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Particle in a Box
Many introductory chemistry textbooks introduce the Schrodinger
Equation, but students don't understand what it means. This section
is optional; if you want to know where orbitals come from, it can
help you understand. It will
be easier to follow this section if you know a little calculus
(basically, what a derivative is).
The Schrodinger Equation is the starting point for describing the
motions of electrons as waves. De Broglie suggested that their
stable "orbits" in the Bohr model were standing waves analogous to
those in a guitar string. Schrodinger extended this theory using the
wave equation and wavefunction. Instead of circular orbits,
Schrodinger's waves were 3D and took up the whole space of the atom,
more like vibration of air in a spherical flute than the vibration
of a circular string. The wavefunction Ψ(x,y,z,t) describes the
amplitude of the electron vibration at each point in space and
time. Oddly, Schrodinger seems to have
proposed the wavefunction without fully understanding
what it means, but it worked! Here we will describe the
time-independent Schrodinger equation for simplicity, which describes
the standing waves. We will also consider only a 1-dimensional system,
such as a particle that only moves linearly, also for
simplicity. Thus, we will find Ψ(x) for a very simple
situation.
Schrodinger proposed that a standing wave is described by
the wavefunction Ψ when the it fits the following differential
equation
HΨ = EΨ
where H is the Hamiltonian operator, which finds the total energy of
the system E. (This approach uses the linear algebra concept of an
eigenfunction and eigenstate, but don't worry if you don't know what
these are.) Kinetic energy KE is given by
KE = p2/2m
where p is the momentum (p = mv). For a particle moving in 1D (along x) Schrodinger assumed
that a permissible general form of Ψ is
Ψ(x, t) = Aei(px - Et)/ħ
where A is a constant and i is the imaginary number (i2 =
-1). (This comes from the equations E = hν and the de Broglie
relationship λ = h/p. These equations connect energy to time and
distance to momentum through Planck's constant. These are also the quantities that are mutually limited by
the Uncertainty Principle.) If this is true, the derivative of the
wavefunction with respect to x is
Notice that this is kind of like the equation HΨ = EΨ in that
we get the original wavefunction multiplied by some important
quantity, like energy or momentum. So the momentum operator
p (like
the Hamiltonian operator, which gives the energy) gives the
momentum p, and can be written like this:
We can
write the Schrodinger equation using this Hamiltonian (which gives total energy, KE + PE)
H Ψ(x) = − | ħ2 | |
d2Ψ | + V(x) Ψ(x) = E Ψ(x) |
2m | dx2 |
The potential energy is given by V(x), which just depends on the
position. The kinetic energy is calculated using the equation above,
using the square of the momentum operator (thus, the first
derivative in the momentum operator becomes a second derivative when
the operator is squared).
Now, if we choose a function V(x) we can find the wavefunctions that
fit!
We will use a simple example: a particle in a box (in 1D). The
potential is 0 inside the box and infinite outside the box. So we
will just know that the particle has to be inside the box, but use V
= 0. Then our Schrodiner Equation looks like this
H Ψ(x) = − | ħ2 | |
d2Ψ | = E Ψ(x) |
2m | dx2 |
or basically the second derivative of Ψ is a constant times
Ψ. There are different forms of the solution, but we'll just
choose a simple one.
Thus,
d2 | sin(ax) = −a2 sin(ax) |
dx2 |
So we can pick sin(ax) or cos(ax) or a sum of these for the
wavefunction:
Ψ(x) = sin(ax) + cos(bx)
So far, there is no quantization! The coefficient a can have any value. But just
like a string on a guitar, the amplitude of Ψ has to be 0 at the
edges of the box. If we just use Ψ(x) = sin(ax), then if the
box is from x=0 to x=L, we need to have an integer number of
half-wavelengths in the box. So
a = nπ/L
so that
Ψ(0) = Ψ(L) = 0
To summarize,
Ψ = sin(nπx/L)
is a solution of the Schrodinger Equation for the 1D
particle-in-a-box system. Try putting this in and see what the
energy is! You should get
There are an infinite number of solutions, or wavefunctions that
satisfy the Schrodinger Equation, corresponding to n = 1,
2, 3... and any sum of these wavefunctions is also a solution. What
do they mean? The amplitude of the particle wave is given by
Ψ. The next section explains the meaning of the wavefunction in
more detail, now that you have been introduced to the math.
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Last modified: Wed Apr 2 15:13:53 KST 2014