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CHEM 2060 Lecture 18: Particle in a Box L18-1
Atomic Orbitals
If electrons moved in simple orbits, p and x could be determined, but this
violates the Heisenberg Uncertainty Principle.
We can only talk about the probability of finding electrons in a region of space.
• Electron Density or Probability Density.
We can think of a spherical “cloud” around the nucleus that is “darker” near the
center and grows “paler” as the distance from the center increases. This
describes the probability of finding the electron.
(darker cloud = higher probability)
(a)
(a)
(b)
Spherical electron density “cloud”
(b)
Circular cross-section of the “cloud”
with a spherical boundary surface enclosing
90-99% probability region of a 1s orbital.
CHEM 2060 Lecture 18: Particle in a Box L18-2
The Wave Equation – here comes the math!
Mathematically, what are we trying to do?
• We need to CONFINE an electron to an atom (i.e., set spatial boundaries)
We need to describe the electron’s position using its wave-like properties.
• We will have to choose a wavefunction of some kind. (sine? cosine?)
We can expect something that looks like normal modes of vibration
(fundamental and overtones … lowest frequency and higher frequencies…)
As energy is related to frequency (recall: E = hν), it is no surprise that we find
the electron can only have certain energies (that correspond to “normal modes”)
• Energy is quantized!
Before we try to set up the wave mechanics for the simplest atomic problem
(i.e., H atom), we’ll look at a simpler problem…particle (or electron) in a box.
CHEM 2060 Lecture 18: Particle in a Box L18-3
Particle in a Box – (3D example: golf ball in a sealed box in zero gravity)
“Box” – region of space where potential energy V of particle is zero.
∴ Energy of the particle inside the box is entirely kinetic T.
Recall: E = T + V
and T = 12 mv
2
The walls of the box are “infinitely high”…
in other words, V = ∞.
• The particle cannot
€ get out of the box!
This scenario defines a
“Square Well Potential”
1D square well potential
• “linear” description of space
…also possible to have a 2D box (plane) or
a 3D box (cube, sphere)
CHEM 2060 Lecture 18: Particle in a Box L18-4
(Quantum) Particle in a Box – an electron, not a golf ball
We must describe the (quantum) particle in a box using Wave Mechanics.
Let’s look at some examples of wave equations…progressively more complex…
• A simple sine function: y = sinx
• Linear combination of wavefunctions: y = asinx + bcosx
• Harmonic oscillator wavefunction: x(t) = x + a sin(ωt) + b cos(ωt)
where x(t) is the position x as a function of time t
and xo is the resting position
and ω is the frequency of oscillation (and has units of reciprocal time)
and the constants a and b€have units of length because x(t) is the position
Note that the Harmonic oscillator wavefunction can be written another way:
x(t) = x + asin(ωt) + bcos(ωt) = x + Ae −iωt + Beiωt
CHEM 2060 Lecture 18: Particle in a Box L18-5
In quantum mechanics, we use the symbol ψ (psi) to indicate a wavefunction.
In this example, ψ is the amplitude A as function of
position x.
% 2πx (
ψ (x) = A sin'
*
& λ )
CHEM 2060 Lecture 18: Particle in a Box L18-6
Waves (think: atom) in a box will form as standing waves:
So this is what we want to
describe mathematically…
€
CHEM 2060 Lecture 18: Particle in a Box L18-7
Back to the Particle in a Box … i.e., electron confined in 1D (linear space)
This is a very simple model designed to illustrate how to use a wavefunction to
describe the spatial distribution (position) of a quantum particle, like an electron.
Reminder: “rules” of this model, for a 1D box of length L in the x direction:
• Particle (mass m) is free to move along x-axis between x = 0 and x = L.
• No change in potential V anywhere inside the box (set V = 0 for 0 < x < L).
• At x = 0 and x = L and beyond, the particle meets an infinite barrier (V = ∞).
• We now describe the particle with a wavefunction (“matter waves”)…
…let’s just start with a simple sine wave and see where that gets us…
% 2 πx (
ψ (x) = a sin'
*
& λ )
where λ =
h Recall: This is just tracking the
p amplitude as a function of x.
CHEM 2060 Lecture 18: Particle in a Box L18-8
Eigenvalues and Eigenfunctions … extracting useful info from the wavefn.
% 2 πx (
ψ (x) = a sin'
*
& λ )
where λ =
h
p
Now, what can this wavefunction do for us?
• One feature of a wavefunction is that the wavelength λ may be extracted from
the function as an eigenvalue.
In other words, if we perform an appropriate mathematical operation on the
wavefunction, the “answer” is a constant multiplied by the original
wavefunction. The wavelength is part of the constant…therefore an eigenvalue.
Easier to show you than to try to explain further…
& 2 π )2
& 2 πx ) If we take the 2nd derivative w.r.t. x of our
d2
ψ (x) = −( + a sin(
+
2
'λ *
' λ * wavefunction, we get back our original
dx

2
= ψ ( x)
wavefunction multiplied by –(2π/λ) .
CHEM 2060 Lecture 18: Particle in a Box L18-9
We say that the function ψ(x) is an
eigenfunction of the second derivative with
respect to position x.
& 2 π )2
& 2 πx )
d2
ψ (x) = −( + a sin(
+
2
'λ *
' λ *
dx


= ψ ( x)
All right, were does that get us?
Well, we have figured out a way of extracting the wavelength λ (as an
eigenvalue) from the wavefunction. This is very important because:
€
• wavelength of a particle is related to its momentum by de Broglie’s equation:
h
p=
λ
1 2 p2
h2
• energy of the particle in the box is entirely kinetic: T = mv =
=
2
2m 2mλ2
€
• so we can figure out the energy! This means that energy is an eigenvalue, too!
HOMEWORK: Make sure you can walk your
€ way through this math!
CHEM 2060 Lecture 18: Particle in a Box L18-10
Okay, the constant that has “popped out” of the wavefunction when the 2nd
derivative w.r.t. x was applied is actually:
% 2 π (2
−' *
&λ )
h2
We have figured out the energy as a function of the wavelength: E =
2mλ2
€
Let’s put these pieces together and rewrite the constant in terms of energy rather
than wavelength.
2π )
(
−
λ2
€
2
2mE (2 π )
= −
h2
2
€
2mE
= − 2

h
Recall: The reduced Planck’s constant  =
2π
CHEM 2060 Lecture 18: Particle in a Box L18-11
The Schrödinger Equation …fundamental equation for Quantum Mechanics
Okay, so we’ve figured out that if we take d2/dx2 of our sine function ψ, we can
extract the wavelength λ. From this we get the kinetic energy of the particle T,
which in our model is equal to the total energy E.
& 2 π )2
& 2 πx )
& 2mE )
d2
ψ (x) = −( + a sin(
+ = −( 2 +ψ (x)
2
'λ *
' λ *
'  *
dx


= ψ ( x)
In fact, we have just derived the Schrödinger Equation for the particle in a box:
− 2 d 2
⋅ 2 ψ ( x ) = Eψ ( x )
2m dx
€
More generally:
€
Hˆ ψ = Eψ
(yikes! weird symbols!)
CHEM 2060 Lecture 18: Particle in a Box L18-12
Operators Ô … let’s look at this weird symbol (Ĥ) before we continue…
One of the things that can make quantum mechanics seem so difficult is the
incorporation of mathematical symbols that you haven’t seen before…like Ĥ.
The little “hat” over the letter means it is an operator.
[DEF]: An operator is a mathematical “operation” that “spits out” a function
when applied to another function.
CHEM 2060 Lecture 18: Particle in a Box L18-13
The Hamiltonian Operator Ĥ and the Schrödinger Equation
In the Schrödinger equation, the Hamiltonian operator Ĥ is whatever
mathematical operator that is necessary in order to get the total energy of the
system E from the wavefunction ψ.
In our particle in a box example, we solved the Schrödinger equation as:
− 2 d 2
⋅ 2 ψ( x) = Eψ( x)
2m

dx


Hˆ
…so the Hamiltonian Operator Ĥ is simply (–ħ2/2m)(d2/dx2)
In other words, this is the operation we have to perform on our wavefunction in
€ the energy E.
order to extract
The particle in the box is a very simple model, so the actual form of the
Hamiltonian is easy to write down.
It isn’t much more complex for models of real atoms or molecules!
CHEM 2060 Lecture 18: Particle in a Box L18-14
Why is the Schrödinger Equation so Important?
In classical mechanics, the differential equations that prescribe the motion of a
particle or system of particles are call equations of motion. In Newtonian
mechanics, the equations of motion are given by:


F = ma
We can derive everything we need to know about the system from this.
Hamilton (and €
others) came up with other ways of formulating the equations of
motion. Hamilton’s version is:
H=T+V
What this means is that if we can figure out the kinetic and potential energy of a
system, we know everything about it!
The Schrödinger equation says this is possible for ANY quantum
system so long as you choose/design an appropriate wavefunction!
Hˆ ψ = Eψ