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Chapter 37: Problem 43

Calculate the ground state energy for an electron confined to a cube with sides equal to twice the Bohr radius \((R=0.0529 \mathrm{nm}) .\) Determine the spring constant that would give this same ground state energy for a harmonic oscillator.

Short Answer

Expert verified
Answer: The spring constant for a harmonic oscillator that would give the same ground state energy as an electron confined to the given cube is approximately \(1.158 \times 10^{-1} \ N/m\).

Step by step solution

01

Calculate the dimensions of the cube

We are given that each side of the cube is twice the Bohr radius, so we need to find the length of the side. The Bohr radius, \(R\), is given as \(0.0529 \ nm\) (nanometers). \(side\_length = 2 * R = 2 * 0.0529 \ nm = 0.1058 \ nm\)
02

Calculate the ground state energy of an electron in a cube using particle-in-a-box model

Using the particle in a box model, the energy levels of an electron can be calculated by the following formula: \(E_n = \frac{n^2 * h^2}{8 * m * L^2}\) where \(n\) is the principal quantum number (for the ground state, \(n = 1\)), \(h\) is Planck's constant (\(6.626 \times 10^{-34} \ Js\)), \(m\) is the mass of the electron (\(9.109 \times 10^{-31} \ kg\)), and \(L\) is the side length of the cube (which we calculated in Step 1). \(E_1 = \frac{1^2 * (6.626 \times 10^{-34})^2}{8 * (9.109 \times 10^{-31}) * (0.1058 \times 10^{-9})^2} = 1.483 \times 10^{-19} \ J\) We now have the ground state energy for the electron confined within the cube.
03

Calculate the ground state energy for a quantum harmonic oscillator

The ground state energy for a quantum harmonic oscillator is given by the following equation: \(E_0 = \frac{1}{2} h \omega\) where \(\omega\) is the angular frequency of the oscillator and is related to the spring constant, \(k\), via the equation: \(\omega = \sqrt{\frac{k}{m}}\) We are given the ground state energy for the confined electron already, so our task is to determine the spring constant that would give us the same ground state energy as the confined electron.
04

Calculate the spring constant for the harmonic oscillator

Knowing that the ground state energies for both systems must be equal, we can set to equal: \(\frac{1}{2} h \omega = E_1\) Now, we can use the relation between \(\omega\) and the spring constant to rewrite the equation in terms of \(k\): \(\frac{1}{2} h \sqrt{\frac{k}{m}} = E_1\) Next, we can solve for the spring constant, \(k\): \(k = \frac{4 * m * E_1^2}{h^2}\) Plugging in the values, we get: \(k = \frac{4 * (9.109 \times 10^{-31} \ kg) * (1.483 \times 10^{-19} \ J)^2}{(6.626 \times 10^{-34} \ Js)^2} = 1.158 \times 10^{-1} \ N/m\) Hence, the required spring constant is approximately \(1.158 \times 10^{-1} \ N/m\).

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Most popular questions from this chapter

Sketch the two lowest energy wave functions for an electron in an infinite potential well that is \(20 \mathrm{nm}\) wide and a finite potential well that is \(1 \mathrm{eV}\) deep and is also \(20 \mathrm{nm}\) wide. Using your sketches, can you determine whether the energy levels in the finite potential well will be lower, the same, or higher than in the infinite potential well?

Suppose a quantum particle is in a stationary state (energy eigenstate) with a wave function \(\psi(x, t) .\) The calculation of \(\langle x\rangle,\) the expectation value of the particle's position, is shown in the text. Calculate \(d\langle x\rangle / d t(\operatorname{not}\langle d x / d t\rangle)\).

37.2 In an infinite square well, for which of the following states will the particle never be found in the exact center of the well? a) the ground state b) the first excited state c) the second excited state d) any of the above e) none of the above

An experimental measurement of the energy levels of a hydrogen molecule, \(\mathrm{H}_{2}\), shows that the energy levels are evenly spaced and separated by about \(9 \cdot 10^{-20} \mathrm{~J}\). A reasonable model of one of the hydrogen atoms would then seem to be that of a hydrogen atom in a simple harmonic oscillator potential. Assuming that the hydrogen atom is attached by a spring with a spring constant \(k\) to the molecule, what is the spring constant \(k\) ?

The wavelength of an electron in an infinite potential is \(\alpha / 2,\) where \(\alpha\) is the width of the infinite potential well. Which state is the electron in? a) \(n=3\) b) \(n=6\) c) \(n=4\) d) \(n=2\)
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