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Recap: Two Slit Experiment. Notice that at some places there are fewer electrons arriving with both open than there were with only one!!!. An Implication. Electrons (and other objects at this scale) do not follow definite paths through space

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Recap: Two Slit Experiment

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Recap: Two Slit Experiment

  • Notice that at some places there are fewer electrons arriving with both open than there were with only one!!!

An Implication

  • Electrons (and other objects at this scale) do not follow definite paths through space

  • They can be represented by a kind of wave, that exhibits interference like water waves

  • But they also behave like particles, in the sense that they are indivisible “lumps”

  • “Wave-particle duality”: Is it a wave or a particle? It’s both! And neither…

Another Implication

  • Observing a system always has some effect on it

  • This effect cannot be eliminated

    • There is a minimum effect that cannot be removed, no matter how clever we are at designing experiments

    • With baseballs, e.g., the effect is too small to be noticeable

  • The observer is part of the observation!

  • The precise statement of this idea is known as Heisenberg’s Uncertainty Principle

Werner Heisenberg

We have to remember that what

we observe is not nature in itself,

but nature exposed to our method

of questioning. – Heisenberg

The Uncertainty Principle

  • In QM, particles are described by waves

    • Usually called the “wave function”

  • Waves for a faster-moving particle have shorter wavelength

  • Those for a slower-moving particle have longer wavelength




  • The wave is spread out in space – the particle can be found wherever the wave is not zero

  • There is an “uncertainty” in the location x of the particle

    (Think of this as the size of the region in space where the particle is likely to be found.)

  • A wave spread out over all space would have infinite uncertainty – not a real particle

Real Waves for Real Particles

  • To make a wave for a real particle, we can add many of these “pure” waves together:

  • With many, we can get a wave that looks like this:

Real Waves, continued

  • But now we don’t have a single speed (wavelength), it’s a mixture!

  • So for a real particle there is an uncertainty in the speed as well:

    If we measure the speed we will get a range of possible results, with a variation of about s

  • Both the speed and location are uncertain

    • Remember: No definite trajectories!

Heisenberg Uncertainty Principle

  • For any particle

    where h is a fundamental constant of nature (“Planck’s constant”) and m is the mass of the particle

    • Strictly speaking, the above is h/m at a minimum; it can be larger

  • What does it mean??







The “Realm of Possibilities”

  • Let’s call the product (x)(s) the particle’s “realm of possibilities”

  • HUP says the area of the rectangle is fixed, equal to h/m









Localizing a Particle

  • Say we make (x) smaller; then (s) must get larger:

  • And vice versa, of course

Rectangle must

have the same

area as before

Baseball RoP (not to scale!!)

Electron RoP

Proton RoP

What it Means

  • The HUP means that the more precisely we localize a particle (know where it is), the more uncertain is its speed, and vice versa

  • Note that heavier particles have a smaller realm of possibility

    • Shows why e.g. baseballs do appear to have a precise location and speed!

Area of the rectangle is reduced if m is large!


Arrange these objects in order, beginning with the object having the largest “realm of possibilities” and ending with the one having the smallest: proton; glucose molecule C6H12O6; helium atom; baseball; electron; grain of dust; water molecule; automobile.

Stability of Atoms in QM

  • If the electron were to spiral in close to the nucleus, its x would get smaller

  • Hence its s wouldget bigger

  • But in this case s itself must get bigger on average!

    • Say s is 1000 m/s; then the range of possibilities with the lowest values for s would be the range from 0 to 1000 m/s

    • The average s would then be around 500 m/s

    • If we make s bigger, the average s will also get bigger

  • The increased speed tends to throw the electron back away from the nucleus

    • Like a merry-go-round 

The Quantum Atom

  • Quantum theory allows a detailed understanding of the chemical properties of the elements

  • Explains the regularities exhibited in the Periodic Table!

  • We can explore these using a simple qualitative model, the Bohr Model (1913)

    • An early, pre-quantum picture

    • But helped “show the way”

Niels Bohr

Bohr’s Model

  • Assumption I: Electrons move in certain special circular or elliptical orbits around the nucleus

    • Still pictured electrons moving in orbits, but only in special ones

    • He had rules for figuring out the allowed orbits

  • Assumption II: When in one of these special orbits, the electrons do not emit any light (EM waves)

    • Although conventional electrodynamics says they do!

Hydrogen: Some Details

  • Allowed orbits are labeled by an integer n = 1, 2, 3, …

  • For each n there is a circular orbit with radius

    where r0 is a minimum radius Bohr could calculate

    • For hydrogen r0 = 0.5  10–10 m

  • An electron in orbit n has energy

C = a constant

Quantum Jumps

  • Assumption III: An electron can “jump” from a higher-energy allowed orbit to a lower-energy one, in which case the energy difference is carried away in the form of light

  • For light, energy corresponds to wavelength (i.e., color)

  • Since the electron can only have particular energies, the emitted light has only particular wavelengths!

  • In the lowest orbit, there is nowhere left to go – it is stable

Spectral Lines

  • Each element produces a characteristic set of frequencies, or “spectrum” of light

  • The “fingerprints” of the elements

  • Completely mysterious using Newtonian physics!

Other Allowed Orbits

Circular orbit

  • Some other, elliptical, orbits are also allowed

  • For a given n, there are

    • one circular orbit, and

    • n2 – 1 elliptical ones

  • Hence a total of n2 allowed orbits

  • They have different shapes but all of them with the same n have the same energy:

    • One orbit has energy E1

    • 4 orbits have energy E2

    • 9 orbits have energy E3

    • Etc.

Elliptical orbit

More Complicated Atoms

  • Have several (perhaps many!) electrons

  • They like to “jump” down to lower energies

    • The lowest orbit is stable, though

  • They should all just pile up in the lowest allowed orbit

  • Problem: For higher atomic number, the nucleus has more (positive) charge

    • So the attractive force between it and the electrons is stronger

    • These atoms should be smaller

    • Not observed to be the case!

The Pauli Exclusion Principle

  • To fix this, Wolfgang Pauli introduced a new rule, the “exclusion principle”:

  • Turns out to have a deep origin – arises from the theory of relativity!

  • Now the electrons cannot all settle into the lowest orbit

At most two electrons can have

the same orbit

Wolfgang Pauli

Building Atoms

  • Now start filling the orbits with electrons

  • They naturally want to get as low as they can

  • But at most two per orbit!

  • When all the orbits with a given energy are occupied we have a “closed shell”

  • Note that as the nuclear charge increases, the electron orbits shrink

    • Overall size of atoms remains constant on average, as observed

The first two

  • Helium

  • Lithium

Closed shell: The n = 1

orbit is fully occupied

The n = 1orbit occupied;

one left over with n = 2

Neon and Sodium

  • We can put 8 electrons in orbits with n = 2

    • There are n2 = 4 orbits

    • Two electrons can go in each

  • So the next closed shell has 10 electrons total

    • The 8 plus the 2 in the lower n = 1 orbit

  • This is Neon, the next ideal (noble) gas!

  • Sodium then has a closed shell plus one, just like Lithium!

  • Patterns of periodic table reproduced

Chemical Properties

  • Atoms with an “extra” electron outside a closed shell are happy to give it up

  • Atoms that are “missing” an electron would like to acquire one

  • Thus e.g. Sodium wants to give its outermost electron up, while Chlorine would like to acquire one

    • They bind, “sharing” the outer electron and forming an NaCl molecule (table salt)

  • Oxygen is missing two electrons; it wants to adopt two from other atoms

    • E.g. one from each of two hydrogen atoms, to form H2O

    • Or it can bind with a single bivalent atom, e.g. MgO

Going Further

  • It gets complicated 

  • Pattern of closed shells does not exactly match with the different n values

    • The energies of all the possible orbits are intricately ordered

  • E.g. the next closed shell (Ar) occurs with 8 electrons in certain n = 3 orbits

  • Repulsion among the electrons plays an increasingly important role

    • We have ignored this so far!

  • But detailed calculations show that everything works beautifully!

The Real Deal

  • The Bohr Model is not really correct, though it does give a simple way to think about atoms that captures some of the essentials

  • The full QM analysis of atoms involves calculating the wave patterns of the electrons

A Partial Connection

  • We can picture the Bohr orbits as “standing waves”, like the waves on a plucked violin string

  • They travel along the orbits and close on themselves

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