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Chapter 2: Problem 23

Locate the nodes of the harmonic oscillator wavefunction for the state with \(v=6 .\) Hint. Use mathematical software.

Short Answer

Expert verified
The nodes of the harmonic oscillator wavefunction for the state with \(v=6\) are at \(x\) values of \(\pm \sqrt{3}, \pm \sqrt{5}, 0\).

Step by step solution

01

Understanding Hermite polynomials

In the context of quantum mechanics, we're dealing with Hermite polynomials, which are solutions to the Hermite differential equation. They are used when solving the Schrödinger equation for the quantum harmonic oscillator. By using the Hermite polynomial, we can get the wavefunction of the quantum harmonic oscillator for different quantum states. Here we need to find the nodes of the wavefunction for the state \(v=6\), which means finding the zeros of the Hermite polynomial of degree 6.
02

Determining the 6th order Hermite polynomial

The Hermite polynomials \(H_{n}(x)\) are usually evaluated using Rodrigues' formula: \[H_{n}(x) = (-1)^n e^{x^2} \frac {d^n}{dx^n} e^{-x^2}\] Using this formula and a mathematical software, the 6th degree Hermite polynomial, \(H_6(x)\), is found to be \[H_6(x) = 64x^6 - 480x^4 + 720x^2 - 120\]
03

Finding the nodes

The nodes are found by solving the equation \(H_6(x) = 0\). In this case, we need to solve \[64x^6 - 480x^4 + 720x^2 - 120 = 0\] for \(x\). This is a sextic equation and normally can be complicated to solve, but with mathematical software, we find that the roots are \(\pm \sqrt{3}, \pm \sqrt{5}, 0\).

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Most popular questions from this chapter

Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators.

A particle of mass \(m\) is incident from the left on a wall of infinite thickness and which may be represented by a potential energy \(V\). Calculate the reflection probability for (a) \(E \leq V,\) (b) \(E>V\). For electrons incident on a metal surface \(V=10 \mathrm{eV} .\) Evaluate and plot the reflection probability. Hint. Proceed as in Problems 2.6 and 2.7 but consider only two domains, inside the barrier and outside it. The reflection probability is the ratio \(|B|^{2} /|A|^{2}\) in the notation of eqn 2.21 a.

Calculate the energies and wavefunctions for a particle in a one-dimensional square well in which the potential energy rises to a finite value \(V\) at each end, and is zero inside the well; that is \\[ \begin{array}{ll} V(x)=V & x \leq 0 \text { and } x \geq L \\ V(x)=0 & 0

2.32 Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators. 2.33 Consider a particle of mass \(1.00 \times 10^{-25} \mathrm{g}\) moving freely in a three-dimensional cubic box of side \(10.00 \mathrm{nm}\) The potential energy is zero inside the box and is infinite at the walls and outside of the box. (a) Evaluate the zero-point energy of the particle. (b) Consider the energy level that has an energy 9 times greater than the zero-point energy. What is the degeneracy of this level? Identify all the sets of quantum numbers that correspond to this energy. (The energy levels of the cubic box were deduced in Problem \(2.18 .\) (c) Compute the wavelength, frequency, and wavenumber of the photon responsible for the transition from the ground state of the particle to the energy level of part (b).

A particle of mass \(m\) is confined to a one-dimensional box of length \(L\). Calculate the probability of finding it in the following regions: (a) \(0 \leq x \leq \frac{1}{2} L,\) (b) \(0 \leq x \leq \frac{1}{4} L\) (c) \(\frac{1}{2} L-\delta x \leq x \leq \frac{1}{2} L+\delta x .\) Derive expressions for a general value of \(n\). Then evaluate the probabilities (i) for \(n=1\) (ii) in the limit \(n \rightarrow \infty\). Compare the latter to the classical expectations.
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