Intro to Statistical Mechanics

Question 1

What are the two approaches to thermal physics?

Answer 1

Thermodynamics and statistical mechanics.

Question 2

What is thermodynamics?

Answer 2

Assumes a unique normal state for any system called thermal equilibrium.

Question 3

What are 2 thing in thermodynamics?

Answer 3

• Can state 4 laws relating quantities p,T,u and s

- Laws are empirical and are justified experimentally

Question 4

What is statistical mechanics?

Answer 4

Defines the concepts of thermal equilibrium and entropy in terms of microscopic probabilities.

Question 5

What are 4 things in statistical mechanics?

Answer 5

• Gives probability of a particular state
• Determines measurable properties using averages of these states
• Derives the results of thermodynamics
• Makes predictions about quantum systems

Question 6

What is an isolated system?

Answer 6

System which is not connected to a reservoir so E,N,V fixed.

Question 7

What is a model system?

Answer 7

The simple harmonic oscillator.

Question 8

What is the equation for the potential energy of the harmonic oscillator?

Answer 8

V = 1/2 mω^2 *x^2

Question 9

What is the equation for En of the harmonic oscillator?

Answer 9

En = (n+1/2)ћω, where n is a quantum number

Question 10

If we have multiple SHOs, what is the total energy equal to?

Answer 10

E = E1+E2+E3+E4

Question 11

What is the equation for the zero point energy?

Answer 11

Nћ*ω/2, where N is the number of SHOs

Question 12

What is the zero point energy?

Answer 12

• Energy of the ground state

- Evidence that things move even at zero temperature

Question 13

What is a microstate?

Answer 13

Any microscopic configuration of the system e.g. for 1l of N2 gas this would specify the position and velocity of every molecule (very difficult to measure)

Question 14

What is a macrostate?

Answer 14

Any configuration with particular measurable properties e.g. for 1l of N2 gas this would specify u & N & V (can be made up of very many microstates)

Question 15

What ways can 3SHOs with E=2ћω be arranged?

Answer 15

2-0-0 with E(1) = 2ћω, 1-1-0, 1-0-1 with E(1) = 1ћω, 0-1-1 , 0-0-2, 0-2-0 with E(1) = 0ћω -> 1 macrostate (E = 2ћω) made up of 6 microstates

Question 16

What is the symbol for microstates?

Answer 16

Ω

Question 17

How does the measure ment effect the number of macrostates?

Answer 17

Normal measurement = 1 macrostate, sensitive to E(1) = 3 macrostates (3 values of E(1), sensitive to E(1), E(2) and E(3) = 6 macrostates

Question 18

What is the first assumption of SM?

Answer 18

Each one of the possible microstates of a system in thermal equilibrium is equally likely to occur.

Question 19

What is the second assumption of SM?

Answer 19

The system’s dynamics are such that the microstate is constantly changing.

Question 20

What is the third assumption of SM?

Answer 20

Given enough time, the system will explore all possible microstates and spend equal time in each of them -> known as the Ergodic Hypothesis.

Question 21

What are the 2 consequences of our assumptions in SM?

Answer 21

• System will appear to choose macroscopic configuration that maximises number of microstates
• measure value of an observable is equal to the average over the microstates.

Question 22

For the 3 SHOs with E = 2ћω, how do we find the expected value of E(1)?

Answer 22

= sum over i of Ei(1)P(i) = 1/Ω * sum over i of Ei(1) = 1/6(2+1+1+0+0)ћω = 2/3 *ћω

Question 23

What is the micro-canonical ensemble?

Answer 23

An ensemble of isolated systems with fixed E & N.

Question 24

What is the canonical ensemble?

Answer 24

Ensemble of systems each of which can exchange energy with a large reservoir.

Question 25

What is the grand-canonical ensemble?

Answer 25

Ensemble of system each of which can exchange both energy and particles with a large reservoir.