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Chapter 2: Problem 11

An electron is confined to a one-dimensional box of length \(L .\) What should be the length of the box in order for its zero-point energy to be equal to its rest mass energy \(\left(m_{\mathrm{e}} c^{2}\right) ?\) Express the result in terms of the Compton wavelength, \(\lambda_{\mathrm{C}}=h / m_{\mathrm{e}} c\)

Short Answer

Expert verified
The length of the box for the zero-point energy to be equal to its rest mass energy is \(\frac{1}{\sqrt{2}}\) times the Compton wavelength of the electron, i.e. \(L = \dfrac{1}{\sqrt{2}} \lambda_C\).

Step by step solution

01

Write down Given Information

We know that \(E_0 = m_ec^2\) and \( \lambda_C=\frac{h}{m_ec}\).
02

Express Zero-Point Energy

The zero-point energy (ground state energy) for a particle in a box of length \(L\) is \(E_0 = \dfrac{\pi^2 \hbar^2}{2mL^2}\). Equate this to \(m_ec^2\) to get \(\dfrac{\pi^2 \hbar^2}{2mL^2} = m_ec^2\).
03

Solve for Length L

On rearranging the equation we get, \(L = \dfrac{\pi \hbar}{\sqrt{2}m_ec}\). Replace \(\hbar = \dfrac{h}{2\pi}\) and rearrange the terms to get \(L\).
04

Express in terms of Compton wavelength

We are asked for the result in terms of the Compton wavelength. Replace \(h/m_ec\) by \(\lambda_C\) in our derived expression to get \(L = \dfrac{1}{\sqrt{2}} \lambda_C\).

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Most popular questions from this chapter

Energy is required to compress the box when a particle is inside: this suggests that the particle exerts a force on the walls. (a) On the basis that when the length of the box changes by dL the energy changes by \(\mathrm{d} E=-F \mathrm{d} L,\) find an expression for the force. (b) At what length does \(F=1 \mathrm{N}\) when an electron is in the state \(n=1 ?\)

Confirm that the completeness relation, eqn 1.25, may be expressed in terms of wavefunctions as \\[ \sum_{n} \psi_{n}(r) \psi_{n}^{*}\left(r^{\prime}\right)=\delta\left(r-r^{\prime}\right) \\] where \(\delta\left(r-r^{\prime}\right)\) is the Dirac \(\delta\) -function described in Section 2.1

The oscillation of the atoms around their equilibrium positions in the molecule HI can be modelled as a harmonic oscillator of mass \(m \approx m_{\mathrm{H}}\) (the iodine atom is almost stationary \()\) and force constant \(k_{\mathrm{f}}=313.8 \mathrm{N} \mathrm{m}^{-1} .\) Evaluate the separation of the energy levels and predict the wavelength of the light needed to induce a transition between neighbouring levels.

For a particle in a box, the mean value and mean square value of the linear momentum are given by \(\int_{0}^{L} \psi^{*} p \psi \mathrm{d} x\) and \(\int_{0}^{L} \psi^{*} p^{2} \psi \mathrm{d} x,\) respectively. Evaluate these quantities. Form the root mean square deviation \(\Delta p=\left\\{\left\langle p^{2}\right\rangle-\langle p\rangle^{2}\right\\}^{1 / 2}\) and investigate the consistency of the outcome with the uncertainty principle. Hint. Use \(p=(\hbar / \mathrm{i}) \mathrm{d} / \mathrm{d} x .\) For \(\left\langle p^{2}\right\rangle\) notice that \(E=p^{2} / 2 m\) and we already know \(E\) for each \(n\). For the last part, form \(\Delta x \Delta p\) and show that \(\Delta x \Delta p \geq \frac{1}{2} \hbar,\) the precise form of the principle, for all \(n\) evaluate \(\Delta x \Delta p\) for \(n=1.\)

Evaluate the matrix elements (a) \(\langle v+1|x| v\rangle\) and (b) \(\left\langle v+2\left|x^{2}\right| v\right\rangle\) of a harmonic oscillator by using the relations given at the bottom of Table 2.1 for the Hermite polynomials.
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