/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Consider the integral $$ E(x... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 3

Consider the integral $$ E(x)=\int_{0}^{x} \mathrm{e}^{-t^{2}} \mathrm{~d} t . $$ a) Write a program to calculate \(E(x)\) for values of \(x\) from 0 to 3 in steps of \(0.1 .\) Choose for yourself what method you will use for performing the integral and a suitable number of slices. b) When you are convinced your program is working, extend it further to make a graph of \(E(x)\) as a function of \(x\). If you want to remind yourself of how to make a graph, you should consult Section 3.1, starting on page 88 . Note that there is no known way to perform this particular integral analytically, so numerical approaches are the only way forward.

Short Answer

Expert verified
Use a numerical method like the trapezoidal rule to approximate \(E(x)\). Compute values from 0 to 3 in steps of 0.1 and plot the result.

Step by step solution

01

Choose a Numerical Integration Method

Select a method such as the trapezoidal rule or Simpson's rule for the numerical integration. The trapezoidal rule approximates the area under the curve as a series of trapezoids.
02

Divide the Interval

Divide the interval \([0, 3]\) into smaller intervals (or slices) with a step size of 0.1. This will give you \(\frac{3-0}{0.1}=30\) slices.
03

Implement the Numerical Integration

Write a program using your chosen numerical method. For example, in Python, this could be achieved using the following code for the trapezoidal rule:```pythonimport numpy as np# Define the integrandf = lambda t: np.exp(-t**2)# Define the function to calculate E(x)def E(x, step_size=0.1): n_slices = int(x / step_size) result = 0.5 * f(0) + 0.5 * f(x) for i in range(1, n_slices): result += f(i * step_size) result *= step_size return result# Calculate E(x) for values from 0 to 3 in steps of 0.1x_values = np.arange(0, 3.1, 0.1)E_values = [E(x) for x in x_values]```This code calculates \(E(x)\) using the trapezoidal rule.
04

Verify Your Program

Run your program and verify if the values make sense. You can compare some known values of the error function to see if your results are reasonable.
05

Plot the Graph

Extend your program to plot the graph of \(E(x)\) as a function of \(x\). Here is how you can do it in Python using matplotlib:```pythonimport matplotlib.pyplot as plt# Plot the graphplt.plot(x_values, E_values, label='E(x)')plt.xlabel('x')plt.ylabel('E(x)')plt.title('Graph of E(x) as a function of x')plt.legend()plt.grid(True)plt.show()```This code generates a graph for \(E(x)\) based on the computed values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trapezoidal rule
The Trapezoidal Rule is a simple method for numerical integration, ideal for approximating the definite integral of a function when an analytical solution is difficult or impossible to achieve.
The basic idea involves breaking the area under the curve into a series of trapezoids rather than rectangles. These trapezoids better approximate the curve, especially for linear or gently curving functions.
To implement the trapezoidal rule, follow these steps:
  • Divide the interval \([a, b]\) into smaller slices of width \(h\).
  • Calculate the area for each trapezoid using the formula: \(\text{Area} = \frac{h}{2} (y_0 + 2y_1 + 2y_2 + ... + 2y_{n-1} + y_n)\).
In our example, for integrating \(E(x) = \int_{0}^{x} e^{-t^2} \mathrm{d} t\), the interval \[0, 3\] is divided into 30 slices as \(h = 0.1\). The integration is then carried out using numpy and Python.
The trapezoidal rule offers a balance between accuracy and computational load, but it may not handle highly nonlinear functions as accurately as other methods like Simpson's rule.
Simpson's rule
Simpson's Rule is another numerical method for integration that generally offers greater accuracy than the Trapezoidal Rule, especially for functions that are quadratic or better approximated by parabolas.
It requires the function values at both the endpoints and the midpoints of the intervals.
This method approximates the area under the curve by fitting a parabola through every three adjacent points, providing a more precise estimate.
Here's how you implement Simpson's Rule:
  • Divide the interval \[a, b\] into an even number of slices, \(n\).
  • Calculate the integral with \(I \approx \frac{h}{3}(y_0 + 4y_1 + 2y_2 + 4y_3 + ... + 2y_{n-2}+4y_{n-1} + y_n)\).
In our integral example, for \(E(x)\), the interval would still be divided, but Simpson's Rule would require calculations for both endpoints and midpoints, enhancing the precision.
Though slightly more complex, Simpson's Rule is particularly valuable for smoother functions and can reduce the error considerably compared to the Trapezoidal Rule.
error function
The Error Function, denoted as \(\text{erf}(x)\), is a special mathematical function that arises in probability, statistics, and partial differential equations related to Gaussian distributions.
It is essentially a scaled integral of the Gaussian function from 0 to a point \(x\).
Mathematically, it can be expressed as: \[ \text{erf}(x) = \frac{2}{\root{\root{}}{\root{}} \root{}} \root{\root{\root{}}}\int_{0}^{x} e^{-t^2} \mathrm{d} t \]Since there's no simple closed form for the integral of \(e^{-t^2}\), which appears in our exercise for computing \(E(x)\), numerical methods like the Trapezoidal and Simpson's Rules are crucial.
These methods allow us to approximate the integral \(E(x)\), giving us a way to compute the error function values iteratively.
By understanding and using the properties of the error function, we can apply these numerical techniques effectively across many fields where Gaussian distributions are common.

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Most popular questions from this chapter

Quantum uncertainty in the harmonic oscillator In units where all the constants are 1 , the wavefunction of the \(n\)th energy level of the one-dimensional quantum harmonic oscillator-i.e., a spinless point particle in a quadratic potential well-is given by $$ \psi_{n}(x)=\frac{1}{\sqrt{2^{11} n ! \sqrt{\pi}}} \mathrm{e}^{-x^{2} / 2} H_{n}(x), $$ for \(n=0 \ldots \infty\), where \(H_{n}(x)\) is the \(n\)th Hermite polynomial. Hermite polynomials satisfy a relation somewhat similar to that for the Fibonacci numbers, although more complex: $$ H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) . $$ The first two Hermite polynomials are \(H_{0}(x)=1\) and \(H_{1}(x)=2 x\). a) Write a user-defined function \(\mathrm{H}(\mathrm{n}, \mathrm{x})\) that calculates \(H_{n}(x)\) for given \(x\) and any integer \(n \geq 0\). Use your function to make a plot that shows the harmonic oscillator wavefunctions for \(n=0,1,2\), and 3, all on the same graph, in the range \(x=-4\) to \(x=4\). Hint: There is a function factorial in the ath package that calculates the factorial of an integer. b) Make a separate plot of the wavefunction for \(n=30\) from \(x=-10\) to \(x=10\). Hint If your program takes too long to run in this case, then you're doing the calculation wrong-the program should take only a second or so to run. c) The quantum uncertainty in the position of a particle in the \(n\)th level of a harmonic oscillator can be quantified by its root-mean-square position \(\sqrt{\left\langle x^{2}\right\rangle}\), where $$ \left\langle x^{2}\right\rangle=\int_{-\infty}^{\infty} x^{2}\left|\psi_{n}(x)\right|^{2} \mathrm{~d} x . $$ Write a program that evaluates this integral using Gaussian quadrature on 100 points, then calculates the uncertainty (i.e., the root-mean-square position of the particle) for a given value of \(n\). Use your program to calculate the uncertainty for \(n=5\). You should get an answer in the vicinity of \(\sqrt{\left\langle x^{2}\right\rangle}=2.3 .\)

The Stefan-Boltzmann constant The Planck theory of thermal radiation tells us that in the (angular) frequency interval \(\omega\) to \(\omega+d \omega\), a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to \(I(\omega)\) d \(\omega\), where $$ I(\omega)=\frac{\hbar}{4 \pi^{2} c^{2}} \frac{\omega^{3}}{\left(\mathrm{e}^{\hbar \omega / k_{a} T}-1\right)} $$ Here \(h\) is Planck's constant over \(2 \pi, c\) is the speed of \(h\) ght, and \(k_{B}\) is Boltzmann's constant. a) Show that the total rate at which energy is radiated by a black body per unit area, over all frequencies, is $$ W=\frac{k_{B}^{4} T^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x . $$ b) Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is. c) Even before Planck gave his theory of thermal radiation around the turn of the 20 th century, it was known that the total energy \(W\) given off by a black body per unit area per second followed Stefan's law: \(W=\sigma T^{4}\), where \(\sigma\) is the StefanBoltzmann constant. Use your value for the integral above to compute a value for the Stefan-Boltzmann constant (in SI units) to three significant figures. Check your result against the known value, which you can find in books or on-line. You should get good agreement.

Heat capacity of a solid Debye's theory of solids gives the heat capacity of a solid at temperature \(T\) to be $$ C_{V}=9 V \rho k_{B}\left(\frac{T}{\theta_{D}}\right)^{3} \int_{0}^{\otimes_{D} / T} \frac{x^{4} \mathrm{e}^{x}}{\left(e^{x}-1\right)^{2}} \mathrm{~d} x $$ where \(V\) is the volume of the solid, \(\rho\) is the number density of atoms, \(k_{B}\) is Boltzmann's constant, and \(\theta_{D}\) is the so-called Debye temperature, a property of solids that depends on their density and speed of sound. a) Write a Python function \(\mathrm{cv}(\mathrm{T})\) that calculates \(C_{V}\) for a given value of the temperature, for a sample consisting of 1000 cubic centimeters of solid aluminum, which has a number density of \(\rho=6.022 \times 10^{28} \mathrm{~m}^{-3}\) and a Debye temperature of \(\theta_{D}=428 \mathrm{~K}\). Use Gaussian quadrature to evaluate the integral, with \(N=50\) sample points. b) Use your function to make a graph of the heat capacity as a function of temperature from \(T=5 \mathrm{~K}\) to \(T=500 \mathrm{~K}\).

5.17 The gamma function: A commonly occurring function in physics calculations is the gamma function \(\Gamma(a)\), which is defined by the integral $$ \Gamma(a)=\int_{0}^{\infty} x^{n-1} \mathrm{e}^{-x} \mathrm{~d} x . $$ There is no closed-form expression for the gamma function, but one can calculate its value for given \(a\) by performing the integral above numerically. You have to be careful how you do it, however, if you wish to get an accurate answer. a) Write a program to make a graph of the value of the integrand \(x^{n-1} \mathrm{e}^{-x}\) as a function of \(x\) from \(x=0\) to \(x=5\), with three separate curves for \(a=2,3\), and 4 , all on the same axes. You should find that the integrand starts at zero, rises to a maximum, and then decays again for each curve. b) Show analytically that the maximum falls at \(x=a-1\). c) Most of the area under the integrand falls near the maximum, so to get an accurate value of the gamma function we need to do a good job of this part of the integral. We can change the integral from 0 to \(\infty\) to one over a finite range from 0 to 1 using the change of variables in Eq. (5.67), but this tends to squash the peak towards the edge of the \([0,1]\) range and does a poor job of evaluating the integral accurately. We can do a better job by making a different change of variables that puts the peak in the middle of the integration range, around \(\frac{1}{2}\). We will use the change of variables given in Eq. (5.69), which we repeat here for convenience: $$ z=\frac{x}{c+x} . $$ For what value of \(x\) does this change of variables give \(z=\frac{1}{2}\) ? Hence what is the appropriate choice of the parameter \(c\) that puts the peak of the integrand for the gamma function at \(z=\frac{1}{2}\) ? d) Before we can calculate the gamma function, there is another detail we need to attend to. The integrand \(x^{n-1} \mathrm{e}^{-x}\) can be difficult to evaluate because the factor \(x^{2-1}\) can become very large and the factor \(\mathrm{e}^{-x}\) very small, causing numerical overflow or underflow, or both, for some values of \(x\). Write \(x^{n-1}=\mathrm{e}^{(a-1) \ln x}\) to derive an alternative expression for the integrand that does not suffer from these problems (or at least not so much). Explain why your new expression is better than the old one. e) Now, using the change of variables above and the value of \(c\) you have chosen, write a user-defined function gamma (a) to calculate the gamma function for arbitrary argument \(a\). Use whatever integration method you feel is appropriate. Test your function by using it to calculate and print the value of \(\Gamma\left(\frac{3}{2}\right)\), which is known to be equal to \(\frac{1}{2} \sqrt{\pi} \simeq 0.886\). f) For integer values of \(a\) it can be shown that \(\Gamma(a)\) is equal to the factorial of \(a-\) 1. Use your Python function to calculate \(\Gamma(3), \Gamma(6)\), and \(\Gamma(10)\). You should get answers closely equal to \(2 !=2,5 !=120\), and \(9 !=362880\).

Rearranging Eq. (5.19) into a slightly more conventional form, we have: $$ \int_{a}^{b} f(x) \mathrm{d} x=h\left[\frac{1}{2} f(a)+\frac{1}{2} f(b)+\sum_{k=1}^{N-1} f(a+k h)\right]+\frac{1}{12} h^{2}\left[f^{\prime}(a)-f^{\prime}(b)\right]+\mathrm{O}\left(h^{4}\right) . $$ This result gives a value for the integral on the left which has an error of order \(h^{4}-a\) factor of \(h^{2}\) better than the error on the trapezoidal rule and as good as Simpson's rule. We can use this formula as a new rule for evaluating integrals, distinct from any of the others we have seen in this chapter. We might call it the "Euler-Maclaurin rule." a) Write a program to calculate the value of the integral \(\int_{0}^{2}\left(x^{4}-2 x+1\right) \mathrm{d} x\) using this formula. (This is the same integral that we studied in Example 5.1, whose true value is \(4.4\).) The order- \(h\) term in the formula is just the ordinary trapezoidal rule; the \(h^{2}\) term involves the derivatives \(f^{\prime}(a)\) and \(f^{\prime}(b)\), which you should evaluate using central differences, centered on \(a\) and \(b\) respectively. Note that the size of the interval you use for calculating the central differences does not have to equal the value of \(h\) used in the trapezoidal rule part of the calculation. An interval of about \(10^{-5}\) gives good values for the central differences. Use your program to evaluate the integral with \(N=10\) slices and compare the accuracy of the result with that obtained from the trapezoidal rule alone with the same number of slices. b) Good though it is, this integration method is not much used in practice. Suggest a reason why not.
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