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Foundations for
quantum mechanics
Atomic Spectra
A little history
Roentgen: discovered x-rays in 1895
More history
• X-rays were discovered to ionize air
• Until then, only ions observed were in solution (e.g., Na+, Cl-)
• Implications: Neutral atoms must possess small electric charges as constituents
• Atom is a complex structure
• Electrical charges enter into its make-up
Thomson: identified electron in 1897
• Early experimenter with mass spectrometer
• Determined mass-to-charge ratio of electron
• Proposed ‘plum-pudding’ model of atom, in which electrons move throughout a solid
atom with evenly-distributed positive charge.
A little more history
Rutherford conducts gold foil experiment in 1911
• Directed high-energy, positively-charged alpha particles (helium ions from radon) at
gold foil surrounded by cylindrical detector
• Thomson’s model = no massive concentration of positive charge = nothing to significantly
deflect ions
• Results:
• Most ions passed through as if foil were empty space
• Few deflected at very large angles
• Conclusion: relatively small nucleus with positive charge around which electrons orbit
Electrons are negatively-charged and
relatively far away from positivelycharged nucleus.
If electrons were stationary, they
would quickly be attracted to the
Easy explanation:
They must be orbiting!
• Electrons orbit stable
• Moving charges emit
energy in the form of
electromagnetic waves.
• Energy is conserved.
• As electrons orbit the nucleus,
their energy must decrease.
• As their energy decreases, they
spiral into the nucleus.
• As they spirals inward, their
speed increases and so would
frequency of light emitted.
• The process repeats.
• The universe does not last very
If our assumptions are true, we would expect to see light emitted from
atoms shift from red (low-energy) to orange to yellow to green to blue
(high-energy) relatively quickly.
(and also, we would expect not to exist as the electrical potential in atoms would have evened out
long ago, making life as we know it impossible.)
Instead, we see this
(Also, we exist.)
Spectral lines
We find that each element (as a gas) emits a
unique set of spectral lines when heated or
subjected to high voltage.
Why these lines and only these lines? (this
uniqueness extends into the infrared and ultraviolet ranges as well)
Why don’t electrons quickly lose their energy?
Energy is proportional to frequency
Photoelectric effect demands that light come in discrete quantities.
Energy is proportional to frequency
𝐸 = ℎ𝑓
Absorption lines demand that
• matter absorbs and emits energy in ONLY discrete quantities
• electrons do not behave the way macroscopic particles do
Quantization of angular momentum
Neils Bohr offered a math trick in 1912
Angular momentum of electrons is quantized
Photon of light is emitted when electron drops from ‘excited’ energy
state to less energetic state, stopping at ‘ground state’
And… Frequency determine by difference in energy levels
ℎ𝑓 = 𝐸𝑢𝑝𝑝𝑒𝑟 − 𝐸𝑙𝑜𝑤𝑒𝑟
Reasons we trust the model
Does a nice job explaining
spectral lines for hydrogen
Reasons we question the model:
Math trick without a very good
• Angular momentum
conserved?? Why?
• Ground state? What?
• Moving charged particles
don’t emit light? Sorry,
Wave nature of matter
Louis de Broglie proposed particles (e.g., electrons) also have wave nature
• Bohr’s quantum numbers correspond to number of full ‘electron’ wavelengths
• ONLY resonant ‘standing waves’ can persist.
= =
𝑝 𝑚𝑣
ℎ = Planck’s constant
𝑝 = momentum = mass x velocity (at non-relativistic speeds)
Wave nature of matter
Louis de Broglie proposed
particles (e.g., electrons) also have
wave nature (1923)
• Bohr’s quantum numbers
correspond to number of full
‘electron’ wavelengths
• ONLY resonant ‘standing waves’
can persist.
Falsifiable predictions
Electrons diffract?
• Yes: Demonstrated 1927
• Davison – Germer
Electrons interfere with each other in double-slit experiment?
• Yes: Demonstrated in 1961