Wave Nature of Matter

Light/photons have both wave & particle behaviors.

Waves – diffraction & interference, Polarization.

Acts like Particles – photoelectric effect, E = hf.

de Broglie/Matter waves 1924

If light behaves as a particle, then particles should behave like waves. Right?

Particles also have , related to their momentum.

Where m = rest mass of the particle

Derive Eq Using E = mc2.what is the wavelength of matter

E = hf E = mc2 = mv2. hf = mv2. but f = v/ and v2. hv/ = mv2.Cancel v.h/ = mv mv = p. h/ = p = h/p

1: Find the of an electron accelerated through a p.d. of 30-V.

Find the e- velocityqV = ½ mv2.v = 3.2 x 106 m/s

Calculate . = h/p2.3 x 10-10 m.

Handy Equation

KE e- = 1/2 mv2 = p2/2m

For e- accelerated through pd eV = KE = p2/2m

De Broglie wavelength or “matter waves” are not physical.

They are not EM or mechanical waves but determine the probability of finding a particle in a particular place.

Evidence

Electrons diffracting through 2 slitsWhat does this pattern look like?

Electron diffraction

Davisson-Germer experiment: similar to xray diffraction

They know the e- speed thus know the deBroglie

Maximum intensity from wave diffraction pattern

Maxima observedFor e-. Diffraction pattern.

Can calc using position of min & max.

agrees with deBroglie from equation.

Results of Davisson-Germer experiment:Proof of deBroglie

2. A 70kg person is running 5 m/s. Find . How does the compare with the on the EM spectrum?

3. Find for an e- moving at 107 m/s. How does the compare with the on the EM spectrum?

Hwk Read Hamper 243 – 246 IB Set

Electron in a Box

Bohr Model of Atom

Electrons jump “oscillate” up & down to different energy levels absorbing or releasing

photons.

Bohr explains H well, not effective for larger atoms.

The atomic orbits of Bohr can better be visualized as e- oscillating in a box closed at both ends.

Picture that the de Broglie waves for e- are standing waves.

This helps explain why energy is quantized.

Electron in a Box

If e- viewed as standing waves the orbit model works better.

2L =

2L/2 =

2L/3 =

Since p = h/:

E = n2h2

8mL2.

Orbit n=1 ground Planck

Circular Diameter

Mass e-

De Broglie & e- in a box

The de Broglie of e- are the‘s of the standing allowed by the box;

since λ = 2L/n where n is an integerenergy is quantized;

If e- are standing waves. Only ’s that fit certain orbits are possible.

Fit a standing wave into a circular orbit

Circumference = 2r = n

deBroglie’s equation for the electron:

= h/mv

You get the equation for quantized angular momentum:

mvr = nh/2

’s that don’t fit circumference undergoes destruction interference & cannot exist.

IB Prb Electron in a Box

Schrodinger Model

Schrodinger used deBroglie’s wave hypothesis to develop wave equations to describe matter waves. Electrons have undefined positions but do have probability regions he called “electron clouds”. The probability of finding an e- in a given region is described by a wave function .

Schrodinger’s model works for all atoms.

Electron cloud

http://www.youtube.com/watch?v=-YYBCNQnYNM&feature=related

The structure of atoms

Heisenberg Uncertainty.

1927 Cannot make simultaneous measurements of position & momentum on particle with accuracy.

The act of making the measurement changes something.

The more certain we are of 1 aspect, the less certain we are of the other.

The total uncertainty will always be equal to or greater than a value:

x = Uncertainty in positionp =Uncertainty in momentum

If you know the momentum exactly, then you have no knowledge about position.

Another aspect to uncertainty is:

Et ≥ h/4.

E = energy J. t = time (s)

If a mass remains in a state for a long time, it can have a well defined E.

Example Problem

The velocity of an electron is 1 x 106 m/s ± 0.01 x 106 m/s. What is the maximum precision in its position?

5.8 x 10-9 m.

http://www.youtube.com/watch?v=hZ8p7fIMo2k

Heisenberg.

Mechanical universe.

The End for now.Minute Physics Heisenberg

http://www.youtube.com/watch?v=7vc-Uvp3vwg

http://www.youtube.com/watch?v=hZ8p7fIMo2k

http://www.youtube.com/watch?v=groBKtfZfsA

HL stuff.

Constructive interference of e- waves scattered from two atoms occurs when d sin = m (m = 1, = 50o, solve for )

The angle depends on the voltage used to accelerate the electrons!Positions of max/min were similar to xray diffraction

KE of electron = 1/2 mv2 = eV = p2/2m

= the same that was found via the diffraction equation

Confirms the wave nature of electrons!

39.3 Probability and uncertainty

QM: a particle’s position and velocity cannot be precisely determined

Single-slit diffraction: << a 1 = angle between central max. and first minimumif 1 is very small, 1 = / a (RADIANS!)

Interpret this result in terms of particles:

tan1 = py / px So 1 = py / px py / px = / a

There is uncertainty in py = py

Can we fix this by making the slit width = a smaller?

py a > h

No, because making the slit smaller makes central max wider

Wide slit, py is well defined (~0)

narrow slit, py could be anything

h = h/2

Slit width a is an uncertainty in position, now called x

y = 1/x

The longer the lifetime t of a state, the smaller its spread in energy E.

A state with a “well-defined” energy

A state with a “poorly-defined” energy

Two-slit interference

With light…

Electrons diffracting through 2 slits

39.4 Electron microscope

Better resolution because e- wavelengths << optical photons

Microscope resolution ~ 2 x wavelength

Scanning electron microscope:• e- beam sweeps across a specimen• e- are knocked off and collected• Specimen can be thick• Image appears much more 3-D than a

regular microscope

SEM image

TEM image of a bacterium

Two waves with different wave numbers k = 2

In reality, wave functions are localized: combinations of 2 or more sin & cos functions

ph

k

A wave packet: particle & wave properties

(x,y,z) A(k)e ikx

dk

Does a wave packet represent a stationary state?

A stationary state

• Has a definite energy (meaning, no uncertainty, only 1 value of E)

• * is independent of time• * = |(x,y,z)|2

(x,y,z) A(k)e ikx

dk