An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Question
Chapter 2.4, Problem 24P
(a)
To determine
To Find: The height of the peak in the multiplicity function using Stirling’s method.
(b)
To determine
To Find: Derive the formula for the multiplicity function in the vicinity of the peak.
(c)
To determine
To Find: Width of the peak in the multiplicity function.
(d)
To determine
To Explain: Explain while flipping of coins getting head and tails.
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Students have asked these similar questions
If V0 = 4 eV, E = 1 eV and L = 0.01 nm, determine the probability of a quantum-mechanical
electron making its way through this barrier. Express your answer as a percentage.
Note: You are being asked to provide a precise calculation, using correct boundary conditions at x = 0 and x = L, and not to use an approximation.
Hints for part d) of question 4: The required calculation is very similar to the calculation of the
transmission coefficient T for the Finite Potential Barrier given in lectures, but with a different
wavenumber in the region to the right of the barrier.
After applying the boundary conditions at x = 0 and x = L, you may choose to simplify the
expression you find for the transmission coefficient, T. In doing this a useful result is that the
modulus-squared of
f = αβ e−kLˆ
− α
∗β
∗
e
kLˆ
,
where α ≡ ˆk + ik and β ≡ ˆk + iq, is given by
|f|
2 = 4(ˆk
2 + k
2
)(ˆk
2 + q
2
) sinh2
(
ˆkL) + 4ˆk
2
(k + q)
2
.
You may use this result to obtain your answer.
Consider the "step" potential:
V(x) =
(a) Calculate the reflection coefficient, for the case E 0.
(b) Calculate the reflection coefficient for the case E > Vo.
(c) For a potential such as this, which does not go back to zero to the right of
the barrier, the transmission coefficient is not simply |F12/A2 (with A the
-Vo
AV(x)
Scattering from a "cliff"
incident amplitude and F the transmitted amplitude), because the transmitted
wave travels at a different speed. Show that
T =
E-Vo F1²
E |A|²
X
for E> Vo. Hint: You can figure it out using Equation
gantly, but less informatively-from the probability current (
What is T, for E Vo, calculate the transmission coefficient for the step potential, and
check that T + R = 1.
(A) Consider a particle in a cubic box. What is the degeneracy of the level it hasenergy three times greater than that of the lowest level? (Explain the combinations of n that led you to the answer given).
(B) The addition of sodium to ammonia generates a solvated electron that is trapped in a cavity of 0.3 nm in diameter, formed by ammonia molecules. The solvated electron can be modeled as a particle that moves freely inside the cubic box with ammonia molecules in the cube surface. If the length of the box is 0.3 nm, what energy is needed for the electron undergo a transition from a lower energy state to the subsequent state?
Chapter 2 Solutions
An Introduction to Thermal Physics
Ch. 2.1 - Prob. 1PCh. 2.1 - Prob. 2PCh. 2.1 - Prob. 3PCh. 2.1 - Prob. 4PCh. 2.2 - For an Einstein solid with each of the following...Ch. 2.2 - Prob. 6PCh. 2.2 - Prob. 7PCh. 2.3 - Prob. 8PCh. 2.3 - Use a computer to reproduce the table and graph in...Ch. 2.3 - Use a computer to produce a table and graph, like...
Ch. 2.3 - Use a computer to produce a table and graph, like...Ch. 2.4 - Prob. 12PCh. 2.4 - Fun with logarithms. (a) Simplify the expression...Ch. 2.4 - Write e1023 in the form 10x, for some x.Ch. 2.4 - Prob. 15PCh. 2.4 - Prob. 16PCh. 2.4 - Prob. 17PCh. 2.4 - Prob. 18PCh. 2.4 - Prob. 19PCh. 2.4 - Suppose you were to shrink Figure 2.7 until the...Ch. 2.4 - Prob. 21PCh. 2.4 - Prob. 22PCh. 2.4 - Prob. 23PCh. 2.4 - Prob. 24PCh. 2.4 - Prob. 25PCh. 2.5 - Prob. 26PCh. 2.5 - Prob. 27PCh. 2.6 - How many possible arrangements are there for a...Ch. 2.6 - Consider a system of two Einstein solids, with...Ch. 2.6 - Prob. 30PCh. 2.6 - Fill in the algebraic steps to derive the...Ch. 2.6 - Prob. 32PCh. 2.6 - Use the Sackur-Tetrode equation to calculate the...Ch. 2.6 - Prob. 34PCh. 2.6 - According to the Sackur-Tetrode equation, the...Ch. 2.6 - For either a monatomic ideal gas or a...Ch. 2.6 - Using the Same method as in the text, calculate...Ch. 2.6 - Prob. 38PCh. 2.6 - Compute the entropy of a mole of helium at room...Ch. 2.6 - For each of the following irreversible process,...Ch. 2.6 - Describe a few of your favorite, and least...Ch. 2.6 - A black hole is a region of space where gravity is...
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- In free spoce, E (2,t) = 10. sin (wt - 82) ox, (V/m) %3D Show that the overage power po ssing through circular disk with a radius of 15.5 m in the Constant plane z= is IW. (Please write down , explaining all the Steps)arrow_forwardAt what displacements is the probability density a maximum for a state of a harmonic oscillator with v = 1? (Express your answers in terms of the coordinate y.)arrow_forwardGiven 8 hydrogen molecules form a perfect quantal gas. The system is put into a heat bath at constant temperature T. (a) Is the mentioned hydrogen gas a boson gas or fermion gas? Find all the distinct energy levels of the system, Es, and the corresponding degeneracies, gs (or number of microstates of Es). Tabulate your work out and answers. (b) Find the partition function, Z, Helmholtz free energy, F, mean energy E, and entropy, S, of the gas. Note: Your answers in Parts (a) and (b) should be in terms of e and kBT, you cannot use the number given to kÊT in Part (c). (c) = Given KBT 3ɛ. Sketch the probability distributions of Es in a bar-chart, and determine the most probable energy level.arrow_forward
- A proton is confined in box whose width is d = 750 nm. It is in the n = 3 energy state. What is the probability that the proton will be found within a distance of d/n from one of the walls? Include a sketch of U(x) and ?(x). Sketch the situation, defining all your variablesarrow_forwardLet's consider the two-qubit state 3 |) = 100)+101) +110). a) Find the expectation values for the values of both qubits separately. b) The product of qubit values is represented by the operator b₁b2 = (ô× 1) (I Øô) = (ô ❀ô), where bn is the observable for the value of qubit n. Find the expectation value for the product. For statistically independent quantities the expectation value of their product is the product of their expectation values. Are the values of the qubits correlated in state |V)? c) Show that the state cannot be expressed as a product state, i.e., it is an entangled state.arrow_forwardA neutron of mass m of energy E a, V(x) = V, ) II. Calculate the total probability of neutron tunneling through the barrier?; from region I to II.arrow_forward
- Consider a particle with function : (image) Normalize this function. Qualitatively plot the probability distribution function in terms of x. Where is it most likely to find a particle? Find the expected value of the place and compare it with part C.arrow_forwardProblem # 2. In the two-level system, estimate the emission line full width at half maximum (FWHM) for spontaneous emission at 650 nm if the spontaneous radiative lifetime of the upper state is about 3,000 nanoseconds.arrow_forward(a) Write down the wave functions for the three regions of the potential energy barrier (Figure 5.25) for E < U₁. You will need six coefficients in all. Use complex exponential notation. (b) Use the boundary conditions at x = 0 and at x = L to find four relationships among the six coeffi- cients. (Do not try to solve these relationships.) (c) Sup- pose particles are incident on the barrier from the left. Which coefficient should be set to zero? Why?arrow_forward
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