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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

Period of an anharmonic oscillator The simple harmonic oscillator crops up in many places. Its behavior can be studied readily using analytic methods and it has the important property that its period of oscillation is a constant, independent of its amplitude, making it useful, for instance, for keeping time in watches and clocks. Frequently in physics, however, we also come across anharmonic oscillators, whose period varies with amplitude and whose behavior cannot usually be calculated analytically. A general classical oscillator can be thought of as a particle in a concave potential well. When disturbed, the particle will rock back and forth in the well: The harmonic oscillator corresponds to a quadratic potential \(V(x) \propto x^{2}\). Any other form gives an anharmonic oscillator. (Thus there are many different kinds of anharmonic oscillator, depending on the exact form of the potential.) One way to calculate the motion of an oscillator is to write down the equation for the conservation of energy in the system. If the particle has mass \(m\) and position \(x\), then the total energy is equal to the sum of the kinetic and potential energies thus: $$ E=\frac{1}{2} m\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)^{2}+V(x) $$ Since the energy must be constant over time, this equation is effectively a (nonlinear) differential equation linking \(x\) and \(t\). Let us assume that the potential \(V(x)\) is symmetric about \(x=0\) and let us set our anharmonic oscillator going with amplitude \(a\). That is, at \(t=0\) we release it from rest at position \(x=a\) and it swings back towards the origin. Then at \(t=0\) we have \(\mathrm{d} x / \mathrm{d} t=0\) and the equation above reads \(E=V(a)\), which gives us the total energy of the particle in terms of the amplitude. a) When the particle reaches the origin for the first time, it has gone through one quarter of a period of the oscillator. By rearranging the equation above for \(\mathrm{d} x / \mathrm{d} t\) and then integrating with respect to \(t\) from 0 to \(\frac{1}{4} T\), show that the period \(T\) is given by $$ T=\sqrt{8 m} \int_{0}^{a} \frac{\mathrm{d} x}{\sqrt{V(a)-V(x)}} $$ b) Suppose the potential is \(V(x)=x^{4}\) and the mass of the particle is \(m=1 .\) Write a Python function that calculates the period of the oscillator for given amplitude a using Gaussian quadrature with \(N=20\) points, then use your function to make a graph of the period for amplitudes ranging from \(a=0\) to \(a=2\). c) You should find that the oscillator gets faster as the amplitude increases, even though the particle has further to travel for larger amplitude. And you should find that the period diverges as the amplitude goes to zero. How do you explain these results?

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 \(T=500 \mathrm{~K}\).

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.

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"

The diffraction limit of a telescope Our ability to resolve detail in astronomical observation is limited by the diffraction of light in our telescopes. Light from stars can be treated effectively as coming from a point source at infinity. When such light, with wavelength \(\lambda\), passes through the circular aperture of a telescope (which we'll assume to have unit radius) and is focused by the telescope in the focal plane, it produces not a single dot, but a circular diffraction pattern consisting of central spot surrounded by a series of concentric rings. The intensity of the light in this diffraction pattern is given by $$ I(r)=\left(\frac{J_{1}(k r)}{k r}\right)^{2} $$ where \(r\) is the distance in the focal plane from the center of the diffraction pattern, \(k=2 \pi / \lambda\), and \(J_{1}(x)\) is a Bessel function. The Bessel functions \(J_{m}(x)\) are given by $$ J_{m}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (m \theta-x \sin \theta) \mathrm{d} \theta, $$ where \(m\) is a nonnegative integer and \(x \geq 0\). a) Write a Python function \(J(m, x)\) that calculates the value of \(J_{m}(x)\) using Simpson's rule with \(N=1000\) points. Use your function in a program to make a plot, on a single graph, of the Bessel functions \(J_{0}, J_{1}\) and \(J_{2}\) as a function of \(x\) from \(x=0\) to \(x=20\). b) Make a second program that makes a density plot of the intensity of the circular diffraction pattern of a point light source with \(\lambda=500 \mathrm{~nm}\), in a square region of the focal plane, using the formula given above. Your picture should cover values of \(r\) from zero up to about \(1 \mu \mathrm{m}\).
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