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Chapter 38

The sub-atomic world behaves very differently from the world of our ordinary experiences. Quantum physics deals with this strange world and has successfully answered many questions in the sub-atomic world, such as: Why do stars shine? Why do elements order into a periodic table? How do we manipulate charges in semiconductors and metals to make transistors and other microelectronic devices? Why does copper conduct electricity but glass does not?

In this chapter we explore the strange reality of quantum mechanics. Although many topics in quantum mechanics conflict with our common sense world view, the theory provides a well-tested framework to describe the sub-atomic world.

Photons and Matter Waves

38-

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Quantum physics:

•Study of the microscopic world

• Many physical quantities found only in certain minimum (elementary) amounts, or integer multiples of those elementary amounts

•These quantities are "quantized"

•Elementary amount associated with this a quantity is called a "quantum" (quanta plural)

Analogy example: 1 cent or $0.01 is the quantum of U.S. currency

Electromagnetic radiation (light) is also quantized, with quanta called photons. This means that light is divided into integer number of elementary packets (photons).

The Photon, the Quantum of Light

38-

3

The energy of light with frequency f must be an integer multiple of hf. In the previous chapters we dealt with such large quantities of light, that individual photons were not distinguishable. Modern experiments can be performed with single photons.

So what aspect of light is quantized? Frequency and wavelength still can be any value and are continuously variable, not quantized

The Photon, the Quantum of Light, cont'd

38-

(photon energy)E hf=

cfλ

=

However, given light of a particular frequency, the total energy of that radiation is quantized with an elementary amount (quantum) of energy E given by:

Where the Planck constant h has a value:34 156.63 10 J s 4.14 10 eV s h − −= × ⋅ = × ⋅

where c is the speed of light 3x108 m/s

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When short wavelength light illuminates a clean metal surface, electrons are ejected from the metal. These photoelectrons produce a photocurrent.

First Photoelectric Experiment:

Photoelectrons stopped by stopping voltage Vstop. The kinetic energy of the most energetic photoelectrons is

The Photoelectric Effect

38-Fig. 38-1

max stopK eV=Kmax does not depend on the intensity of the light!

→ single photon ejects each electron

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The Photoelectric Effect, cont’d

38-

Fig. 38-2

Second Photoelectric Experiment:

Photoelectric effect does not occur if the frequency is below the cutoff frequency fo, no matter how bright the light!

→ single photon with energy greater than work function Φejects each electron

slope abbc

=

6

The Photoelectric Effect, cont’d

38-

Photoelectric Equation

The previous two experiments can be summarized by the following equation, which also expresses energy conservation

max (photoelectric equation)hf K= + Φ

max stop stophK eV V fe e

Φ = → = −

Using

equation for a straight with slope h/e and intercept –Φ/e

( )-15

14 14

2.35 V 0.72 Vslope 4.1 10 V s11.2 10 7.2 10 Hz

h abe bc

−= = = = × ⋅

× − ×Multiplying this result by e

( )( )-15 -19 -344.1 10 V s 1.6 10 C 6.6 10 J sh = × ⋅ × = × ⋅

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Photons Have Momentum

38-

Fig. 38-3

Fig. 38-4

h= (photon momentum)hfpc λ

=

8

Photons Have Momentum, Compton shift

38-

Fig. 38-5

'hf hf K= +Conservation of energy

( )2 1K mc γ= −

Since electron may recoil at speed approaching cwe must use the relativistic expression for K

where γ is the Lorentz factor

( )2

1

1 v cγ =

−

Substituting K in the energy conservation equation

( ) ( )2' 1 1'

f c h hhf hf mc mcλ

γ γλ λ

=

= + − = + −→

Conservation of momentum along x cos cos'

h h mvφ γ θλ λ

= +

Conservation of momentum along y 0 sin sin'

h mvφ γ θλ

= −

p mvγ=

9

Photons Have Momentum, Compton shift cont’d

38-

'λ λ λ∆ = −Want to find wavelength shift

Conservation of energy and momentum provide 3 equations for 5 unknowns (λ, λ’, v, φ, and θ ), which allows us to eliminate 2 unknowns v and θ.

( )1-cos (Compton shift)hmc

λ φ∆ =

λ, λ’, and φ, can be readily measured in the Compton experiment

is the Compton wavelength and depends on 1/m of the scattering particleh

mcLoose end: Compton effect can be due to scattering from electrons bound loosely to atoms (m=me→ peak at θ ≠ 0) or electrons bound tightly to atoms

(m ≈ matom >>me → peak at θ ≈ 0)

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How can light act both as a wave and as a particle (photon)?

Light as a Probability Wave

38-

Fig. 38-6

Standard Version: photons sent through double slit. Photons detected (1 click at a time) more often where the classical intensity:

2rms

0

EIcµ

= is maximum.

The probability per unit time interval that a photon will be detected in any small volume centered on a given point is proportional to E2 at that point.

Light is not only an electromagnetic wave but also a probability wave for detecting photons

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Light as a Probability Wave, cont'd

38-

Single Photon Version: photons sent through double slit one at a time. First experiment by Taylor in 1909.

1. We cannot predict where the photon will arrive on the screen.

2. Unless we place detectors at the slits, which changes the experiment (and the results), we cannot say which slit(s) the photon went through.

3. We can predict the probability of the photon hitting different parts of the screen. This probability pattern is just the two slit interference pattern that we discussed in Ch. 35.

The wave traveling from the source to the screen is a probability wave, which produces a pattern of "probability fringes" at the screen.

1238-

Fig. 38-7

Light as a Probability Wave, cont'd

Single Photon, Wide-Angle Version: More recent experiments (Lai and Diels in 1992) show that photons are not small packets of classical waves.

1. Source S contains molecules that emit photons at well separated times.2. Mirrors M1 and M2 reflect light that was emitted along two distinct paths, close to 180o apart. 3. A beam splitter (B) reflects half and transmits half of the beam from Path 1, and does the same with the beam from Path 2.4. At detector D the reflected part of beam 2 combines (and interferes) with the transmitted part of beam 1.

5. The detector is moved horizontally, changing the path length difference between Paths 1 and 2. Single photons are detected, but the rate at which they arrive at the detector follows the typical two slit interference pattern.

Unlike the standard two slit experiment, here the photons are emitted in nearly opposite directions! Not a small classical wavepacket!

1338-

Light as a Probability Wave, cont'd

Conclusions from the previous three versions/experiments:

1. Light generated at source as photons.

2. Light absorbed at detector as photons.

3. Light travels between source and detector as a probability wave.

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If electromagnetic waves (light) can behave like particles (photons), can particles behave like waves?

Electrons and Matter Waves

38-Fig. 38-9

(de Broglie wavelength)hp

λ = where p is the momentum of the particle

Electronsθ

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Consider a particle that is detected at Point I and then is detected at Point F. Can we think of this as the propagation of a wave. Imagine any non-straight path connecting I and F. This path will have a neighboring path that will destructively interfere with it. For the straight (direct) path connecting I and Fthe neighboring paths will constructively interfere to reinforce the probability of traveling in a straight line. Of course, if the particle is charged and in the presence of an electric or magnetic field, the interference conditions on neighboring paths will change and the path may become curved.

Waves and Particles

38-

Fig. 38-11

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For light E(x, y, z, t) characterizes its wavelike nature, for matter the wave function Ψ(x, y, z, t) characterizes its wavelike nature.

Schrödinger’s Equation

38-

Like any wave, Ψ(x, y, z, t) has an amplitude and a phase (it can be shifted in time and or position), which can be conveniently represented using a complex number a+ib where a and b are real numbers and i2 = -1.

On the situations that we will discuss, the space and time variables can be grouped separately:

( ) ( ), , , , , , i tx y z t x y z t e ωψ −Ψ =

where ω=2πf is the angular frequency of the matter wave.

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What does the wave function mean? If the matter wave reaches a particle detector that is small, the probability that that a particle will be detected there in a specified period of time is proportional to Iψ|2, where Iψ| is the absolute value (amplitude) of the wave function at the detector’s location.

Schrödinger’s Equation, cont’d

38-

The probability per unit time interval of detecting a particle in a small volume centered on a given point in a matter wave is proportional to the value Iψ|2 at that point.

Since ψ is typically complex, we obtain Iψ|2 by multiplying ψ by its complex conjugate ψ* . To find ψ* we replace the imaginary number i in ψ with –iwherever it occurs.

( )( )2 * a ib a ibψ ψψ= = + −

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How do we find (calculate) the wave function? Matter waves are described by Schrödinger’s equation. Light waves a described Maxwell’s equations, matter waves are described by Schrödinger’s equation.

For a particle traveling in the x direction through a region in which forces on the particle cause it to have a potential energy U(x), Schrödinger’s equationreduces to:

Schrödinger’s Equation, cont’d

38-

where E is the total energy of the particle.

If U(x) = 0, this equation describes a free particle. In that case the total energy of the particle is simple its kinetic energy (1/2)mv2 and the equation becomes:

( )2 2

2 2

8+ - =0 (Schrodinger's eq. in 1D)d m E U xdx h

ψ π ψ &&

22 2 2 2

2 2 2

8+ =0 + 2 =02

d m mv d pdx h dx h

ψ π ψψ π ψ →

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In the previous equation, since λ=h/p, we can replace the p/h with 1/λ, which in turn is related to the angular wave number k=2π/λ.

Schrödinger’s Equation, cont’d

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2 + =0 (Schrodinger's eq., free particle)d kdx

ψ ψ &&

( ) = ikx ikxx Ae Beψ −+

The most general solution is:

Leading to:

( ) ( ) ( )( ) ( )

, =

i t ikx ikx i t

i kx t i kx t

x t x e Ae Be e

Ae Be

ω ω

ω ω

ψ − − −

− − +

Ψ = +

= +

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Finding the Probability Density Iψ|2

38-Fig. 38-12

= cos sin and = cos sin i ie i e iθ θθ θ θ θ−+ −

Choose arbitrary constant B=0 and let A= ψ0

( ) 0= ikxx eψ ψ ( )2 22 20 0= ikx ikxe eψ ψ ψ=

( )( )2* 1ikx ikx ikx ikx ikx ikx ikxe e e e e e− −= = = =

( )22 22 20 0 0= 1 (a constant)ikxeψ ψ ψ ψ= =

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In the previous example, the momentum (p or k) in the x-direction was exactly defined, but the particle’s position along the x-direction was completely unknown. This is an example of an important principle formulated by Heisenberg: Measured values cannot be assigned to the position r and the momentum p of a particle simultaneously with unlimited precision.

Heisenberg’s Uncertainty Principle

38-

where 2 (Heisenberg's uncertainty principle)

x

y

z

x p hy p

z p

π∆ ⋅ ∆ ≥ =∆ ⋅ ∆ ≥

∆ ⋅ ∆ ≥

h hh

h

Note that if 0Then , , and

x y zp p px y z

∆ = ∆ = ∆ =

∆ → ∞ ∆ → ∞ ∆ → ∞

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As puck slides up hill, kinetic energy K is converted to gravitational potential energy U. If the puck reaches the top its potential energy is Ub. The puck can only pass over the top if its initial mechanical energy E> Ub. Otherwise the puckeventually stops its climb up left side of hill and slides back top left. For example, if Ub=20 J and E=10 J, the puck will not make pass over the hill, which acts as a potential barrier.

Barrier Tunneling

38-

Fig. 38-13

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Fig. 38-16

38-

Fig. 38-14

What about an electron approaching an electrostatic potential barrier?

Fig. 38-15Due to the nature of quantum mechanics, even if E< Ub there is a non-zero transmission probability (transmission coefficient T) that the electron will get through (tunnel) to the other side of the electrostatic potential barrier!

( )22

2

8 where bbL m U E

T e bh

π− −≈ =

Barrier Tunneling, cont’d

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As tip is scanned laterally across the surface, the tip is moved up or down to keep the tunneling current (tip to surface distance L) constant. As a result the tip maps out the contours of the surface with resolution on the scale of 1 nm instead of >300 nm for optical microscopes!

The Scanning Tunneling Microscope (STM)

38-

Fig. 38-17