An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Question
Chapter 7.6, Problem 70P
(a)
To determine
The expression for total energy of gas of bosons confined to volume
(b)
To determine
Evaluate the total energy for
(c)
To determine
The specific heat at high temperature is given as
(d)
To determine
To find the expression of energy for
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Students have asked these similar questions
Problem 1:
This problem concerns a collection of N identical harmonic oscillators (perhaps an
Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf,
and so on.
a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving
1-x
the formula by long division. Prove it by first multiplying both sides of the
equation by (1 – x), and then thinking about the right-hand side of the resulting
expression.
b) Evaluate the partition function for a single harmonic oscillator. Use the result of
(a) to simplify your answer as much as possible.
c) Use E = -
дz
to find an expression for the average energy of a single oscillator.
z aB
Simplify as much as possible.
d) What is the total energy of the system of N oscillators at temperature T?
Suppose a particle of mass m and charge q is in a one-dimensional harmonic oscillator potential with natural frequency
wo. For times t > 0 a time-dependent potential of the form
V₁(x,t) = εx cos(wt)
is turned on. Assume the system starts in an initial state In).
1. Find the transitionn probability from initial state (n) to a state \n') with n' ‡ n.
2. Find the transition rate (probability per unit time) for the transition (n) → \n').
Note: (n'|x|n)=
=
ħ
2mwo
(√√n +18n',n+1 + √ñdn',n−1).
Consider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on.
A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators?
B) Obtain the average energy of N oscillators at temperature T from the partition function.
C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this?
D) Find the Helmholtz free energy from this system.
E) Derive the expression that gives the entropy of this system for the temperature.
Chapter 7 Solutions
An Introduction to Thermal Physics
Ch. 7.1 - Prob. 1PCh. 7.1 - Prob. 3PCh. 7.1 - Prob. 4PCh. 7.1 - Show that when a system is in thermal and...Ch. 7.1 - Prob. 7PCh. 7.2 - Prob. 8PCh. 7.2 - Prob. 9PCh. 7.2 - Prob. 11PCh. 7.2 - Prob. 12PCh. 7.2 - Prob. 13P
Ch. 7.2 - Prob. 14PCh. 7.2 - Prob. 15PCh. 7.2 - Prob. 16PCh. 7.2 - Prob. 17PCh. 7.2 - Prob. 18PCh. 7.3 - Prob. 19PCh. 7.3 - Prob. 20PCh. 7.3 - Prob. 21PCh. 7.3 - Prob. 22PCh. 7.3 - Prob. 24PCh. 7.3 - Prob. 25PCh. 7.3 - Prob. 26PCh. 7.3 - Prob. 29PCh. 7.3 - Prob. 32PCh. 7.3 - Prob. 33PCh. 7.3 - Prob. 34PCh. 7.4 - Prob. 37PCh. 7.4 - Prob. 38PCh. 7.4 - Prob. 39PCh. 7.4 - Prob. 40PCh. 7.4 - Prob. 41PCh. 7.4 - Prob. 42PCh. 7.4 - Prob. 43PCh. 7.4 - Prob. 44PCh. 7.4 - Prob. 45PCh. 7.4 - Prob. 46PCh. 7.4 - Prob. 47PCh. 7.4 - Prob. 48PCh. 7.4 - Prob. 49PCh. 7.4 - Prob. 50PCh. 7.4 - Prob. 51PCh. 7.4 - Prob. 52PCh. 7.4 - Prob. 53PCh. 7.4 - Prob. 54PCh. 7.4 - Prob. 55PCh. 7.4 - Prob. 56PCh. 7.5 - Prob. 57PCh. 7.5 - Prob. 58PCh. 7.5 - Prob. 59PCh. 7.5 - Prob. 60PCh. 7.5 - The heat capacity of liquid 4He below 0.6 K is...Ch. 7.5 - Prob. 62PCh. 7.5 - Prob. 63PCh. 7.5 - Prob. 64PCh. 7.6 - Prob. 65PCh. 7.6 - Prob. 66PCh. 7.6 - Prob. 67PCh. 7.6 - Prob. 68PCh. 7.6 - If you have a computer system that can do...Ch. 7.6 - Prob. 70PCh. 7.6 - Prob. 71PCh. 7.6 - Prob. 72PCh. 7.6 - Prob. 73PCh. 7.6 - Prob. 75P
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- Suppose an Einstein Solid is in equilibrium with a reservoir at some temperature T. Assume the ground state energy is 0, the solid is composed of N oscillators, and the size of an energy "unit" is e. (a) Find the partition function for a single oscillator in the solid, Z1. Hint: use the general series summation formula 1+ x + x? + x³ + ... = 1/ (1- x) (b) Find an expression for , the average energy per oscillator in the solid, in terms of kT and e. (c) Find the total energy of the solid as a function of T, using the expression from part (b). (d) Suppose e = 2 eV and T = 25°C. What fraction of the oscillators is in the first excited state, compared to the ground state (assuming no degeneracies of energy levels)?arrow_forwardwrite the solution step by step and clearly.arrow_forwardThe intensities of spectroscopic transitions between the vibrational states of a molecule are proportional to the square of the integral ∫ψv′xψvdx over all space. Use the relations between Hermite polynomials given in Table 7E.1 to show that the only permitted transitions are those for which v′ = v ± 1 and evaluate the integral in these cases.arrow_forward
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