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...

Thermodynamic

...

variables

...

are

...

time

...

averages

...

of

...

their

...

microscopic

...

counterparts.

...

There

...

is

...

an

...

enormous

...

complexity

...

with

...

quantum

...

mechanics,

...

but

...

there

...

are

...

few

...

variables

...

in

...

thermodynamics.

...

The

...

two

...

worlds

...

are

...

connected

...

by

...

posulating

...

that

{
Latex
} \[ \overline E = \langle E(t) \rangle \] {latex}

.

...

The

...

function

...

on

...

the

...

right-side

...

of

...

the

...

equation

...

can

...

be

...

a

...

many

...

body

...

wavefunction.

...

It

...

is

...

possible

...

to

...

compute

...

the

...

wavefunction

...

but

...

it

...

can

...

be

...

very

...

complicated.

...

A

...

major

...

postulate

...

involves

...

a

...

weighted

...

average

...

over

...

all

...

possible

...

states,

...

and

...

major

...

pursuit

...

is

...

to

...

find

...

the

...

probabilities,

{
Latex
} \[ P_v \] {latex}

,

...

involved

...

in

...

the

...

weighted

...

average.

...

  • Thermodynamic

...

  • variables

...

  • =

...

  • time

...

  • averages

...

  • of

...

  • their

...

  • microscopic

...

  • counterparts

...

  • Latex
     \[ U = \overline E = \langle E(t) \rangle = \frac {1}{\Delta t} \int_{\Delta t} \langle \Psi (q,t) \mid \hat H \mid \Psi^*(q,t) \rangle \]

...

  •  where 
     Latex
...
  •  \[ q \] 
...
  •  is 
...
  •  the 
...
  •  quantum 
...
  •  number.
...
Major 
...
 Postulate
...
A 
...
 major 
...
 postulate 
...
 is 
...
 that 
...
 the 
...
 the 
...
 time 
...
 average 
...
 is 
...
 the 
...
 weighted 
...
 average 
...
 over 
...
 all 
...
 possible 
...
 states 
...
 the 
...
 system 
...
 can 
...
 be 
...
 in 
...
 for 
...
 a 
...
 given 
...
 set 
...
 of 
...
 boundary 
...
 conditions.


{
 Latex
} \[ E = \sum_V E_V P_V \] {latex}

If 
If 
 Latex
 {latex} \[P_V \] {latex} 
 is 
...
 found, 
...
 everything 
...
 can 
...
 be 
...
 calculated. 
...
 Any 
...
 variable 
...
 that 
...
 can 
...
 fluctuate 
...
 can 
...
 be 
...
 determined 
...
 in 
...
 this 
...
 way.
...
Math 
...
 Review
...
Below 
...
 is 
...
 a 
...
 listing 
...
 of 
...
 topics 
...
 reviewed. 
...
 Read 
...
 McQuarrie 
...
 Ch. 
...
 1 
...
 and 
...
 see 
...
 the 
...
online 
...
 lecture 
...
notes for additional information
  • Time-dependent Schrodinger equation
  • wavefunction
  • The time-dependence is removed when dealing with equilibrium.
  • The solution without time-dependence is the stationary state.
  • Hamiltonian
  • The concept of degeneracy involves many states with the same energy
  • Boundary conditions give specific\Psi (t), E
  • Many-body problems are the sum of one-particle systems. 
    • Assume that degrees of freedom can be decoupled.
    • Decouple the Hamiltonian and write as a sum.
    • Energies of particular Hamiltonians can be superimposed
  • Symmetry of wave functions is related to indistinguishability. 
    • Given an N particle wavefunction, 
       Latex
       \[ \Psi (1, 2,..., N) \] 
...
    •  Latex
       \[ \Psi (2, 1, 3,..., N) = \pm \Psi (1, 2, 3,..., N) \] 
    • Indistinguishable particles are dealt with in this course
Examples of Simple Quantum Mechanical Systems
Write what interactions are assumed and solutions.
Particle in a 1-D Infinite Well Potential
A physical example of a 1-D infinite well potential is a particle in a box. Below is a schematic of the potential.
Image Added
Write the Hamiltonian, 
 Latex
{latex}
** Indistinguishable particles are dealt with in this course

h1. Examples of Simple Quantum Mechanical Systems

Write what interactions are assumed and solutions.

h2. Particle in a 1-D Infinite Well Potential

A physical example of a 1-D infinite well potential is a particle in a box. Below is a schematic of the potential.

!Infinite_potential.PNG!

Write the Hamiltonian, {latex} \[ \hat H \] {latex}
, 
...
 define 
...
 the 
...
 potential, 
...
 and 
...
 find 
...
 the 
...
 energy 
...
 eigenvalues.

{
 Latex
} \[ \hat H = \frac{-\hbar^2}{2m}\frac{\partial}{\partial x^2} + U(x) \] {latex}
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ U(x)= \begin{cases} \infty, & |x| > \frac{a}{2} \\ 0, & \frac{-a}{2} < x < \frac{a}{2} \end{cases} \] 
...
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ \varepsilon_n = \mbox{energy eigenvalues} \] 
...
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ \varepsilon_n = \frac{h^2 n^2}{8ma^2},n=1,2,
...
... \] 
Simple Harmonic Oscillator (1-D)
...
In 
...
 the 
...
 case 
...
 of 
...
 a 
...
 simple 
...
 harmonic 
...
 oscillator, 
...
 a 
...
 system 
...
 moved 
...
 from 
...
 equilibrium 
...
 feels 
...
 a 
...
 restoring 
...
 force.
Image Added
The energy eigenvalues are discrete.
 Latex


!Simple_harmonic_oscillator.PNG!

The energy eigenvalues are discrete.
{latex} \[ \hat H = \frac{\hbar}{2m}\frac{\partial}{\partial x^2} + U(x) \] {latex}
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ U(x) = \frac{1}{2} k x^2 \] 
...
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ \varepsilon_n = (n + \frac{1}{2})\hbar \omega(k), n = 0, 1, 2, ... \] 
The Concept of Degeneracy
Hydrogen Atom
Consider the hydrogen atom. The energy eigenvalues are proportional to the inverse of the square of the principal quantum number, 
 Latex
{latex}

h1. The Concept of Degeneracy


h2. Hydrogen Atom

Consider the hydrogen atom. The energy eigenvalues are proportional to the inverse of the square of the principal quantum number, {latex} \[ n \] {latex}
. 
...
 For 
...
 every 
 {
 Latex
} \[ n \] {latex}
, 
...
 there 
...
 are 
 {
 Latex
} \[ s \] {latex}, {latex}
, 
 Latex
 \[ p \] {latex}

, 
...
 and 
 {
 Latex
} \[ d \] {latex} 
 states 
...
 that 
...
 are 
...
 dependent 
...
 on 
...
 the 
...
 angular 
...
 momentum. 
...
 They 
...
 are 
...
 all 
...
 degenerate 
...
 in 
...
 energy, 
...
 but 
...
 there 
...
 are 
...
 different 
...
 wavefunctions 
...
 associated 
...
 with 
...
 each. 
...
 Consider 
...
 a 
...
 table 
...
 of 
...
 degeneracy. 
...
 The 
...
 degeneracy, 
 {
 Latex
} \[ w \] {latex}
, 
...
 is 
...
 equal 
...
 to 
 {
 Latex
} \[ 2n^2 \] {latex}
, 
...
 where 
...
 the 
...
 factor 
...
 of 
...
 two 
...
 is 
...
 due 
...
 to 
...
 spin.


{
 Latex
} \[ n=1, 2, ... \] {latex}
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ 0 \le l \le n-1 \] 
...
 Wiki Markup
{html}
<P></P>{html}
...
 Latex
...
 \[ -l \le m \le l \] 
...
 Wiki Markup
{html}{html}
...
 Latex
 \[ \mbox {n} \] 
...
 Latex
...
 \[ \mbox{states} \] 
...
 Latex
 \[ \mbox{degeneracy
...
} 
...
\] 
 Latex
 \[ \mbox {1} \] 
...
 Latex
...
 \[ 1s \] 
...
 Latex
 \[ \mbox{1} \] 
...
 Latex
 \[ \mbox {2} \] 
...
 Latex
 \[ 2s, 2p_x, 2p_y, 2p_z \] 
...
 Latex
 \[ \mbox{4
...
} 
...
\] 
 Latex
 \[ \mbox{3} \] 
...
 Latex
...
 \[ 3s, 3p_x, 3p_y, 3p_z \] 
...
 Latex
 \[ \mbox{9} \] 
{
 Latex
 } |

{latex} \[ 3d_{xy}, 3d_{yz}, 3d_{xz} \] {latex}
{
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ 3d_{x^2-y^2}, 3d_{z^2-r^2} \]
...
 
Degeneracy of one particle in a 3-D 
...
 Infinite 
...
 Well 
...
 Potential
...
This 
...
 is 
...
 a 
...
 generalization 
...
 of 
...
 the 
...
 one-dimensional 
...
 case. 
...
 Assume 
...
 that 
...
 the 
...
 three 
...
 directions 
...
 are 
...
 independent, 
...
 and 
...
 write 
...
 the 
...
 energy 
...
 eigenvalues.
 Latex



{latex} \[ E_{n_x,n_y,n_z}=\frac{h^2}{8ma^2} (n_x^2+n_y^2+n_z^2) \] {latex}
{
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ n_x,n_y,n_z=1, 2, 3 ,... \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ n_x^2+n_y^2+n_z^2 = R^2 \] 
...
How 
...
 many 
...
 ways 
...
 can 
...
 we 
...
 get 
...
 the 
...
 same 
 {
 Latex
} \[ R \] {latex}
? 
...
 Consider 
...
 an 
...
 example 
...
 of 
 {
 Latex
} \[ R=6 \] {latex}
. 
...
 Create 
...
 a 
...
 table 
...
 of 
...
 possibilities 
...
 and 
...
 find 
...
 that 
 {
 Latex
} \[ w=3 \] {latex}.

|| {latex} 
.
 Latex
 \[ n_x \] 
...
 Latex
 \[ n_y \] 
...
 Latex
 \[ n_z \] 
 1 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 1 
Generalized for any R in 3-D
How do we find the generalized 
 Latex
{latex} ||
| 1 | 1 | 2 |
| 1 | 2 | 1 |
| 2 | 1 | 1 |

h2. Generalized for any _R_ in _3-D_

How do we find the generalized {latex} \[ w(\varepsilon) \] {latex} 
 in 
...
 3D? 
...
 Show 
...
 is 
...
 two 
...
 dimensions 
...
 and 
...
 envision 
...
 in 
...
 three 
...
 dimensions. 
...
 Degeneracy 
...
 is 
...
 how 
...
 many 
...
 dots 
...
 land 
...
 on 
...
 the 
...
 arc 
...
 of 
...
 R 
...
 in 
...
 the 
...
 n 
...
 space. 
...
 Below 
...
 is 
...
 a 
...
 diagram 
...
 in 
...
 the 
...
 case 
...
 of 
...
 two 
...
 dimensions. 
...
 Look 
...
 at 
...
 positive 
...
 vales 
...
 of 
 {
 Latex
} \[ n_x \] {latex} and 
 and 
 Latex
{latex} \[ n_y \] {latex}
. 
...
 For 
...
 small 
...
 quantum 
...
 numbers, 
...
 there 
...
 is 
...
 an 
...
 irratic 
...
 step 
...
 function, 
...
 but 
...
 the 
...
 function is smooth for large functions.
Image Added
The degeneracy in three dimensions is equal to the number of points on the sphere with radius R in the first quadrant.
 Latex
 is smooth for large functions.


!R_versus_nx_and_ny.PNG!

The degeneracy in three dimensions is equal to the number of points on the sphere with radius R in the first quadrant.

{latex} \[ R^2 = n_x^2+n_y^2+n_z^2 \] {latex}

 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ R^2 = \frac{8ma^2 \varepsilon}{h^2} \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ \varepsilon > 0 \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ \varepsilon = \varepsilon_x + \varepsilon_y + \varepsilon_z \] 
When 
 Latex
{latex}

When {latex} \[ R \] {latex} or {
 or 
 Latex
latex} \[ E \] {latex} 
 is 
...
 large, 
...
 it 
...
 can 
...
 be 
...
 treated 
...
 as 
...
 a 
...
 continuous 
...
 variable. 
...
 Determine 
...
 the 
...
 number 
...
 of 
...
 lattice 
...
 points 
...
 between 
 {
 Latex
} \[ R \] {latex} 
 and 
 Latex
and {latex} \[ R + dR \] {latex} or 
 or 
 Latex
{latex} \[ \varepsilon \] 
 and 
 Latex
{latex} and {latex} \[ \varepsilon + d \varepsilon \] {latex}
. 
...
 Look 
...
 at 
...
 the 
...
 number 
...
 of 
...
 points 
...
 within 
...
 the 
...
 sphere; 
...
 consider 
...
 the 
...
 number 
...
 of 
...
 points 
...
 with 
...
 energy 
...
 less 
...
 than 
 {
 Latex
} \[ \epsilon \] {latex}
. 
...
 If 
...
 the 
...
 number 
...
 of 
...
 points 
...
 is 
...
 dense, 
...
 it 
...
 can 
...
 be 
...
 set 
...
 to 
...
 the 
...
 volume, 
...
 and 
...
 below 
...
 is 
...
 an 
...
 expression 
...
 for 
 {
 Latex
} \[ \phi (\epsilon) \] {latex}
, 
...
 which 
...
 is 
...
 defined 
...
 as 
...
 the 
...
 number 
...
 of 
...
 points 
...
 within 
 {
 Latex
} \[ R \] {latex}.

{latex}
.
 Latex
 \[ \phi (\epsilon) = \frac{1}{8} \left ( \frac{4 \pi R^3}{3} \right ) \] {latex}
{] 
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ \phi (\epsilon) = \frac{\pi}{6} \left ( \frac{8 m a^2 \epsilon}{h^2} \right )^{\frac{3}{2}} \] 
...
The 
...
 number 
...
 of 
...
 states 
...
 in 
...
 a 
...
 slice, 
...
 or 
...
 the 
...
 number 
...
 of 
...
 states 
...
 between 
 {
 Latex
} \[ \epsilon \] {latex} and
 and 
 Latex
 {latex} \[ \epsilon + \delta \epsilon \] {latex}
, 
...
 is 
...
 of 
...
 interest. 
...
 A 
...
 formula 
...
 is 
...
 below 
...
 for 
...
 the 
...
 number 
...
 of 
...
 states 
...
 that 
...
 become 
...
 available 
...
 when 
...
 increasing 
...
 energy 
...
 by 
...
 a 
...
 small 
...
 amount. 
...
 There 
...
 is 
...
 an 
...
 assumption 
...
 that 
 {
 Latex
} \[ \delta \epsilon \] {latex} 
 is 
...
 very 
...
 small, 
...
 and 
...
 there 
...
 is 
...
 a 
...
 Taylor 
...
 expansion.


{
 Latex
} \[ w( \epsilon, \Delta \epsilon )= \phi (\epsilon + \Delta \epsilon) - \phi (\epsilon) \] {latex}
{
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ w( \epsilon, \Delta \epsilon ) = \frac {\pi}{6} \left ( \frac{m a^2}{h^2} \right )^{\frac{3}{2}} \left ( \left (\epsilon + \Delta \epsilon \right )^{\frac{3}{2}} - \epsilon^{\frac{3}{2}} \right ) \]
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ w( \epsilon, \Delta \epsilon ) = \frac{\pi}{6}\left ( \frac{8 m a^2 \epsilon}{h^2} \right )^{\frac{3}{2}} \left ( \left (1 + \frac{\Delta \epsilon}{\epsilon} \right )^{\frac{3}{2}} - 1 \right ) \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ w( \epsilon, \Delta \epsilon ) = \frac{\pi}{4} \left ( \frac{8 m a^2 \epsilon}{h^2} \right )^{\frac{3}{2}} \epsilon^{\frac{3}{2}} \Delta \epsilon \] 
Look at the order of magnitude of this. Consider just the kinetic energy in three dimensions. The formula for kinetic energy is 
 Latex
{latex}


Look at the order of magnitude of this. Consider just the kinetic energy in three dimensions. The formula for kinetic energy is {latex} \[ \epsilon = \frac{3}{2} k T \] {latex}
. 
...
 Temperature 
...
 can't 
...
 be 
...
 assigned 
...
 to 
...
 just 
...
 one 
...
 particle. 
...
 Consider 
...
 one 
...
 particle 
...
 in 
...
 contact 
...
 with 
...
 heat 
...
 bath, 
...
 a 
...
 gas 
...
 particle 
...
 in 
...
 a 
...
 box.


{
 Latex
} \[ T=300K \] {latex}
{
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ m=10^{-22} g \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ a=10 cm \]
...
 
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ \Delta \epsilon = 0.01 \epsilon \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ w( \epsilon, \Delta \epsilon ) \approx 10^{28} \] 
...
There 
...
 is 
...
 a 
...
 huge 
...
 number 
...
 of 
...
 additional 
...
 points. 
...
 The 
...
 number 
...
 of 
...
 states 
...
 that 
...
 can 
...
 be 
...
 accessed 
...
 is 
...
 enormous. 
...
 The 
...
 numbers 
...
 are 
...
 very 
...
 dense 
...
 using 
...
 room 
...
 temperature. 
...
 When 
...
 calculating 
...
 with 
...
 interacting 
...
 particles 
...
 the 
...
 results 
...
 are 
...
 about 
...
 the 
...
 same.
...
Summary
  • for small R --> 
...
  •  w 
...
  •  is 
...
  •  erratic
...
  • for 
...
  •  large 
...
  •  R 
...
  •  --> 
...
  •  w 
...
  •  is 
...
  •  more 
...
  •  smooth
...
For 
...
 3D 
...
 case


{
 Latex
} \[ n_x^2+n_y^2+n_z^2 = R^2 = \frac{8ma^2 \varepsilon}{h^2} \] {latex} 

 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ \varepsilon > 0 \] 
...
 Wiki Markup
{html}<p>{html}
...
 Latex
...
 \[ \varepsilon = \varepsilon_x + \varepsilon_y + \varepsilon_z \] 
...
Additional 
...
 math 
...
 topics
...
A 
...
 listing 
...
 is 
...
 below 
...
 of 
...
 additional 
...
 math 
...
 topics 
...
 covered. 
...
 Additional 
...
 information 
...
 is 
...
 posted 
...
 at 
...
 thecourse 
...
 website.
...
  • Average 
...
  •  of 
...
  •  Latex
...
  •  \[ u \] 
...
  • Mean of a function 
     Latex
     \[ f(u) \] 
...
  •  Latex
...
  •  \[ m^{th} \] 
...
  •  moment 
...
  •  of 
...
  •  distribution
...
  •  Latex
     \[ m^{th} \] 
...
  •  central 
...
  •  moment 
...
  •  of 
...
  •  a 
...
  •  distribution
...
  • Integration 
...
  •  of 
...
  •  function 
...
  •  with 
...
  •  probability 
...
  •  density
...
  • Gaussian 
...
  •  distribution
...
  • Stirling's 
...
  •  approximation
...
  • Binomial/multinomial 
...
  •  distribution
...
October 
...
 20, 
...
 2006
...