Harmonic Oscillator: Energy Levels Explained

In summary, the energy levels of the harmonic oscillator are given by the equation En = (N+1/2)hf and this is derived by solving Schrödinger's equation for a potential of the harmonic oscillator form. This derivation is not found in the textbook and can be done using the Power Series Technique or by using raising and lowering operators. Both methods are analytical techniques, with the latter being easier.
  • #1
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why is the energy levels of the harmonic oscillator En=(N+1/2)hf?
 
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  • #2
Is it derived in your textbook? It's not a particulary short derivation regardless of the method you use do I'd rather not type it out here. As for why the energy levels are like that because that's what you get when you solve Schrödinger's equation for a potential of the harmonic oscillator form.
 
  • #3
no it's not derived in my textbook... thank you~
 
  • #4
That expression for E arises when solving the Schroedinger Equation analytically using the Power Series Technique (I don't know of any other analytical technique - if anyone else does let me know).
The solution is not hard but it is long.

An easier way to derive E is to use raising and lowering operators.
 
  • #5
thank you very much! :)
 

FAQ: Harmonic Oscillator: Energy Levels Explained

1. What is a harmonic oscillator?

A harmonic oscillator is a system that exhibits periodic motion, where the restoring force is directly proportional to the displacement from equilibrium. Examples of harmonic oscillators include a mass on a spring and a pendulum.

2. How are energy levels determined in a harmonic oscillator?

The energy levels in a harmonic oscillator are determined by the quantum mechanical principles of wave-particle duality and the Schrödinger equation. The energy levels are quantized, meaning they can only take on certain discrete values.

3. What is the relationship between energy levels and frequency in a harmonic oscillator?

The energy levels in a harmonic oscillator are directly proportional to the frequency of the oscillation. This means that as the frequency increases, the energy levels also increase.

4. Can the energy levels in a harmonic oscillator have negative values?

No, the energy levels in a harmonic oscillator cannot have negative values. The lowest energy level, also known as the ground state, has an energy of zero, and all other energy levels are positive multiples of this ground state energy.

5. How does temperature affect the energy levels in a harmonic oscillator?

As temperature increases, the average energy of the particles in the harmonic oscillator also increases. This results in a broadening of the energy levels, meaning there is a greater range of energy values that the particles can occupy.

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