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Chapter 35: Problem 26

Your roommate is taking Newtonian physics, while you've moved on to quantum mechanics. He claims that QM can't be right, because he didn't see any evidence of quantized energy levels in a mass-spring harmonic oscillator experiment. You reply by calculating the spacing between energy levels of this system, which consists of a \(1-\mathrm{g}\) mass on a spring with \(k=80 \mathrm{N} / \mathrm{m}\) What is that spacing, and how does this help your argument?

Short Answer

Expert verified
The spacing between energy levels is very small and thus could not have been noticed in a classical physics experiment. The experiment supports quantum theory which posits that energy is quantized.

Step by step solution

01

Identify the energy equation

For a quantum harmonic oscillator, the energy levels are given by the equation \(E_n = \hbarω(n+1/2)\) where \(E_n\) is the energy of the n-th level, \(\hbar\) is the reduced Planck's constant, ω is the natural frequency of the oscillator, and n is the principal quantum number.
02

Calculate the angular frequency

The natural frequency \(w\) of a spring-mass system is \(\sqrt{k/m}\), where \(k = 80 N/m\) is the spring constant and m = \(1g = 1 × 10^-3 kg\) is the mass. By substituting these values into the formula, the angular frequency ω can be found out.
03

Substituting the values into the energy equation

After calculating the angular frequency of the system, the values of ω and \(\hbar\) can be substituted into the energy equation \(E_n = \hbarω(n+1/2)\). This will provide the energy levels of the system.
04

Calculate the energy difference

To comment on the spacing between the energy levels of the system, the difference between consecutive energy levels, say for n=0 and n=1, needs to be calculated. This difference \(ΔE = E_1 - E_0\) will represent the 'spacing' between levels.
05

Final explanation

After calculating the energy difference or the 'spacing', this value can be used to counter the roommate's argument that he did not observe quantized energy levels in a mass-spring harmonic oscillator experiment. This calculation shows that the energy levels for a simple harmonic oscillator are, indeed, quantized and that 𝛥E is so small that it might not be noticeable in a classical physics experiment but it does exist in the quantum world.

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Most popular questions from this chapter

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