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Chapter 5: Problem 13

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 \(\sqrt{\left\langle x^{2}\right\rangle}=2.3 .\)

Short Answer

Expert verified
Calculate Hermite polynomial, plot wavefunctions for given \(n\) values, use Gaussian quadrature to find root-mean-square position for \(n=5\), expect \( \sqrt{\left\langle x^{2} \right\rangle} \approx 2.3 \).

Step by step solution

01

- Define Hermite Polynomial Function

Create a user-defined function that calculates the Hermite polynomial. Use the relation \[ H_{n+1}(x)=2xH_{n}(x) - 2nH_{n-1}(x) \] with the initial conditions \( H_{0}(x)=1 \) and \( H_{1}(x)=2x \).
02

- Calculate and Plot Wavefunctions for Given n Values

Use the Hermite polynomial function to calculate the wavefunctions for \( n = 0, 1, 2, 3 \). For each \(n\), use the wavefunction formula \[ \psi_{n}(x)=\frac{1}{\sqrt{2^{n} n ! \sqrt{\pi}}} e^{-x^{2} / 2} H_{n}(x) \] Plot these wavefunctions over the range \( x = -4 \) to \( x = +4 \) on the same graph.
03

- Plot Wavefunction for n = 30

Using the same wavefunction formula, calculate and plot the wavefunction for \( n = 30 \) over the range \( x = -10 \) to \( x = +10 \). Ensure the program runs efficiently in a second or so.
04

- Define Gaussian Quadrature Method

Write a program that utilizes Gaussian quadrature with 100 points to compute integrals. This method helps in accurately computing the integral of \( \left\langle x^{2} \right\rangle\)
05

- Calculate Root-Mean-Square Position

Calculate the integral \[ \left\langle x^{2} \right\rangle = \int_{-\infty}^{\infty} x^{2}|\psi_{n}(x)|^{2} \, dx \] using the Gaussian quadrature method. Then, obtain the uncertainty as \[ \sqrt{\left\langle x^{2} \right\rangle} \]
06

- Compute Uncertainty for n = 5

Use the program from Step 5 to calculate the uncertainty for \( n = 5 \). According to the problem, you should expect a value around \( 2.3 \).

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Key Concepts

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

Hermite Polynomials
Hermite polynomials are a set of orthogonal polynomials used frequently in physics, particularly in quantum mechanics. They are solutions to the Hermite differential equation, which appears in the context of the quantum harmonic oscillator. These polynomials are given by the recurrence relation: \[ H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x) \] Where the first two Hermite polynomials are \(H_0(x) = 1\) and \(H_1(x) = 2x\). This simple recurrence relation shows how each polynomial builds upon the previous ones.
Hermite polynomials play an essential role in shaping the quantum harmonic oscillator's wavefunctions. Without them, the calculations and representations of these wavefunctions would be far more complicated.
Gaussian Quadrature
Gaussian Quadrature is an efficient numerical method for evaluating integrals, particularly useful when integrating functions over a specific range. Instead of summing up the function values at equally spaced points (like in the midpoint or trapezoidal rules), Gaussian quadrature uses specially chosen points and weights within the integration range. This method provides more accurate results with fewer points. In this context, Gaussian Quadrature is applied to compute the integral for the root-mean-square position \[ \left\langle x^2 \right\rangle = \int_{-\infty}^{\infty} x^2 | \psi_n(x) |^2 \, dx \]. Using 100 points in our calculations boosts the precision and ensures that we get an accurate approximation of the integral. This accuracy is crucial when determining properties like quantum uncertainty in a harmonic oscillator.
Wavefunction Plotting
Plotting the wavefunctions of a quantum harmonic oscillator involves calculating \(\psi_n(x)\) for different energy levels \(n\). The wavefunction formula for the nth energy level of the harmonic oscillator is given by: \[\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} \mathrm{e}^{-x^2 / 2} H_n(x)\]. \Where \( H_n(x) \) are the Hermite polynomials. \When plotting, choose the range of \(x\) values you want to include. For lower energy levels, \(x = -4\) to \(x = 4\) is a typical range. For higher energy levels, you might need to extend this range. For instance, plotting \(\psi_{30}(x)\) may require \(x = -10\) to \(x = 10\). \Plotting these wavefunctions shows their shapes and helps visualize their probabilistic nature. It demonstrates that the wavefunctions have specific symmetries and nodes that increase with higher energy levels.
Quantum Uncertainty
Quantum Uncertainty is a fundamental concept in quantum mechanics, stemming from Heisenberg's uncertainty principle. It states that certain pairs of physical properties, like position and momentum, cannot be simultaneously known to arbitrary precision. For the harmonic oscillator, the uncertainty in position for the nth energy level can be quantified by the root-mean-square (RMS) position, represented as \(\sqrt{ \left\langle x^2 \right\rangle }\). That RMS value is derived from: \[ \left\langle x^2 \right\rangle = \int_{-\infty}^{\infty} x^2 \left| \psi_n(x) \right|^2 \, dx \]Using Gaussian quadrature to evaluate this integral accurately allows for a precise calculation of the uncertainty. For \(n = 5\r\), you should find \( \sqrt{ \left\langle x^2 \right\rangle } \) approximately equal to 2.3, showcasing the growth of uncertainty with higher energy levels.

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Most popular questions from this chapter

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.

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.

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 \(9 !=362880\).

Gravitational pull of a uniform sheet A uniform square sheet of metal is floating motionless in space: The sheet is \(10 \mathrm{~m}\) on a side and of negligible thickness, and it has a mass of 10 metric tonnes. a) Consider the gravitational force due to the plate felt by a point mass of \(1 \mathrm{~kg}\) a distance \(z\) from the center of the square, in the direction perpendicular to the sheet, as shown above. Show that the component of the force along the \(z\)-axis is $$ F_{z}=G \sigma z \iint_{-L / 2}^{L / 2} \frac{\mathrm{d} x \mathrm{~d} y}{\left(x^{2}+y^{2}+z_{2}^{2}\right)^{3 / 2}}, $$ where \(G=6.674 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant and \(\sigma\) is the mass per unit area of the sheet. b) Write a program to calculate and plot the force as a function of \(z\) from \(z=0\) to \(z=10 \mathrm{~m}\). For the double integral use (double) Gaussian quadrature, as in Eq. (5.82), with 100 sample points along each axis. c) You should see a smooth curve, except at very small values of \(z\), where the force should drop off suddenly to zero. This drop is not a real effect, but an artifact of the way we have done the calculation. Explain briefly where this artifact comes from and suggest a strategy to remove it, or at least to decrease its size. This calculation can thought of as a model for the gravitational pull of a galaxy. Most of the mass in a spiral galaxy (such as our own Milky Way) lies in a thin plane or disk passing through the galactic center, and the gravitational pull exerted by that plane on bodies outside the galaxy can be calculated by just the methods we have employed here.

Electric tield of a charge distribution: Suppose we have a distribution of charges and we want to calculate the resulting electric field. One way to do this is to first calculate the electric potential \(\phi\) and then take its gradient. For a point charge \(q\) at the origin, the electric potential at a distance \(r\) from the origin is \(\phi=q / 4 \pi \epsilon_{0} r\) and the electric field is \(\mathbf{E}=-\nabla \phi\). a) You have two charges, of \(\pm 1 \mathrm{C}, 10 \mathrm{~cm}\) apart. Calculate the resulting electric potential on a \(1 \mathrm{~m} \times 1 \mathrm{~m}\) square plane surrounding the charges and passing through them. Calculate the potential at \(1 \mathrm{~cm}\) spaced points in a grid and make a visualization on the screen of the potential using a density plot. b) Now calculate the partial derivatives of the potential with respect to \(x\) and \(y\) and hence find the electric field in the \(x y\) plane. Make a visualization of the field also. This is a little trickier than visualizing the potential, because the electric field has both magnitude and direction. One way to do it might be to make two density plots, one for the magnitude, and one for the direction, the latter using the "hsv"
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