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 <math>\overline E = \langle E(t) \rangle</math>. 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, <math>P_v</math>, involved in the weighted average.

• Thermodynamic variables = time averages of their microscopic counterparts
• <math>U = \overline E = \langle E(t) \rangle = \frac
  Unknown macro: {1}

  Unknown macro: {Delta t}

  \int_

  \langle \Psi (q,t) \mid \hat H \mid \Psi^*(q,t) \rangle</math> where 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.

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  <math>E = \sum_V E_V P_V</math>

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  If <math>P_V</math> 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 <math>\Psi (t), E</math>
  • 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, <math>\Psi (1, 2,..., N)</math>
    • <math>\Psi (2, 1, 3,..., N) = \pm \Psi (1, 2, 3,..., N)</math>
    • 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.

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  Write the Hamiltonian, <math>\hat H</math>, define the potential, and find the energy eigenvalues.

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  <math>\hat H = \frac{-\hbar^2}

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  \frac

  Unknown macro: {partial}

  Unknown macro: {partial x^2}

  + U
  </math>

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  <math>U= \begin

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  \infty, & |x| > \frac

  Unknown macro: {a}

  Unknown macro: {2}

  
  0, & \frac{-a}

  < x < \frac

  Unknown macro: {2}

  \end

  </math>

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  <math>\varepsilon_n = \mbox

  Unknown macro: {energy eigenvalues}

  </math>

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  <math>\varepsilon_n = \frac

  Unknown macro: {h^2 n^2}

  Unknown macro: {8ma^2}

  </math>, <math>n=1,2,...</math>

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  Simple Harmonic Oscillator (1-D)

  In the case of a simple harmonic oscillator, a system moved from equilibrium feels a restoring force.

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  The energy eigenvalues are discrete.

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  <math>\hat H = \frac

  Unknown macro: {hbar}

  \frac

  Unknown macro: {partial}

  Unknown macro: {partial x^2}

  + U</math>

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  <math>U = \frac

  Unknown macro: {2}

  k x^2 </math>

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  <math>\varepsilon_n = (n + \frac

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  )\hbar \omega(k), n = 0, 1, 2, ...</math>

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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, <math>n</math>. For every <math>n</math>, there are <math>s</math>, <math>p</math>, and <math>d</math> 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 deneracy, <math>w</math>, is equal to <math>2n^2</math>, where the factor of two is due to spin.

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<math>n=1, 2, ...</math>

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<math>0 \le l \le n-1</math>

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<math>-l \le m \le l</math>

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<table cellpadding = 5>

<tr>

<td>
<center><math>\mbox

</math></center>
</td>

<td>
<center><math>\mbox

</math></center>
</td>

<td>
<center><math>\mbox

</math></center>
</td>

</tr>

<tr>

<td>
<math>\mbox

</math></center>
</td>

</tr>

<tr>

<td>
<math>\mbox

</math>

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The number of states in a slice, or the number of states between <math>\epsilon</math> and <math>\epsilon + \delta \epsilon</math>, 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 <math>\delta \epsilon</math> is very small, and there is a Taylor expansion.

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<math>w( \epsilon, \Delta \epsilon )= \phi (\epsilon + \Delta \epsilon) - \phi (\epsilon)</math>

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<math>w( \epsilon, \Delta \epsilon ) = \frac

\left ( \frac

, \varepsilon > 0</math>, <math>\varepsilon = \varepsilon_x + \varepsilon_y + \varepsilon_z</math>

Additional math topics

A listing is below of additional math topics covered. Additional information is posted at the course website.

• Average of <math>u</math>
• Mean of a function of <math>u</math>, <math>f(u)</math>
• _m_th moment of distribution
• _m_th central moment of a distribution
• Integration of function with probability density
• Gaussian distribution
• Stirling's approximation
• Binomial/multinomial distribution

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October 20, 2006
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