Apr 4, 2007 PHYS 117B.02 1

PHYS 117B.02 Lecture Apr 4 The last few lectures we’ve been switching

gears from classical to quantum physics

This way: The quantum leap

Apr 4, 2007 PHYS 117B.02 2

“If, in some cataclysm , all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words ? I believe it is the atomic hypothesis ( or the atomic fact) that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.”

Richard Feynman

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To understand atoms we need to understand the quantum behavior.

The theme of today’s lecture is particles and waves.

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We already learned some things about particles and waves:

Photoelectric effect – light behaves as a particle … but previously we have shown that it also behaves as a wave.

Atomic spectra : Bohr’s model explained many features of the hydrogen atom spectra (and hydrogen-like atoms) by assuming that angular momentum is quantized and that photons are emitted and absorbed when electrons jump from one orbit to another

Today we’ll talk about particle waves and quantum behavior in general

One “lucky break” – electrons behave just like light! I’ll mostly follow R. Feynman, but our textbook also has nice

discussion on the topic in ch 38.9 and ch 39.1-3

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The Bohr atom: another perspective

Angular momentum is quantized: L = hn/2, n=1,2,3 ….

You can also think of the quantization of L as requiring the electrons to form standing waves along the classical orbit:

2r = n ut, you need to assume that the electron is a wave ! Now if = h/p

2r = n h/( mv) => L = mvr = n h/2

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Particles and waves: de Broglie’s hypothesis ( 1924) Nature is beautifully symmetric. If light behaves both as a particle and a

wave, then matter ( electrons, protons, etc) should also have wave properties. = h/p , where h is Planck’s constant and p is the

momentum of the particle

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What is the wavelength of the electron ?

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Double slit experiment revisited: Shooting solid objects (particles) through double slit

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Let’s analyze the distribution of bullets at the backstop

The assumptions: Machine gun shoots at constant rate Only whole bullets arrive at the

detector Move the detector and count how

many bullets are collected at each position for some fixed amount of time

Define probability: P = N (x)/Ntotal

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The results of the bullets experiment

Close slit 2

Measure P1

Close slit 1

Measure P2

Both slits open:Measure P = P1 + P2

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Do an experiment with water waves

Now we measure intensity of the wave:

I ~ E2

I12 ≠ I1 + I2

Interference: I12 = I1 + I2 + 2 √(I1I2) *cos φ

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But what will happen if our bullets are microscopic particles ?

Two slit experiment using electron gun:

http://www.colorado.edu/physics/2000/applets/twoslitsb.html

If you take PHYS225 you can do this experiment yourself !

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Electrons produce an interference pattern!

Even if we shoot electrons one-by-one - we get an interference pattern !

Each electron interferes with itself !

How can this be ? How do the electrons pass

through the slits ?

bullets

water

electrons

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Watching the electrons

Let’s try to observe through which slit the electron will go Shine light near the slits: photons scattered from the electron will come to our

eyes – bingo ! We know which way the electron went! Hmm… it looks like we are disturbing the electrons with the light! Of course – we know that light has E and B field, carries energy, exerts

pressure !

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Let’s be more sneaky

Reduce the intensity of the light source or Change the wavelength

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Let’s first reduce the intensity: OK here we go:

Three groups of electrons A group that we see going through slit 1 A group that we see going through slit 2 A group that we don’t see, but we detect with our

detector

And the result is: Group 1 and 2 perfectly behaved (no interference) Group 3 produces an interference pattern!

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Let’s now try to increase the wavelength Longer wavelength means smaller momentum: = h/p , so we

will disturb the electrons less Gradually change the photon wavelength ( make the light

“redder” In the beginning – all looks the same. We can tell the position of

the electrons at the slits and there is no interference Remember:

in order to resolve the position of the electrons, the wavelength of the photon has to be of the order of the distance between the slits

Once we get to longer wavelengths: we’ll get a fuzzy spot The interference pattern will reappear, but we will no longer be able

to tell through which slit the electron went Heisenberg’s uncertainty principle:

It is impossible to design an experiment in which we can measure which one of two possible paths was taken without destroying the interference pattern!

∆px ∆x ≥ ћ ∆E ∆t ≥ ћ

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Probability in classical and quantum physics Assume we have an ideal experiment:

We know the initial conditions (electron leaves the gun headed to the slits)

There are no external influences We can NOT predict the final state of the system

exactly (don’t know where it will land) We can only predict the odds !

This is different from classical physics We have given up the idea that the world is

deterministic