CHEM 2060 Lecture 17: Intro to QM L17-1

PART FOUR: Introduction to Quantum Concepts in Chemistry

At the beginning of the 20th century, it was well known that there were certain physical phenomena that could not be explained using CLASSICAL (NEWTONIAN) MECHANICS.

E.g., the observation that light sometimes behaves as a stream of particles

1905: Albert Einstein famously proposed that light is quantized… …in other words, it comes in discrete quanta, what we now call photons.

E.g., The behavior of electrons in the famous “double slit experiment” shows that objects with measurable mass can sometimes behave as waves.

1924: De Broglie hypothesized that all matter possesses wave-like properties.

1905-1938 saw the rapid development of Quantum Mechanics. The math in this new type of mechanics accounts for both the quantization of light and the wave-like nature of matter and is partly built on well-understood wave mechanics (e.g., sine waves, etc.) CHEM 2060 Lecture 17: Intro to QM L17-2

Light is Quantized – Photons! (i.e., light behaves like a particle with no mass)

Einstein proposed that the energy of a single quantum of light (i.e., a photon) is:

E = hν where h is a constant, called Planck’s constant = 6.63 × 10-34 J⋅s and ν (nu) is the frequency of the light; related to the wavelength λ by € c = λν where c is the speed of light = 3.00 × 108 m/s

Photons not only€ carry energy but also linear momentum, p Einstein’s most famous equation states E = mc2, or more accurately, including the relativistic relationship between momentum p and total energy E:

E2 = (pc)2 + (mc2 )2 h so for a photon whose mass m = 0, the photon momentum is p = λ €

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Matter has Wave-like Properties – Electrons are Diffracted!

De Broglie made the bold suggestion that the same equation can be used to describe the relationship between a particle of matter (m>0) and its wavelength.

For a particle of mass m, the momentum p is related to the velocity v (vee) by

p = mv

h Therefore: p = mv = λ € This shows the relationship between the wavelength of a particle and its momentum (i.e., its mass and velocity) € h h λ = = mv p

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The Wave Particle Duality

So… some very simple equations describe some very weird (or at least unintuitive) behavior!

For the moment, we are more interested in the wave-like nature of matter, so let’s use some real values and solve the λ = h/mv equation.

EXAMPLE 1: wavelength of an electron travelling in a cathode ray tube (CRT) -31 6 me = 9.1 × 10 kg; typical velocity is ca. v = 1 × 10 m/s 6.6 ×10−34 J ⋅s o λ = = 7 A atomic scale! (9.1×10−31kg)(1×106 m/s)

EXAMPLE 2: wavelength of a baseball pitched by a MLB pro −34 € 6.6 ×10 J ⋅s −34 -24 λ = =1.2 ×10 m or 1.2 × 10 Å (0.142 kg)(38 m/s) Quantum effects are negligible!

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1927: Clinton, Davisson & Germer demonstrated (experimentally) that electrons are diffracted by crystals just like X-rays

It is interesting to note that G.P. Thompson, who shared the 1937 Nobel Prize with Davisson for these experiments, which proved that electrons are waves, is the son of J.J. Thompson who received the Nobel Prize in 1906 for proving that cathode rays were actually particles - electrons! And the amazing thing is that they were both right.

CHEM 2060 Lecture 17: Intro to QM L17-6

Light:

When light from a point source passes through a small circular aperture, it does not produce a bright dot as an image, but rather a diffuse circular disc (known as Airy's disc) surrounded by much fainter concentric circular rings.

This diagram shows the phenomenon of light interference. A screen with a double slit, equally separated, is illuminated with a bright light, the two slits cause diffraction of the light waves, producing two sets of wave fronts which overlap. This results in Constructive Interference (bright areas) and Destructive Interference (dark areas). CHEM 2060 Lecture 17: Intro to QM L17-7

The Heisenberg Uncertainty Principle

Before we try to understand it, let’s start by looking at what the Heisenberg Uncertainty Principle actually states.

There are a number of ways of stating the Uncertainty Principle. The most important to us (as Chemists) are:

 Ø It is impossible to know simultaneously both the σ σ ≥ momentum p and position x of a particle. x p 2

where σx is the standard deviation of position and σp is the standard deviation of momentum € h and the reduced Planck’s constant  = 2π

Ø It is impossible to know simultaneously both the energy E and time t, which describes the fact that a quantum state with a short lifetime has a range of energies… this is less important to us now,€ so we’ll leave it at that. CHEM 2060 Lecture 17: Intro to QM L17-8

Impossible to know both position and momentum simultaneously

This seems like a hard concept to grasp, but it actually appears in classical wave mechanics! Let’s look at this in order to understand the concept itself…

Take a sine wave.

It has one wavelength and therefore one momentum (remember p = h/λ).

It is infinitely long and therefore it is not possible to identify the wave as being in any specific position. (i.e., It is unbound.)

So, we know the momentum exactly, but the position of anything defined by this wavefunction IS ANYWHERE ALONG THE INFINITE LENGTH.

Question: What happens when we add two identical sine waves? sinθ + sinθ Question: …when we subtract two identical sine waves? sinθ + (-sinθ) Question: …when we add a sine wave and a cosine wave? sinθ + cosθ CHEM 2060 Lecture 17: Intro to QM L17-9

Boundary Conditions (i.e., confining the wave to a specific area in space)

We understand that a linear combination (i.e., addition) of wavefunctions (e.g. sine, cosine, etc.) generates a new wavefunction, which is the result of constructive and/or destructive interference.

In principle, a linear combination of a set of wavefunctions (called a wavetrain) can generate ANY wavefunction, so long as the wavetrain is big enough.

Consider a violin string (or a skipping rope, or any other finite length of rope):

It has several normal modes of vibration. These are the fundamental mode and all the overtones.

The transverse standing wave in any one of the normal modes is sinusoidal along the string but is zero outside the fixed endpoints. It is bound. The wave exists in a specific region of space. CHEM 2060 Lecture 17: Intro to QM L17-10

The Smaller the Boundary, the Greater the Range of Momenta

A transverse standing wave of a normal mode of vibration may look like a sine wave, but it isn’t! (A pure sine wave is infinitely long.)

It has a single frequency (we hear it as a note on the violin string)… …BUT, in order to mathematically describe the standing wave including the fact that it is zero outside the fixed endpoints of the violin string, we need a wavetrain that is the linear combination (superposition) of a LOT of individual wavefunctions (waves)….millions? trillions? I don’t know how many.

Each of these waves in the wavetrain has a unique wavelength λ and therefore a unique wave momentum p.

SO, AS SOON AS A WAVE IS CONFINED IN SPACE, IT MUST BE DESCRIBED BY A WAVETRAIN WITH A RANGE OF MOMENTA!!

** The smaller the area in which the wave is confined, the greater the number of individual waves needed to make the wavetrain, and the greater the range of wavelengths and momenta. ** CHEM 2060 Lecture 17: Intro to QM L17-11

How does the Uncertainty Principle affect chemical models?

Chemistry is essentially the study/manipulation of the interaction of electrons belonging to atoms or ions. It is the electrons that are involved in bonding, lone pairs, redox properties, etc.

Because all matter has wave-like properties, and these become very apparent with small, fast-moving objects such as electrons, it is impossible to ignore these properties when designing meaningful chemical models.

If we determined exactly the position of an electron (i.e., confine the area of the wave that describes its position to essentially a point) we must make the wavetrain that describes its range of momenta essentially infinite!

The Heisenberg Uncertainty principle is one of the major results of Quantum Mechanics (i.e., of describing matter using a wavefunction). • impossible to specify position & momentum of a particle simultaneously • impossibility to ascribe electrons to orbits like planets around the sun!

⇒ ORBITALS with probability distributions CHEM 2060 Lecture 17: Intro to QM L17-12

Electrons

• Wave-Particle Duality – behave like both waves and particles

• If a wave-motion is confined to a finite region in space a series of patterns of standing waves result.

This is exactly like the normal modes of vibration for a violin string, where the fundamental mode of vibration is the lowest in energy (lowest frequency; we hear it as the note played) and the overtones are progressively higher in energy as the number of nodes increases (higher frequency).

• Electrons in atoms/ions are confined by the Columbic field of the nucleus.

• Orbitals are simply 3D standing waves that describe the electron’s position.

HOMEWORK: What does a finite length 2D sinusoidal standing wave with one node look like? What does a finite length 3D sinusoidal standing wave with one node look like? CHEM 2060 Lecture 17: Intro to QM L17-13

2D Standing Waves – Chladni Plates

A rigid surface (e.g., a guitar body) or a membrane (e.g., drum head) has normal modes of vibration (fundamental and overtones).

Normal modes can be observed by sprinkling a free-flowing powder on the resonating surface. The powder will be bounced around by the wave until it settles in a node.

The result is called a Chladni Plate (after German physicist/musician, Ernst Chladni).

On a round surface, the Chladni Plates of various normal modes look like cross-sections of atomic orbitals! CHEM 2060 Lecture 17: Intro to QM L17-14

3D Standing Waves – Spherical (not linear) – ATOMIC ORBITALS!

Science, Vol. 187, 1975, pp. 605-612. “Of Atoms, Mountains & Stars: A Study In Qualitative Physics” Victor F. Weisskopf.