This is how the time-independent
Schrödinger equation looks like. And this is how the time-dependent Schrödinger equation looks like.
We can already state that the Schrödinger equation is - mathematically speaking - a
partial differential equation of second order. differential equation means, that the
searched quantity is not a variable, but a function and in the equation
there are derivatives of this function. The function we are looking for in the Schrödinger
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(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 = 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 (r(0) = (0,0,0)). Once you have
determined the trajectory r(t) by solving the differential equation, you can then use it to
find out all other relevant quantities, such as the particle's velocity (v = dr/dt), its momentum
(p = m v) or its kinetic energy (W = ½ m v^2). 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 behavour is described
by the uncertainty principle; a fundamental principle of quantum mechanics,
that cannot be bypassed. So we cannot determine a trajectory r(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 Schrödinger equation. In quantum mechanics you do not
calculate a trajectory r(t), but a so-called wave function SAI. 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 Schrödinger equation. It is only through this novel approach to
nature using the Schrödinger 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 Schrödinger 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 applications that the Schrödinger
equation has given us. So let us first find out, where this powerful equation comes from.
Unfortunately it is not possible to derive the Schrödinger 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 and modern technical society
show that the Schrödinger equation works perfectly and is applicable to most quantum mechanical
problems. Let us try to understand the fundamental principles of the Schrödinger 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 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 a potential energy, lets 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 = h/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 * 10^{-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 a 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 the de-Broglie wavelength, but with the wavenumber k. k is
2 pi over lambda. Thus the momentum becomes h times k over 2pi. „h over 2pi“ 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 behaviour 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. The larger the de-Broglie wavelength, the more likely the object behaves
quantum mechanically. 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 SAI. 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 cos(kx - \omega t)
It does not 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 - \omega t) is the real part and A times Sine (kx - \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 a complex plane wave
can also be a possible solution to the Schrödinger 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 - \omegat 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 counterclockwise. 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. The
particle is in the state \Psi. So always remember: When we talk about state in quantum mechanics,
we mean the wave function. Of course, there are different states that different particles can
take under different conditions. The plane wave 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/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 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“ k^2
and a minus sign. The minus sign because „i squared is equal to -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 Schrödinger 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 = \hbar k and replace k^2.
Now, to bring the kinetic energy into play, replace the momentum p^2 with the help of the
relation: „kinetic energy is equal to p^2/2m“. You can easily obtain this form from „one
half m v^2“ by rearranging the momentum „p is equal to m times v“ 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 that 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 Schrödinger equation.
This Schrödinger equation is one-dimensional and time-independent. You can recognize the
one-dimensionality immediately by the fact that only the derivative with respect to a single
space coordinate occurs. In this case with respect to x. And you can recognize the time independence
of the Schrödinger equation by the fact that a constant total energy occurs. In general,
however, the wave function may be time-dependent. Let's recap for a moment. To
derive the Schrödinger 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 Schrödinger 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 Schrödinger
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 can not 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 Schrödinger equation would no
longer agree with the results of experiments. For an experimentalist, however,
such complex 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 the
three-dimensional case a probability per volume. Let's stay with the 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(t):
The integral of the squared magnitude indicates the probability P(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. The 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 particular location „a“, but only for a space region (here
between a and b), because otherwise the integral would be zero. Obviously! Because there
are infinitely many space points on the distance between a and b. If each of these space points had
a finite probability, then the sum (that is the integral) of all the probabilities would be
infinite, which would make no sense at all. Therefore, we always calculate the probability to
find the particle in a specific region of space. In order for the statistical interpretation to
be compatible with the Schrödinger equation, the solution of the Schrödinger 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 one hundred percent somewhere
between „minus infinity“ and „plus infinity“. In other words, the integral for the probability,
integrated over the entire space, must be 1. The normalization condition is a necessary
condition that every physically possible wave function must fulfill. After solving the
Schrödinger equation, the found wave function \Psi must be normalized using the normalization
condition <q>17</q> . 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(konjunktiv was vs were?) not the case, the Schrödinger equation and the statistical
interpretation would be incompatible. There are of course solutions to the Schrödinger equation,
for example \Psi = 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-ingrable functions in mathematics. If you know with one hundred percent 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 Schrödinger equation. 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 that with one hundred percent probability the electron must be between the two electrodes. If the negative electrode is at x=0
and the positive electrode at x=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 |e^{ikx| „the magnitude of e to the i kx 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 die
integration limits. Rearrange for amplitude and you get „one over square root of d“ as amplitude.
In this way you normalize the wave function and determine the amplitude for a given problem.
In the 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 wavefunction
is "1 over squareroot of cubic meter". Once you have determined the wave function
by solving the Schrödinger 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 the wave function as well as you can describe
a classical particle with the trajectory. We can learn something about the behavior of
the wave function from the Schrödinger equation without having solved it already.
In the Schrödinger equation, bring the term with the 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 could
also have a completely different behavior. Here we 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 even solve the Schrödinger equation. According to the law of conservation of energy,
the total energy W is a certain constant value regardless of where the particle is in this
potential. In a diagram it is a horizontal line that intersects our one-dimensional potential
energy function in two points x_1 and x_2. A classical particle can under no circumstances
exceed this total energy! Consequently, it can only move between the reversal points x_1 and
x_2. Between these two points, it can completely convert its kinetic energy into potential energy
and vice versa - without moving outside of x_1 and x_2. The particle is trapped in this region.
If we find the particle outside of x_1 and x_2, its potential energy would be greater than its
total energy. So the kinetic energy W - 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 a classical particle can never be outside of x_1 and x_2. We call the
region outside x_1 and x_2 the classically forbidden region. And the region within x_1
and x_2 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 Schrödinger equation you can
extract interesting information about the behavior of the wave function. This
can be seen when you look at the signs of the energy difference and the wave function.
Their two signs, together with the minus sign on the other side of the equation, determine the
sign of the second spatial derivative of the wave function. The second spatial derivative
is called curvature. You can visualize the curvature as follows: Imagine the wave function is
a road that you want to ride along with a bicycle. A negative curvature means that the wave function
bends to the right. You would have to steer your bicycle to the right. A positive curvature, on
the other hand, means that the wave function 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 wave function at
location x is positive. Then the right hand side with the minus sign in front of it must also
be positive to satisfy the equation. Consequently, the curvature at this location must be negative.
The curvature causes the positive wave function to always bend towards the x-axis.
Even if it rises a little initially, this rise will become smaller and smaller
until the wave function inevitably falls. The second case is: The wave function
at the location x is negative. Then the curvature at this
location must be positive. Again, the curvature causes the negative wave
function to bend from below towards the x-axis. If you compare the sign of the curvature with the
sign of the wave function 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 wave function
around the x-axis. In the classically allowed region between the locations x_1 and x_2 we can
say about each wave function that it oscillates. The higher the total energy W of
the particle, the more the wave function oscillates. If the total energy is
lower, the wave function oscillates less. Now let's look at the forbidden region and
see how the wave function 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 wave function and the curvature
do not always have the opposite sign, but the same sign. Therefore the wave function
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 x_1 and x_2, the total energy of this particle is always
quantized. It can then accept values W_0, W_1, W_2, W_3 and so on, but no energy values
in between. For each of these allowed energies there is a corresponding wave function
Psi_0, Psi_1, Psi_2, Psi_3 and so on. The different possible wave functions
and the corresponding allowed energies are numbered with an integer n. Here
n is a so-called quantum number. We say: A particle with the smallest possible
energy W_0 is in the ground state Psi_0. And a particle that has an energy greater
than W_0 is in the excited state. Overall, we can summarize: In the classically
allowed region the wave function oscillates and in the classically forbidden region the wave
function drops exponentially. And the total energy of the trapped particle described
by this wave function is quantized. You can generalize the one-dimensional
Schrödinger equation <q>15</q> to a three-dimensional Schrödinger equation. We assume
that the wave function \Psi(x,t) depends not only on one spatial coordinate x but on three
spatial coordinates x,y,z: \Psi(x,y,z,t). You can also combine the three space coordinates
more compactly to a vector \bold symbol{r (vectors are shown in bold here): \Psi(\bold symbol ,t).
In the one-dimensional Schrödinger 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 Schrödinger equation. This is what multidimensional analysis
tells you to do, if you want to convert the one-dimensional Schrödinger-equation into
the three-dimensional Schrödinger-equation. This is our time-independent
three-dimensional Schrödinger equation. You can write it more compactly.
Bracket the wave function. The sum of the spatial derivatives in
the brackets form a so-called Laplace operator \nabla^2 (Nabla 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 derivate makes no
sense. Here you apply the Laplace operator to the wave function \Psi.
The result \nabla^2 \Psi gives the second spatial derivative of the wave
function, that is exactly what we had before. Do you see another possible operator on the
right hand side? Bracket the wave function. The operator in the brackets on the right hand
side is called Hamilton operator \hat{H or just Hamiltonian. You can use it to write the
Schrödinger equation very compactly. Using the Hamilton operator, you formulated the
Schrödinger 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 energy conservation law is not
fulfilled? In other words: What if the total energy of the quantum mechanical particle is not
constant in time? This can happen, for example, when the particle interacts with its environment,
thereby changing its total energy. For such problems, the time-independent Schrödinger
equation is not applicable. For this you need a more general form of the Schrödinger equation,
the time-dependent Schrödinger equation. Now we assume that the law of conservation of
energy is generally not fulfilled. So we assume a time dependent total energy W(t). For simplicity,
we assume that the particle is not in an external field and therefore has no potential energy W_pot.
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 expressionW_kin
\Psi looks familiar to you? You've already seen it in the derivation of the
time-independent Schrödinger equation when we were looking at the second spatial derivative
of the plane wave. Let’s use this expression. Now we make some magic. But as I said, nobody
has yet succeeded in deriving the time-dependent Schrödinger equation from fundamental
principles. But the important thing is that it still works perfectly in experiments.
Let’s do it step-by-step. 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 = \hbar \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 “hbar over i“ hbar/i with i. So „i squared “i^2 becomes -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
Schrödinger equation for a special case - for a particle without potential energy.
Erwin Schrödinger now assumed that the equation is also fulfilled if the time-dependent potential
energy W_pot(x,t) is added to the kinetic term. Thus it has a similar form to the time-independent
Schrödinger 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 Schrödinger
equation can be extended to three dimensions. Just replace the second spatial derivative with
the Laplace operator \nabla^2. And you're done! Solving the time-dependent
Schrödinger 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 dependence 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. Make the following variable separation. Divide
the wave function \Psi(x,t) into two parts: Into a part that depends only on the location
x. Let's call this function with a little Greek letter "psi." \psi(x)
And into a part that depends only on time t. Let's call this function
with a small Greek letter "phi" \phi(t) Next, write the original 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 Schrödinger equation is linear, you can form a linear combination of such
solutions and thus obtain all wavefunctions (even those that cannot be separated).
As you can see from the time-dependent Schrödinger equation, the time derivative and
the second spatial derivative occur there. Take this two derivatives independently
from each other using the product rule. So. Take the derivative of the separated
wave function with respect to time t. 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 Schrödinger equation. Also insert the separated wave function
in the term with the potential energy. You can make a little plastic surgery here. We
know that \phi(t) only depends on time and that \psi(x) only depends on space. You can imply this
information by replacing the partial derivatives (noted with curved Del \partial ) with so-called
total derivatives (noted with a non-italic d ). This way you don't have to write \phi(t) or
\psi(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 non-italic 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 \phi .
What is the good of this? If you change 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. A constant that has the unit
of energy. It corresponds to the total energy W which is time-independent. Denote
the right hand side as a constant W. Bring i \hbar to the other
side. Here 1/i becomes -i. The same applies for the space coordinate. If you
change 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 the constant W. If you multiply the differential
equation only by \psi, you get the time independent Schrödinger
equation. In this version, the wave function \psi(x) 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 specify the solution for the
temporal differential equation directly. 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/\phi yields the natural logarithm and on the
right hand side the integration yieldst. 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 Schrödinger equation, without a given potential energy function W_pot
. But it’s ok. 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 Schrödinger
equation, you can solve this time-independent Schrödinger equation instead. This way you
only get the space-dependent part \psi(x) of the whole wave function.
But, if you look at the separation ansatz, you just have to multiply space-dependent
Part psi(x) with time-dependent part phi(t) to get the total wave function. And for phi you
have found that it is an exponential function. This time dependent part is the same for
all problems you will encounter. Perfect! 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 < W>,
constant mean value of momentum < p>, and so on. So that's it! I hope that after watching this
video you have gained a solid basic knowledge of the Schrödinger equation. But remember that the
Schrödinger 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 does not naturally take into account the spin of a particle. All these
problems are only solved by the more general equation of quantum mechanics, by the Dirac
equation. You will learn this in another video.
