Question

A mass-and-spring harmonic oscillator used for classroom demonstrations has the angular frequency \(\omega_{0}=4.45 \mathrm{~s}^{-1}\). If this oscillator has a total (kinetic plus potential) energy \(E=1.00 \mathrm{~J},\) what is its corresponding quantum number, \(n ?\)

Short answer

Answer: The quantum number of the harmonic oscillator is approximately \(n = 2 × 10^{34}\).

Step-by-step solution

Identify and Write down the Known Values

We are given the angular frequency and the total energy of the oscillator: \(\omega_{0} = 4.45 \, \mathrm{s}^{-1}\) \(E = 1.00 \, \mathrm{J}\) The value of the reduced Planck constant is well-known and is approximately: \(\hbar = 1.0545718 × 10^{-34} \, \mathrm{J \cdot s}\)

Rearrange the Energy Level Formula for the Quantum Number

We need to solve for \(n\) in the energy level formula: \(E_n = \hbar \omega_{0} (n + \frac{1}{2})\) Rearranging the equation for \(n\): \(n = \frac{E_n}{\hbar \omega_{0}} - \frac{1}{2}\)

Plug in Known Values and Solve for the Quantum Number

Now, plug in the given values for \(E_n, \hbar, \omega_{0}\), and solve for \(n\): \(n = \frac{1.00 \, \mathrm{J}}{(1.0545718 × 10^{-34} \, \mathrm{J \cdot s})(4.45 \, \mathrm{s}^{-1})} - \frac{1}{2}\) After evaluating the expression, we obtain: \(n \approx 2.16 × 10^{34}\) Since the quantum number must be an integer, we round it to the nearest whole number: \(n \approx 2 × 10^{34}\) The corresponding quantum number for the given mass-and-spring harmonic oscillator is approximately \(n = 2 × 10^{34}\).

Key concepts

Angular Frequency

Angular frequency, denoted by the symbol \(\omega\), is a fundamental parameter in the study of oscillatory systems, such as springs, pendulums, and circuits. Imagine a child swinging back and forth on a swing—the rate at which they complete their to and fro motion can be likened to the angular frequency of a harmonic oscillator.

In the realm of quantum mechanics, angular frequency is crucial in understanding the behavior of quantum systems, such as the quantum harmonic oscillator. The quantum harmonic oscillator is an analogy to the classical system where a mass is attached to a spring, but in the quantum context, it describes particles as waves with quantized energies. The angular frequency \(\omega\) in this scenario is related not just to the time period of the oscillations but also to the energy levels that the quantum system can possess. Specifically, for the quantum harmonic oscillator, the relationship between energy and angular frequency is given by the formula \(E_n = \hbar \omega (n + \frac{1}{2})\), where \(E_n\) is the energy, \(\hbar\) is the reduced Planck's constant, and \(n\) is the quantum number.

Total Energy of an Oscillator

The total energy of an oscillator combines both kinetic and potential energies. In classical mechanics, this total energy remains constant (if we assume no external forces act like friction) as the oscillator moves back and forth. Similarly, in quantum mechanics, the total energy of a quantum harmonic oscillator remains quantized and constant for a given state.

The total energy of a quantum harmonic oscillator is quantified using quantum numbers. The formula \(E_n = \hbar \omega_{0} (n + \frac{1}{2})\) tells us the energy of the nth quantum state, where \(n\) is the quantum number representing which quantized state the system occupies, \(\hbar\) represents the reduced Planck's constant, and \(\omega_{0}\) is the natural angular frequency of the oscillator. Importantly, this energy is quantized, meaning that unlike in a classical system where energy values can be continuous, a quantum oscillator can only have certain discrete energy levels.

Quantum Number

Quantum number is a term that often mystifies students, but it is a cornerstone of quantum physics and crucial to understanding quantum systems. Quantum numbers are similar to the distinct, unchangeable rungs on a ladder—each quantum number corresponds to a unique energy level the system can occupy.

For the quantum harmonic oscillator, the principal quantum number \(n\) labels the energy states. Starting from \(n = 0\), which represents the ground state, each increment in \(n\) corresponds to a higher energy level or ‘excited state’. The levels are evenly spaced, and the energy difference between adjacent levels is given by \(\hbar \omega_{0}\), with the smallest possible energy change occurring when the system moves between adjacent states. This quantization of energy is a fundamental departure from classical mechanics, where a mass on a spring can have a continuous range of energies.