Quantum Mechanics

Weeks One & TwoHTH Senior Physics

Andrew Gloag

Prior to the 20th Century:

• Scientists thought that everything about physics & mechanics would soon be known.

• Newtonian Physics was thought to govern everything in the Universe.

• Determinism ruled - everything in the Universe was predictable if you knew enough about the initial conditions.

The nucleus

• Prior to Rutherford, there was the “plum pudding” model

• In 1909 he bombarded thin gold foil with alpha particles

• A small few of these bounced back in the direction they came from

• This indicated a very tiny, highly concentrated positive charge at center of atom.

Wave Particle Duality

• Electromagnetic waves are made up of a large number of particles called photons.

• The Energy of each photon is given by:

• The position and velocity of particles is described by a wave. The wavelength is:

Planck Constant

ch

E

h = 6.6 x 10-34 Js

Speed of Light c = 3.0 x 108 m/s

Implications (not too bad)

• Blue light has a wavelength of 400 x 10-9 m• How many photons per second come out of a

0.1 mW Blue LED?

Planck Constant h = 6.6 x 10-34 Js

Speed of Light c = 3.0 x 108 m/s

Implications (quite difficult)

• Blue light has a wavelength of 400 x 10-9 m• How fast must an electron be travelling to

have the same wavelength as blue light?

Planck Constant h = 6.6 x 10-34 Js

Electron Mass me = 9.1 x 10-31 kg

Implications (Oh, Lordy!)

• Blue light has a wavelength of 400 x 10-9 m• The flux of sunlight near the earth is

approximately 2,000 Watts per square meter• What is the force on a solar sail near the

earth, assuming all photons in the light are blue?

Planck Constant h = 6.6 x 10-34 Js

Speed of Light c = 3.0 x 108 m/s

The wave-function. Electrons in atoms

• The electron “orbit” is described by a standing wave.

• Only certain wave patterns fit• Each extra wiggle adds energy• Electrons can only occupy

energy levels that correspond to one of these patterns.

Wave-function II. Quantum Tunneling

• In all real cases the edge of the wave is not zero.• The wave function decays exponentially in the

barrier region•

If the barrier is thin enough a little bit of the wave function leaks through

The Scanning Tunneling Microscope

• The tunneling of electrons from the tip of a needle can be used to measure structure at the atomic level.

Week Two: The Uncertainty Principle

• To measure a particle’s position you shine light on it.

• The resolution you get depends on the wavelength (and so the color) you use.

• The photon that interacts with the particle imparts momentum during the collision.

• The momentum changes the particle’s velocity.• So by measuring position you change velocity!

The Uncertainty Principle

• “The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. ”

• Δx∙Δp ≥ h/4π• Works also for Energy and

Time (ΔE∙Δt ≥ h/4π)

Implications• The position of an electron can easily be

narrowed down to the size of a single atom (approximately 1 x 10-10m).

• What is the uncertainty in the velocity of the electron in this case?

Planck Constant h = 6.6 x 10-34 Js

Electron Mass me = 9.1 x 10-31 kg

What it means 1 (ΔpΔx)

• A particle can never be truly “at rest” as we would know the position and momentum exactly.– Electrons, in particular, must always be moving.– We can never cool an atom to absolute zero (but we

can get very close!)– If atoms are sufficiently cold and close, the

uncertainty of position may be more than the inter-atomic spacing. In this case all atoms can behave as if they are one.

– This is called a Bose-Einstein condensate.

What it means II (ΔEΔt)

• You can never make a “perfect clock” – there will always be an uncertainty in time.

• You can never make a perfectly stable energy beam.– Laser physicists have known this for a long time and call

it “Shot Noise”• You can never have truly “empty” space.– Even in the depths of space, the vacuum is not really a

vacuum.– We believe that virtual particle pairs are constantly

being created and annihilated.

The two views• Einstein did NOT like the idea that the universe in

inherently uncertain. He famously said “God does not play Dice.”

• Einstein liked determinism.• Neils Bohr held that the mathematics described the

world perfectly. If a wave-function describes an electron as being both in the box and out, it really was.

• In his “Copenhagen Interpretation” the Universe splits to accommodate both possibilities.

• Bohr liked lots of Universes.

The bigger picture• Atoms consist of a nucleus surrounded by electrons.• The electrons can only exist in certain well defined energies (a

bit like rungs on a ladder)• Electrons normally fill up the lower rungs first, but there are

only a certain number of slots available on each rung.• There is always room on the higher up rungs, but to jump up

an electron must acquire the EXACT amount of energy to make the transition.

• The main ways electrons get this energy are photon absorption and thermal excitation (from heat).

• Photons must give ALL of their energy at once (E=hc/λ) or none at all.

The bigger picture 2• Atoms only absorb light at certain well defined wavelengths,

giving rise to an “absorption spectrum” for each element.• When an electron drops back down to a lower level it emits a

photon.• The energy of this photon (E = hc/λ)is the EXACT energy

difference between the two levels.• The electron may go down in several small steps or 1 big one.• The emission spectrum is slightly different from the

absorption spectrum.• The direction of the emitted photon is random.• Practically all light comes from electron transitions!

Questions for Discussion

• Why do very hot objects glow?

• Why do lasers give light at narrow wavelengths?

• Why is glass transparent?

• Why is wood opaque?

• How can we tell what is in a stellar dust cloud?