This is quantum mechanics 5, the Schrodinger equation. Welcome. The Schrodinger equation,
published in 1926 by Erwin Schrodinger is practically synonymous with quantum mechanics.
It is the primary key which unlocks the secrets of the atom and of chemistry, and which describes
what Schrodinger called an undulatory theory of the mechanics of atoms and molecules.
Now, since this is a video about an equation it will necessarily contain some math, but,
hopefully, you'll find the emphasis is on motivation and illustration of physical concepts.
Schrodinger started with de Broglie's hypothesis that electrons are associated with waves having
wavelength equal to Planck's constant over electron momentum. Recall from the previous
video that this explained the discrete energy level's of the Bohr model of hydrogen as simply
the requirement that the length of an orbit equal an integer number of wavelengths. Schrodinger
sought the general equation governing these waves � a wave equation of the electron.
To understand how he arrived at this we first need to talk about wave equations in general.
Wave equations describe the behavior of fields. A field, which, following Schrodinger, we'll
denote by Greek letter psi, is simply some quantity which exists at all points of a space.
An example is the field of Earth's surface temperature. At any point on Earth this field's
value is simply the temperature at that point. Atmospheric pressure defines another field,
as does ground elevation. For any field Physics seeks an equation to describe its behavior.
In the great majority of cases, Nature quote wants a field to be as uniform as possible.
Usually this means that in equilibrium the field is uniformly constant everywhere. For
example, if we dip a wire rectangle in soapy water the resulting soap bubble will be (neglecting
gravity) uniformly flat. The field of soap bubble elevation will be constant.
But what if the field can't be constant? What then would as uniform as possible mean? Suppose
we bend our wire rectangle into a curve. Rather than trying to stay flat, the soap film will
relax into some smoothly curved shape. What determines this equilibrium shape? The answer
is that the field value at each point equals the average of the neighboring values. This
is what �as uniform as possible� means in general.
Imagine the field psi at some point in three-dimensional space and a small sphere of radius a surrounding
that point. Call the average field value on the sphere psi average. Then equilibrium simply
requires that psi equals psi average. This is so common in physics that we have a special
symbol and operation, called the Laplacian, that measures the difference between psi average
and psi. The Laplacian is denoted by what looks like a triangle to the power two, applied
to the field of interest. Specifically, for a very small radius a, the Laplacian is basically
six over a squared times the difference of psi average and psi. In equilibrium these
values are equal and we have Laplace's equation, which simply tells us that there is no difference
between a field value and the average of its neighbors.
But what if we disturb the equilibrium of a field? Say we use our finger to make an
indentation in an initially flat, stretched membrane. We expect that the indented part
will feel a force pulling it upward. Neighboring points will feel a corresponding force pulling
them downward. As those points move in an attempt to re-establish equilibrium they will
disturb the equilibrium of yet other points which in turn will move in response. The result
is a wave which propagates away from the initial disturbance out to infinity � assuming our
membrane is of infinite extent.
If, instead, the membrane has a boundary the wave will be reflected there and travel until
it's reflected a second time and a third time and on and on. Unless energy is removed from
the membrane, these fluctuations will continue indefinitely. Now, if we look at the displacement
of a single point on the membrane over time we find a fairly complicated curve. The equation
that tells us how to predict this curve is called the wave equation.
To develop the wave equation we'll consider one-dimensional waves of balls connected by
springs. In the situation shown here the middle ball is pulled by two springs. The net force
would be zero if the ball were at the center of the line connecting its neighbors. This
is simply the condition that its position is the average of its neighbor's. This average
position is higher than the actual position so in this case the Laplacian is positive.
We write that the acceleration of the ball is proportional to the Laplacian, that is,
proportional to the difference of the equilibrium position and the actual position. We denote
acceleration with two dots, and the constant of proportionality turns out to be the square
of the wave velocity c � the speed of light or of sound or of whatever it is we're describing.
If we plot the ball's position, which is our field value, through time we get some curve.
At a particular time we're at some point on the curve. We can draw a line through the
curve at this point. This is the velocity or the slope of the field in time and we denote
this with a single dot. We can also draw a circle through the curve at this point and
this is related to the acceleration or the curvature of the field in time that we denote
with two dots. This curvature is also the slope of the slope of the field.
So, the wave equation tells us that when the field value at a point is not in equilibrium
with its neighbors, the plot of that field value vs. time will have a corresponding curvature.
The greater the disequilibrium the greater the curvature. If the field value is less
than the equilibrium value the curvature will be upward (as shown in this figure). Conversely,
if the field value were greater than the equilibrium value the curvature would be downward. Starting
with some initial condition, knowing this curvature at every point allows us to calculate
the field at all future times.
Now, in general this process is pretty complicated. But it's possible to find special waves that
vary in a relatively simple manner such that the displacement at every point is a uniform
sine wave defining a single definite frequency. In quantum mechanics frequency is related
to energy so we expect that such waves will turn out to be very important.
If the field varies as sine of t and at every point we plot the field slope we get another
curve which is the cosine of t. Plotting the slope of the cosine produces minus the sine
of t. For our purposes if the field (or position) is sine of t, the slope (or velocity) is cosine
of t and the curvature (or acceleration) is minus sine of t.
Now imagine we make our sine wave oscillate faster. We write the field as sine 2 pie nu
t where nu is the frequency of oscillation. Since this compresses our curve horizontally
the slopes will increase and our slope function will have a larger amplitude � it's 2 pie
nu times the cosine of 2 pie nu t. Likewise the curvature has an even larger amplitude
� it's minus quantity 2 pie nu squared times sine of 2 pie nu t. This is minus quantity
2 pie nu squared times the field itself.
For waves with this special single-frequency behavior the wave equation has a special form.
Our general wave equation is time-curvature of the field equals wave-speed squared times
the Laplacian of the field. For the single-frequency case the curvature is just minus quantity
2 pie nu squared times the field. If we divide by c squared and use the relation between
wave speed, wavelength and frequency, the wave equation becomes the so-called Helmholtz
equation. This is the wave equation for a single frequency, and it was well-known in
Schrodinger's time having been used to describe light, radio, sound and water waves.
Now, finally, we're ready to develop the Schrodinger equation. Recall de Broglie's hypothesis for
the relation between electron momentum and wavelength � the same relation we have for
photons � p equals Planck's constant over wavelength. From this one over lambda equals
p over h and plugging that into the Helmholtz equation gives us the result shown here. Now
momentum, p, squared, equals m v squared. And we can write this as 2 m times one half
m v squared. This second quantity is just the kinetic energy of the electron, K, so
we have p squared equals 2 m K .
Since the total energy of an electron is its kinetic energy plus its potential energy we
can write the kinetic energy as the total energy E minus the potential energy V. Plugging
2 m times E minus V in for p squared we obtain the Schrodinger equation for a single frequency
in essentially the form Schrodinger presented it in 1926. Incredibly that's basically all
there is to it. Recall that h bar is a shorthand for h over 2 pie.
Since that time people have found it useful to multiply through by h bar squared over
2 m, where, recall, h bar is h over 2 pie, and rearrange to get the following form. Minus
h bar squared over 2 m times the Laplacian of the field plus the potential energy times
the field equals the total energy times the field.
What's attractive about this form is that it corresponds to the classical equation kinetic
energy plus potential energy equals total energy. We see that in quantum mechanics the
Laplacian, the measure of the disequilibrium of the field associated with an electron,
corresponds to the electron's kinetic energy. In the nearly 90 years since it was first
presented, this form of Schrodinger's equation has proven to be one of the most consequential
equations in all of physics. Nearly any scientific inquiry or technological application at the
atomic scale must apply it at some level.
Still, this form of the Schrodinger equation, as the Helmholtz equation from which we obtained
it, is limited to single-frequency fields, corresponding to Bohr's stationary states.
We want to find the general form of the equation. To start, for a frequency nu Planck's relation
gives the energy as h nu, so let's write E psi equals h nu psi.
Now, we've seen that if the field varies as sine 2 pie nu t, the slope in time is 2 pie
nu times cosine 2 pie nu t. That factor of nu means that if we multiply by h bar, the
2 pie cancels and leaves us with h nu. So, h bar times psi dot equals E times the cosine.
This is almost what we're looking for, namely, E times psi, but we have a cosine instead
of a sine. We know that the curvature gives us a sine but it also gives us a nu squared
which would produce a factor of E squared and that's not what we're looking for. Instead,
it seems that in general the E psi term should be replaced by something involving the field's
slope in time, but the nuts and bolts don't quite mesh yet.
The way the nuts and bolts were put together struck many physicists, including Schrodinger,
as rather bizarre and a bit unsettling. It was necessary to make use of the so-called
imaginary unit i . Mathematicians invented i in order to be able to write i squared equals
minus 1, or, i equals the square root of minus 1. The square of any real number cannot be
negative, so i is an imaginary number. Now that's fine for abstract math but it would
be shocking if a theory of physical reality employed an imaginary number. Yet that is
what the general form of Schrodinger's equation does.
Assume the field has a real part, sine 2 pie nu t, plus an imaginary part, i cosine 2 pie
nu t. The field's slope in time, psi dot is then 2 pie nu times the cosine minus i 2 pie
nu times the sine. Multiply this by i times h bar. The 2 pies cancel, h times nu is energy
E, and minus i squared is minus minus 1 equals one. The resulting real part is E times the
sine and the imaginary part is i times E times the cosine. So, i h bar psi dot equals E psi.
Now, we can replace E psi in the single-frequency Schrodinger equation by i h bar psi dot to
get the general Schrodinger equation shown here. Our mathematical description of physical
reality employs unreal, imaginary numbers. Incredible.
Now the steps we just took might seem like they would only work for a field oscillating
at a single frequency nu, so how is this a more general equation, you might ask.
The key is that for the first form of the Schrodinger equation you need to specify the
energy before you solve the equation. This necessarily limits you to a solution at the
single frequency corresponding to the energy E.
In the second form the frequency is extracted by the slope operation so you don't need to
put it into the equation at the start.
Moreover, suppose you have two fields oscillating at different frequencies and you add them
together to get a new field. The slope term, I h bar psi dot operates on the two components
separately and when you plug all of this into the general Schrodinger equation it breaks
up into two, independent single-frequency equations. This idea can be extended to the
sum of any number of components. Using the theory of Fourier, discussed in a previous
video, this means that this second form of Schrodinger's equation is truly general, applying
to any conceivable field.
So we have two forms of the Schrodinger equation. The form for the general case is used when
we need to describe dynamic situations, such as an electron traveling through the double-slit
experiment. The single-frequency form is used to calculate the energy levels of atoms and
the chemical bonds in molecules. In the next part we'll look at animations of Schrodinger
equation solutions to get some idea of what all this math actually says about how the
world works at atomic scales.