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The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful in describing phenomena in the world. Material particles were determinate objects that obeyed the laws of classical mechanics. Electromagnetic waves were traveling waves of electric and magnetic fields, in which the waves were continuous and exhibited phenomena of interference and refraction that could be explained from their wavelength and frequency.
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The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Dec 22, 2015

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Page 1: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

The birth of quantum mechanics

Until nearly the close of the 19th century, classical mechanics and classical electrodynamics had been largely successful in describing phenomena in the world. Material particles were determinate objects that obeyed the laws of classical mechanics. Electromagnetic waves were traveling waves of electric and magnetic fields, in which the waves were continuous and exhibited phenomena of interference and refraction that could be explained from their wavelength and frequency.

Page 2: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

In 1890-1910, there were problems!

There were situations where electromagnetic waves exhibited properties that should be associated with particles!

•Black body radiation

•Photoelectric effect

•Frank-Hertz experiment

•Spectra of emission and absorption by atoms

There were situations where material particles exhibited properties that should be associated with waves!

•Electron diffraction

Page 3: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

We will study several of these paradoxes, and arrive at the wave-particle duality that spawned quantum mechanics:

Light behaves like waves much of the time, but like particles some of the time;

Material particles behave like particles much of the time, but like waves some of the time.

A successful description of both light and matter must somehow weave together both kinds of properties!

Page 4: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

First let’s review the statistical mechanics of material particles.

(section 1.12 in the text)

Suppose we want to calculate the dependence of the density of air as a function of altitude on Earth.

We can get it by using the ideal gas law,

pV=NRT

and the principle of detailed balance

Page 5: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

The ideal gas law relates pressure p (force per unit area)to the number of moles N, the volume V, and the

temperature T. R is the universal gas constant.

For describing a gas, it is more convenient to use the number density of particles n=NaN/V of particles:

where k = 1.4 x 10-23 J/oK = 10-4 eV/oK is Boltzmann’s constant.

nkTp

Page 6: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

How does the atmosphere vary as a function of altitude?

1) A molecule must have work done upon it to elevate by a distance dz:

2) The density of air decreases with altitude: n(z)

dzmgdE

mgF

Page 7: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Detailed Balance

We require that the force on each particle of gas be in balance (otherwise it would rise or fall).

Consider a horizontal slice of the atmosphere at altitude z:

face area A, thickness dz, mass of each particle m

Downward force due to gravity pulling on each particle:

Upward force due to the pressure difference between the top and bottom of the slice:

gmdzAznMgFdown

AzdpAppF topbottomup )()(

Page 8: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Now require Fup = Fdown in each slice of the sky:

Now connect n(z) and p(z) through the ideal gas law:

So the ideal gas law becomes

Solution:

The air density decreases exponentially, with scale length

)( )( zdpdzmgzn

kTznzp )()(

dzznkT

mgzdn )()(

kTmgzenzn /0 )(

mkgsm

KKJmgkTL 400,8

)107.130)(/8.9(

)300)(/104.1(/

272

23

Page 9: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

This result is easily generalized to any situation where material particles are distributed in a volume of space where the potential energy that varies over the region:

This is the Boltzmann distribution function. It governs the distribution of particles in the presence of any interaction potential.

)(rU

kTrUenrn /)(0)(

Page 10: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Black Body Radiation

When a material body is heated, it emits electromagnetic radiation with a broad spectrum.

Mystery #1. It is observed experimentally that the total intensity (power per unit area) radiated by a black body is determined solely by its absolute temperature. There is no way to explain this result by treating light as a wave!

Page 11: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Stephan-Boltzmann law4 TI

= 5.7 x 10-8 W/m2/oK4

Example: A steel rod is heated red hot (T ~ 700 oC ~1,000 oK).

The rod is 1 cm diameter and 1 m long. How much power does it radiate as blackbody radiation?

P = T4 A = (5.7 x 10-8 W/m2/K4)(1,000 K)4 ( x 10-2m)(1 m)

= 1,800 W.

Page 12: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Mystery #2. The spectrum of light from blackbody radiation cannot be explained by assuming that the light is composed of waves.

The spectrum of light is the pattern of intensity as a function of wavelength:

Page 13: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

The spectrum of black-body radiation can be explained (up to a point!) if we consider the radiation to be produced by oscillations of the atoms in the material.

• The light emitted should be proportional to the number of modes in which the oscillations of a given wavelength can be excited:

Consider the modes that can be excited within a cubic cavity of dimension L.

#modes in x: Nx = L/

modes in y: Ny = L

modes in z: Nz = L/

Page 14: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

The power radiated in wavelength interval d is proportional to the fraction of a wavelength: dP ~ d

So the energy density (J/m3) is

geometry spherical ain d

8(

geometry;r rectangula ain )(

4

4

)dλ

ddNNNd zyx

This energy spectrum was derived by Rayleigh and Jeans by assuming that blackbody radiation is emitted from atomic oscillators as a wave process, and that there must be detailed balance between the standing waves that can be supported inside the solid and the emitted radiation that comes out.

Page 15: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

This Rayleigh-Jeans theory is not too bad in its description of the spectrum of long-wavelength light.

Unfortunately, it leads to an ultraviolet catastrophe:

The power radiated at short wavelength (high energy) increases without bound!

High-frequency (low-) cutoff requires factor

/ae

4/1

Page 16: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Plank realized that he could (empirically) obtain the observed spectrum IF he assumed that blackbody radiation behaved as if it were emitted by oscillators that could only change energy by integer multiples of some minimum energy step u: E = muThen he would have from the Boltzmann distribution:

kTmuenun /0)(

Page 17: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

The total energy of all the oscillators emitting this particular energy E=mu is then

kTmuenmuunmu /0 )(

Following the derivation in the book, we calculate the average energy of an oscillator:w

1

/

0

/0

0

/0

kTu

m

kTmu

m

kTmu

e

u

en

enmuw

Page 18: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Put this together with the Rayleigh-Jeans result for the number of oscillators of wavelength :

d

e

ud

kTu 1

8)(

/4

This spectrum matched experimental observation only if the energy u were inversely proportional to wavelength :

hhc

u h = 6.6 x 10-34 J s = 2,000(2) eV Å/c

This only makes sense if light is emitted in quantized packets: it behaves like a particle when it is emitted!

Page 19: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Example: We can estimate the temperature at the surface of a star by determining the wavelength corresponding to the maximum intensity in its spectrum, and assume that the emission is a blackbody spectrum. This wavelength is 550 nm (red) for the Sun, 430 nm (blue) for the North Star, and 290 nm (ultraviolet) for Sirius. Calculate the surface temperatures.

0)1(5)1(

81

8)(

//2/6

/

5

kThckThckThc

kThc

ekT

hce

e

hc

d

de

hc

Page 20: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

e

e

kT

hc

15

0)1(5

This is a transcendental equation. We must obtain an approximate solution. The solution will be near = 5.

hc = 2000 eV Å = 200(2) eV nm

nm

105.2

)/10(5

2200

5

6

4

Kx

KeV

nmeV

k

hcT

Page 21: The birth of quantum mechanics Until nearly the close of the 19 th century, classical mechanics and classical electrodynamics had been largely successful.

Sun pk = 550 nm T = 4,600 oK

North Star pk = 430 nm T = 5,800 oK

Sirius pk = 290 nm T = 8,700 oK