A harmonic oscillator absorbs a photon of wavelength 8.65 ×10^-6 m when it undergoes a transitio

step 1 Okay, so in this problem, we are asked to find the ground state energy of our oscillator, given

A harmonic oscillator absorbs a photon of wavelength 8.65...

A harmonic oscillator absorbs a photon of wavelength $8.65 \times 10^{-6} \mathrm{m}$ when it undergoes a transition from the ground state to the first excited state. What is the ground-state energy, in electron volts, of the oscillator?

Key Concepts

Energy Level Transitions

In quantum systems like the harmonic oscillator, transitions between energy levels occur through the absorption or emission of photons. The energy of the photon absorbed or emitted must equal the difference in energy between the initial and final states. For the oscillator, transitioning from the ground state to the first excited state absorbs a photon whose energy equals the spacing between these levels.

Unit Conversion

Energy calculations in quantum mechanics are often performed in Joules, but it is sometimes necessary to convert these values to electron volts (eV) for convenience. This conversion uses the relation 1 eV = 1.602 × 10?¹? Joules, facilitating comparisons and practical applications in atomic and molecular physics.

Quantum Harmonic Oscillator

This concept refers to a system whose potential energy is quadratic in displacement, leading to quantized energy levels. In a quantum harmonic oscillator, the energy of each state is given by E? = (n + 1/2)??, where n is an integer quantum number, ? is the reduced Planck's constant, and ? is the angular frequency of the oscillator.

Photon Energy Calculation

Photon energy is directly related to its wavelength (or frequency) by the relationship E = hc/?, where h is Planck's constant and c is the speed of light. This equation is used to determine the energy of a photon involved in transitions between energy levels.

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