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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
 \[ 3d










\mbox{3d_{xy}, 3d_{yz}, 3d_{xz} \] 
 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
 \[ 






E_{n_x,n_y,n_z}=\frac{h^2}{8ma^2} (n_x^2+n_y^2+n_z^2)}




 \] 
 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
 \[ R \] 
? Consider an example of 
 Latex
 \[ R=6 \] 
. Create a table of possibilities and find that 
 Latex
 \[ w=3 \] 
.
 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
 \[ w(\varepsilon) \] 
 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
 \[ n_x \] 
 and 
 Latex
 \[ n_y \] 
. 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
 \[ 






How many ways can we get the same R? Consider an example of R=6. Create a table of possibilities and find that w=3.







n_x



n_y



n_z







1



1



2







1



2



1







2



1



1








h2. Generalized for any \_R\_ in \_3-D\_

How do we find the generalized w(\varepsilon) 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 n_x and n_y. For small quantum numbers, there is an irratic step function, but the function 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.






R^2 = n_x^2+n_y^2+n_z^2




 \] 
 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
 \[






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






 \] 
.
 Latex
 \[ \phi (\epsilon) = \frac{1}{8} \left ( \frac{4 \pi R^3}{3} \right )




 \] 
 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
 \[ \epsilon \] 
 and 
 Latex
 \[ </math><br></center>The number of states in a slice, or the number of states between <math>\epsilon</math> and <math>\epsilon + \delta \epsilon \epsilon</math>, is of ] 
, 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
 \[  that <math>\delta \epsilon \epsilon</math>] 
 is 
...
 very 
...
 small, 
...
 and 
...
 there 
...
 is 
...
 a 
...
 Taylor 
...
 expansion.
 Latex
 \[ w<center><br><math>w( \epsilon, \Delta \epsilon )= \phi (\epsilon + \Delta \epsilon) - \phi (\epsilon)</math><br><math>w( \epsilon, \Delta \epsilon] 
 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
 \[ 






Look at the order of magnitude of this. Consider just the kinetic energy in three dimensions. The formula for kinetic energy is\epsilon = \frac{3}{2} k T \] 
. 
...
 Temperature 
...
 can't 
...
 be 
...
 assigned 
...
 to 
...
 just 
...
 one 
...
 particle. 
...
 Consider 
...
 one 
...
 particle 
...
 in 
...
 contact 
...
 with 
...
 heat 
...
 bath, 
...
 a 
...
 gas 
...
 particle 
...
 in 
...
 a 
...
 box.
...
 Latex
 \[ T=300K \] 
 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
 \[ n_x^2+n_y^2+n_z^2 = R^2 = \frac{8ma^2 \varepsilon}{h^2} \] 
 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