Comments on Ch. 2 HW

Chapter 3:Wave Properties of Particles

de Broglie waves

waves of what?

quiz 2

“Two seemingly incompatible conceptions can each represent an aspect of the truth.”—Louis de Broglie

Electrons will be ejected from the silver ball.

max 0K = hf - hf = hf - φ

If the ball is in a vacuum, suspended by a nonconducting string. As electrons are ejected, the ball acquires a positive charge and positive potential.

The ball will build up a positive charge until no electrons have enough KE to escape, at which point a steady-state situation will exist and the potential will not change.

Homework Problem 2-14A silver ball is suspended by a string in a vacuum chamber and ultraviolet light of wavelength 200 nm is directed at it. What electrical potential will the ball acquire as a result.

Electrons stop escaping when

maxK = eV = hf - φ

hf - φV =

e

8

-15-9

3×10 m/s4.14×10 eV×s - 4.7eV

200×10 mV =

e

ch - φ

λV = e

8

-15-9

3×10 m/s4.14×10 eV×s - 4.7eV

200×10 mV =

e

V = 1.51 V

My mathcad solution is rather sloppy.

symbol units

Suppose that a 60 W light bulb radiates primarily at a wavelength of 1000 nm, a number just above the optical range. Find the number of photons emitted per second.

power = total energy/time

power = (energy/photon) (N of photons/time)

(N of photons/time) = power / (energy/photon)

n = power / (energy of a photon)

n = P / hf = P / (hc/)

n = 60 / [(6.63x10-34)(3x108)]/(1000x10-9)

If you stick with SI units, n will be in units of s-1; i.e. photons/s.

Planck, Nobel Prize, energy quanta, 1918.

Einstein, Nobel Prize, law of the photoelectric effect, 1921.

Millikan, Nobel Prize, charge of electron and photoelectric effect, 1923.

The Big Winners, Chapter 2:

Chapter 2 OSE’s

photonE = hf max 0hf = K + hf

c = f λphotonp = hf / c = h / λ

s 2

2GMR =

c

All our other equations are derived from these, the equations of relativity, and the equations of classical physics.I will give you more than these few equations on your quiz.

What kind of a clown would post something like this for the world to see?

Energy! Energy!Potential!NOT energy!

Here’s the trouble with physicists:We all think alike.We don’t think like ordinary humans.We all know what each other means, so we can afford to be sloppy like this.Right?

Chapter 3Wave Properties of Particles

Overview

Einstein introduced us to the particle properties of waves in 1905 (photoelectric effect). Compton scattering of x-rays by electrons (which we skipped in Chapter 2) confirmed Einstein's theories.

One might ask "Is there a converse?" Do particles have wave properties?

Asking such a question these days is likely to get you ignored, but in the 1920’s, so many incredible things were happening in physics that somebody might have listened to you.

In fact, Louis De Broglie postulated wave properties of particles in his thesis in 1924, based partly on the idea that if waves can behave like particles, then particles should be able to behave like waves.

de Broglie

Werner Heisenberg and a little later Erwin Schrödinger developed theories based on the wave properties of particles.

Heisenberg Schrödinger

In 1927, Davisson and Germer confirmed the wave properties of particles by diffracting electrons from a nickel single crystal.

http://hyperphysics.phy-astr.gsu.edu/hbase/davger.html

De Broglie postulated that these equations also apply to particles. In particular, a particle of mass m and momentum p has a de Broglie wavelength whose magnitude is

hλ= .

p

3.1 de Broglie Waves

Recall that a photon has energy E=hf, momentum p=h/, and a wavelength =h/p.

“Didn’t you just write that same equation a couple of lines above?”No, the equation in the first sentence was for waves. The one in the box represents a new idea. It is for particles.

If the particle is moving fast enough that a relativistic calculation is needed, use the relativistic momentum:

hλ= .

mv

What made de Broglie (who was a real-life prince and a brilliant theorist but a real klutz in the lab) propose, seemingly just for kicks, that particles have a wavelength?“As in my conversations with my brother we always arrived at the conclusion that in the case of X-rays one had both waves and corpuscles, thus suddenly - ... it was certain in the course of summer 1923 - I got the idea that one had to extend this duality to material particles, especially to electrons. And I realised that, on the one hand, the Hamilton-Jacobi theory pointed somewhat in that direction, for it can be applied to particles and, in addition, it represents a geometrical optics; on the other hand, in quantum phenomena one obtains quantum numbers, which are rarely found in mechanics but occur very frequently in wave phenomena and in all problems dealing with wave motion.”—de Broglie, 1963

Or, as Einstein put it:

“Mathematics are well and good…

…but nature keeps dragging us around by the nose.”

So the proposed wave nature of particles did not come out of nowhere, but it was certainly a daring hypothesis for a young Ph.D. student of physics.*

h hλ= =

p mv

Now we have this equation that says particles have a wavelength. What are we going to do with it?

Experiment! Find experimental verification!

In order for us to observe a particle's wave properties, the de Broglie wavelength must be comparable to something the particle interacts with; e.g. the spacing between two slits, or the spacing between periodic arrays of atoms in crystals. *Also got him the 1929 Nobel Prize.

Example: find the wavelength of a 46 g golf ball moving with a speed of 30 m/s.

A nonrelativistic calculation will do just fine here.

h hλ= =

mv mvnon-relativistic: =1

-34

-3

6.63×10 J sλ=

(46×10 kg)× 30 m/s

-34λ=4.8×10 m

Tell me something that has a physical dimension on the order of 10-34 m, which the golf ball wave could interact with?Can we do an experiment which would detect the golf ball wave?

Example: find the wavelength of an electron moving with a speed of 107 m/s.

This is a fast electron, but it is moving at about 1/30 the speed of light. A nonrelativistic calculation will suffice. h

λ=mv

-34

-31 7

6.63×10 J sλ=

(9.11×10 kg)× 10 m/s

-11λ=7.3×10 m

The wavelength is small, but roughly comparable to atomic dimensions, so we need to consider the wave nature of electrons when they are moving rapidly through solids.

Some things to think about:

Collisions seem to be instantaneous, so particles are really “there” and the wave associated with a particle isn’t the particle “spread out.”

Later we will see how a particle's wave has a phase velocity greater than the speed of light, c. Thus, the phase velocity cannot have a physical interpretation.

Useful equations; some from this class, some from earlier:E = hf p = h/ = 2f k = 2/ħ = h/2 E = ħ p = ħk

We aren’t going to talk about the experimental verification of the wave nature of particles yet, but we’ll get there eventually.

If matter has a wavelength, there must be some function–a “wave function”—which describes the wave nature of the matter in question.

Do you think that if we could somehow find out what this wave function is, and what mathematical laws it obeys, then maybe we could learn something about the matter that it describes?

Do you think I’d be asking that question of the answer were “no”?

That means we are going to have to spend more time thinking about the mathematics of waves and the functions that describe them.

3.2 Waves of What?

This section foreshadows chapter 5.

In other words, what physical “thing” is it whose variation makes up matter waves.

Sound waves consist of pressure differences in a medium. Water waves consist of different heights of water. E&M waves consist of measurable oscillations of electric and magnetic fields. What about matter waves?

The "thing" whose variations makes up matter waves is the wave function, ("psi", usually pronounced "si").

Huh?

The wave function of a matter wave is not something we can see or sense. It has no “direct ” physical significance. is the solution to Schrödinger's equation. Remember I mentioned Schrödinger above, and said he developed a theory for the wave properties of particles. We'll learn about “his” equation before long.

is, in general, complex. It cannot be directly measured. The time and/or space average of is zero. (That shouldn't bother you--the time/space average of a sine wave is zero but you measured sine waves in your Physics 24 labs.) However, can tell us something about the matter it represents.

In general, is a function of position (x,y,z) and time.

The probability of finding the object described by at the position (xyz) at time t is proportional to the value of * there.

* tells us the probability of finding the body represented by .

Math review: if is complex, then * = 2 is real (and positive). Putting around * would be redundant.

If *=0 at some position in space and time, then the object is not there. If *=1 at some position in space and time, the object is there. Later we will find there are fundamental limits on how precisely you can locate an object.

In general, the value of * is between zero and one. A small value at some position in space and time means a small probability of finding the object there; a large value means a large probability of finding the object there.

Does this way of doing things bother anybody? It has bothered a lot of physicists. It's a radically different way of thinking.

Note the difference between the probability of an event and the event itself.

If we detect an electron, it was "there." It was not 50% "there."

If we have a collection of identical particles then * is proportional to the actual density of particles. We often call * a “probability density” even when we are talking about one particle.*

*As we will see when we study hydrogen, the idea of a probability density for a single electron makes a lot of sense.

If the probability of finding an electron at (xyzt) is 50%, it doesn't mean that the electron is 50% "there." It means that half of our measurements would find the whole electron "there," and the other half would find no electron at all.

Looking ahead a bit more…

For a particle or system of particles described by the wavefunction , *dV is the probability of finding a particle (or the system) in an infinitesimal volume element dV. To find the probability of finding the particle somewhere in space, we integrate the probability over all space. We assume that the probability of finding a particle somewhere in space is 1 (otherwise, no particle, and nothing worth doing), so that

*

all space

Ψ Ψ dV = 1 .

Such a wavefunction is said to be normalized.

Remember, the wavefunction tells us the probability of finding the particle at a particular point in space and time, but the particle is not "spread out" in some wave.

Actually determining is generally a difficult problem. We will often assume an appropriate wavefunction without going into the details of where it came from.

This concludes our brief diversion into the world of quantum mechanics. We will return in chapter 5.

If we are to claim that particles are waves (actually, have wave properties) then we had better understand waves in detail…