solved by the more general equation of the quantum mechanics but the Dirac equation. You will learn this in another video. With this in mind bye and see you next time.