probability of finding electrons in a region of space

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) 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.

(a) (b)


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.


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

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T = 12 mv2

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)


(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:

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x(t) = x + asin(ωt)+bcos(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:

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x(t) = x + asin(ωt) + bcos(ωt) = x + Ae−iωt + Beiωt


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.

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ψ(x) = A sin2πxλ

% & '

( ) *


Waves (think: atom) in a box will form as standing waves:

So this is what we want to describe mathematically…


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…

Recall: This is just tracking the amplitude as a function of x.

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ψ(x) = asin 2πxλ

% & '

( ) * where λ =

hp


Eigenvalues and Eigenfunctions … extracting useful info from the wavefn.

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…

If we take the 2nd derivative w.r.t. x of our wavefunction, we get back our original wavefunction multiplied by –(2π/λ)2.

€

ψ(x) = asin 2πxλ

% & '

( ) * where λ =

hp

€

d2

dx2ψ(x) = −

2πλ

& ' (

) * + 2asin 2πx

λ& ' (

) * +

=ψ(x)


We say that the function ψ(x) is an eigenfunction of the second derivative with respect to position 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:

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p =hλ

• energy of the particle in the box is entirely kinetic:

• 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!

€

d2

dx2ψ(x) = −

2πλ

& ' (

) * + 2asin 2πx

λ& ' (

) * +

=ψ(x)

€

T =12mv2 =

p2

2m=

h2

2mλ2


Okay, the constant that has “popped out” of the wavefunction when the 2nd derivative w.r.t. x was applied is actually:

€

−2πλ

% & '

( ) * 2

We have figured out the energy as a function of the wavelength:

€

E =h2

2mλ2

Let’s put these pieces together and rewrite the constant in terms of energy rather than wavelength.

€

−2π( )λ2

2 = − 2mE 2π( )2

h2 = − 2mE2

Recall: The reduced Planck’s constant

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=h2π


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.

In fact, we have just derived the Schrödinger Equation for the particle in a box: More generally: (yikes! weird symbols!)

€

d2

dx2 ψ(x) = −2πλ

& ' (

) * +

2asin 2πx

λ& ' (

) * +

=ψ(x)

= −2mE2

& ' (

) * + ψ(x)

€

−2

2m⋅d2

dx2ψ(x ) = Eψ(x )

€

ˆ H ψ = Eψ


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.


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: …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 order to extract the energy E. 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!

€

−2

2m⋅

d2

dx2

ˆ H

ψ(x) = Eψ(x)


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:

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F = m a

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!

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ˆ H ψ = Eψ

probability of finding electrons in a region of space

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