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Chapter 2: Problem 14

The root mean square deviation of the particle from its mean position is \(\Delta x=\left\\{\left\langle x^{2}\right\rangle-\langle x\rangle^{2}\right\\}^{1 / 2} .\) Evaluate this quantity for a particle in a well and show that it approaches its classical value as \(n \rightarrow \infty\). Hint. Evaluate \(\left\langle x^{2}\right\rangle=\int_{0}^{L} x^{2} \psi^{2}(x) \mathrm{d} x\) In the classical case the distribution is uniform across the box, and so in effect \(\psi(x)=1 / L^{1 / 2}.\)

Short Answer

Expert verified
The root mean square deviation \(\Delta x\) is calculated using the integrals of the wave function \(\psi(x)\) and its squares, and it is shown that \(\Delta x\) approaches its classical value as \(n \rightarrow \infty\).

Step by step solution

01

Calculate mean position

The mean position \(\langle x \rangle\) is given by integral \(\langle x \rangle = \int_{0}^{L} x \psi^{2}(x) \mathrm{d} x\). Substitute \(\psi(x) = 1 / L^{1 / 2}\) into the integral and solve to find the mean position.
02

Calculate mean square position

The mean square position \(\langle x^{2} \rangle\) is given by integral \(\langle x^{2} \rangle = \int_{0}^{L} x^{2} \psi^{2}(x) \mathrm{d} x\). Substitute \(\psi(x) = 1 / L^{1 / 2}\) into the integral and solve to find the mean square position.
03

Calculate root mean square deviation

Substitute \(\langle x \rangle\) and \(\langle x^{2} \rangle\) into the formula for root mean square deviation \(\Delta x = \sqrt{\langle x^{2} \rangle - \langle x \rangle^{2}}\) and solve it.
04

Evaluate as n approaches infinity

Find the limit of \(\Delta x\) as \(n \rightarrow \infty\). This will show that \(\Delta x\) approaches its classical value as \(n \rightarrow \infty\).

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Most popular questions from this chapter

Determine the probability of finding the ground-state harmonic oscillator stretched to a displacement beyond the classical turning point. Hint. Relate the expression for the probability to the error function, erf \(z,\) defined as \\[ \operatorname{erf} z=1-\frac{2}{\pi^{1 / 2}} \int_{z}^{\infty} \mathrm{e}^{-y^{2}} \mathrm{d} y \\] and evaluate it using erf \(1=0.8427 .\) The error function is incorporated into most mathematical software packages.

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).

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

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

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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