Ch 9 pages 442-444 Lecture 18 – Quantization of energy

Ch 9pages 442-444

Lecture 18 – Quantization of energy

We have found that the spectral distribution of radiation emitted by a heated black body, modeled as a large number of atomic oscillators is predicted to be

Summary of lecture 17

based on classical mechanical considerations. This result would predict that short wavelength radiation should be emitted with high intensity, contradicting experimental observations that short wavelength radiation is emitted with low intensities (bodies do not glow at low temperature).

kTETI44

88),(

Quantization of Energy

In 1901, Max Planck published a quantum theory of radiation to explain the known spectral distribution of black body radiators. Unlike Raleigh, he only allowed the oscillators to adopt certain energy values, not all. Planck’s quantum hypothesis can be constructed as follows. He assumed that the black body is composed of a large number of oscillators whose energies obey the harmonic oscillator equation:

Ep

m

xx 2 2

2 2


The frequency of the harmonic oscillation is given in terms of the constants m (mass) and (spring force constant) as:

Ep

m

xx 2 2

2 2

2

m

Rearrange the equation for the harmonic oscillator as follows

Ep

m

x p

mE

x

E

p

a

x

b

a mE and bE

x x x

2 2 2 2 2

2

2

22 2 2 21 1

22


The expression:

Ep

m

xx 2 2

2 2

Represents an ellipse with semi-axes a and b. Therefore, the trajectory of a harmonic oscillator can be represented in momentum-coordinate space as an ellipse:

Ep

m

x p

mE

x

E

p

a

x

b

a mE and bE

x x x

2 2 2 2 2

2

2

22 2 2 21 1

22

p

a

x

bx2

2

2

21


Classically, when an oscillating atom emits radiation, its trajectory is modified as the momentum and the amplitude of displacement change. The energy emitted by an oscillator has no restricted values. But Planck assumed that in the black body, oscillator trajectories are restricted in such a way that only certain trajectories are possible.


E=nh where n=0,1,2,3…

E

mk

EEmEabnh

/

2/22 2/12/1

Stating that only certain trajectories are allowed means that only ellipses with certain values of a and b may exist. The area of an ellipse is ab and we can express this quantum restriction on the motion and energy of an oscillator:

Where is the oscillator frequency, h is a constant (Planck’s constant), and n is an integer. It follows that, under Plank’s quantum hypothesis, the energy of an oscillator is restricted by the quantization rule:


E=nh where n=0,1,2,3…

We shall see next week that the correct expression for the quantized energy levels of a harmonic oscillator is actually

hvnE )2/1(


The second crucial hypothesis introduced by Planck was that, if an oscillator emits energy, it must pass from E=(n+1)h to say E=nh

The quantum hypothesis restricts energy changes to E=h. This means that energy is emitted into the cavity of the black body in discrete amounts or quanta. These energy particles are called photons and this hypothesis is crucial to explain spectroscopy, as we shall see later.

hvnE )2/1(


To formalize Plank’s equations, we can write his hypothesis to explain the black body phenomenon as follows. The intensity of radiation is still governed by the equation:

hvnE )2/1(

ETI4

8),(

And we can still calculate the energy using the relationship

E kTq

T

2 ln


But q now has the ‘quantized’ form:

hvnE )2/1( ETI4

8),(

Therefore:

E kTq

T

2 ln

q e eE n kT

n

nh kT

n

( )/ /

0 0

0

/

0

/

22 1ln

n

kTnh

n

kTnh

e

enh

T

q

qkT

T

qkTE


If we introduce the general expression:

ETI4

8),(

By expanding in terms of

0

/

0

/

22 1ln

n

kTnh

n

kTnh

e

enh

T

q

qkT

T

qkTE

kThex /

32

32

1

4321

xxx

xxxxhE

rn xrn

rnx

nnnxx

!)!1(

)!1(....

!2

)1(11 2

We find:11)1(

)1(/1

2

kThe

h

x

xh

x

xxhE


Where c is the speed of light so that

ETI4

8),(

Planck’s Quantum Theory of Black Body Radiation is summarized by the following expression for the light emitted as a function of temperature and frequency:

By fitting the equation for I(,T) to experimental data, Planck determined that the constant h=6.626x10-34 J-sec. The constant h is now called Planck’s constant. At high temperatures (kT>>h, Planck’s Radiation Law converges to the classical Jean’s Law.

11)1(

)1(/1

2

kThe

h

x

xh

x

xxhE

18

1

88),( /

5/44

kThc

kThe

hc

e

hETI

c


By fitting the equation for I(,T) to experimental data, Planck determined that the constant h=6.626x10-34 J-sec. The constant h is now called Planck’s constant. At high temperatures (kT>>h, Planck’s Radiation Law converges to the classical Jean’s Law.

18

1

88),( /

5/44

kThc

kThe

hc

e

hETI

Heat Capacities Revisited

Heat capacities of diatomic gas molecules and crystalline solids are predicted to be CV=7R/2 and CV=3R, respectively, at room

temperature

These predictions are based on the assumption that vibrational motions contribute a factor of RT (per dimension) to the energy in accordance with the equipartition principle

However, CV is closer to 5R/2 per mole of gas per diatomic

molecules and CV is almost zero at room temperature for many

solids.

Molecular Partition Function of a Diatomic Molecule

From the discussion of the last class, the classical energy of a diatomic molecule is:

E E E Etrans rotate vibrate

This expression can be used to calculate the molecular partition function. First remember once again that, in Lecture 2, we mentioned the following fundamental property of the partition function

To a high degree of approximation, the energy of a molecule in a particular state is the simple sums of various types of energy (translational, rotational, vibrational, electronic, etc.).


If E E E Etrans rotate vibrate

Using this fact we can rearrange the form for the molecular partition function:

......///vibrottr

kTEkTEkTE qqqeeeq vibrottr then

kT

Rpxpd

lkT

ppdpdp

kT

pppdpdpdpVq Rzyx

zyx 2

/Re

2exp

2exp

2222222

We have partitioned q according to:

q V q q qtrans rotate vibrate


We can calculate the energy from the relationship

Notice that the translational, rotational, and vibrational partition functions all involve integrals of the form:

T

xqxqVxqkT

T

qkT

N

EE vibrottrans

lnln 22

kT

Rpxpd

lkT

ppdpdp

kT

pppdpdpdpVq Rzyx

zyx 2

/Re

2exp

2exp

2222222

adxe ax

0 2

12

Therefore: kTqkTqkTq vibrottrans ;;2/3


From which it immediately follows that:

The energy per mole E and the heat capacity CV are then:

kT

Rpxpd

lkT

ppdpdp

kT

pppdpdpdpVq Rzyx

zyx 2

/Re

2exp

2exp

2222222

kTqkTqkTq vibrottrans ;;2/3

kT

T

TkT

T

xqxqVxqkT

T

qkT

N

EE vibrottrans

2

7lnlnln 2/7222

RTE2

7

2

7R

T

EC

VV


or approximately 29 J/K-mole for a diatomic gas. This expression reflects the equipartition principle, each degree of freedom contributes 1/2R to the heat capacity or 1/2RT to the total energy of a system (per mole).

RTE2

7

2

7R

T

EC

VV


However, almost no diatomic gas obeys this expression. For example, for H2, CV is approximately 20J/K-mole or

approximately 5/2R

An even more serious situation arises when we attempt to calculate the heat capacity of solids. If we regard the solid as a three-dimensional array of atoms, the motions executed by these atoms are vibrations

Therefore, the motions of the atoms may be regarded as harmonic oscillations in three dimensions.


From the equipartition principle, we would expect the vibrational energy to be E=3RT and the contribution to the heat capacity from vibrational motions should be CV=3R

In fact, at room temperature the vibrational heat capacity for crystalline solids is almost 0 and only approaches 3R at high temperatures

These observations indicate serious failures of classical mechanics to accurately account for the behavior of polyatomic gases and solids

These failures contributed to the birth of quantum mechanics.


Plank’s hypothesis can also be used to reexamine the heat capacities and their deviation from classical behavior as well

Let us focus on diatomic gases by defining the average energy as:

E E E Etrans rotate vibrate

If we assume translational and rotational motions obey the equipartition principle, but that the vibrational motions obey quantum mechanical behavior, then we can write:

E E E EkT

Etrans rotate vibrate vibrate

5

2


To be correct, as we shall see next week, the energy levels for a quantum mechanical 1-dimensional oscillator with characteristic frequency v are given by:

Using Planck’s quantization hypothesis for harmonic oscillations and applying it to bond vibrations (homework), we can calculate the partition function to be:

E E E EkT


5

2

1/// )1( kThv

n n

kTnhvkTE eeeq n

hvnEn )2

1( kThv

kThv

e

eq

/

2/

1


If we limit ourselves to Planck’s description at this stage, then the partition function provided in the homework allows you to calculate properties such as the vibrational energy and specific heat:

E E E EkT


5

2

EkT

EkT h

evibrate h kT

5

2

5

2 1

/

2//2

/1

2

5

12

5

kThkThkTh

VV ee

kT

hNk

Nk

e

NhNkT

TT

EC


In the low temperature limit (h>>kT):

2//2

/1

2

5

12

5

kThkThkTh

VV ee

kT

hNk

Nk

e

NhNkT

TT

EC

2

5

2

51

2

52

/2//2

Nk

kT

hNke

Nkee

kT

hNk

NkC kThkThkThV

In the high temperature limit (h<<kT, using kThve lThv /1/

2

7

2

5111

2

51

2

522

2//2

NkNk

Nk

kT

h

kT

h

kT

hNk

Nkee

kT

hNk

NkC kThkThV


2//2

/1

2

5

12

5

kThkThkTh

VV ee

kT

hNk

Nk

e

NhNkT

TT

EC

The high temperature value of CV for diatomic molecules

agrees with the equipartition principle, which is the result obtained using classical statistical mechanics. In this limit: kT>>h (notice that h is the separation between the vibrational energy levels). Classical statistical mechanics correctly predicts the vibrational heat capacity if the separation between the vibrational energy levels (i.e. energy quantization) is negligible compared to kT. But at low temperature, where quantization of energy levels is important, classical statistical mechanics fails and quantum effects become significant.


2//2

/1

2

5

12

5

kThkThkTh

VV ee

kT

hNk

Nk

e

NhNkT

TT

EC

Later, when we develop a theory of quantum wave mechanics, we will show why quantization for vibrational motions is much more important at low temperatures than for translational and rotational motions

Before we do that we will consider another problem that classical physics fails to explain: the stability of the hydrogen atom.

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Ch 9 pages 442-444 Lecture 18 – Quantization of energy