Infinite energy states for an harmonic oscillator?

In summary, the conversation discusses the creation of an infinite-energy state for the harmonic oscillator using a normalizable wave function. The example given is a function that is square-integrable but converges to a non-zero constant for x approaching infinity. This is possible because the Hamiltonian of the harmonic oscillator is unbounded, meaning that the mentioned wave function does not fall within its domain. This concept is related to the fact that most physical observables correspond to unbounded operators.
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So, I've read conference proceedings and they appear to talk about counter-intuitive it was to create an infinite-energy state for the harmonic oscillator with a normalizable wave function (i.e. a linear combination of eigenstates). How exactly could those even exist in the first place?
 
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  • #2
Try ##\Psi(x) = \frac{1}{1+x^2}##. It's clearly square-integrable, but if you apply the harmonic oscilllator Hamiltonian, you get a function that converges to a non-zero constant for ##x\rightarrow\infty##. The reason for why there can be such states is that the Hamiltonian of the harmonic oscillator is unbounded. The ##\Psi## I mentioned doesn't lie in the domain of ##\hat H##. Most physical observables correspond to unbounded operators.
 

1. What is an harmonic oscillator?

An harmonic oscillator is a type of system or object that exhibits oscillatory motion or vibrations around an equilibrium point. It is characterized by a restoring force that is proportional to the displacement from the equilibrium point and is described by a mathematical model known as the harmonic oscillator equation.

2. What are energy states for an harmonic oscillator?

Energy states for an harmonic oscillator refer to the different levels or amounts of energy that the system can have. These energy states are quantized, meaning they can only have discrete values and are determined by the frequency of the oscillations. The higher the energy state, the higher the frequency and the greater the amplitude of the oscillations.

3. What is meant by "infinite" energy states for an harmonic oscillator?

When we say infinite energy states for an harmonic oscillator, we mean that there is no upper limit to the energy that the system can have. As mentioned before, the energy states are quantized, but there is no maximum value that the energy can reach. This is because the potential energy function for an harmonic oscillator is parabolic, which means it continues to increase as the displacement from the equilibrium point increases.

4. Can an harmonic oscillator have negative energy states?

Yes, an harmonic oscillator can have negative energy states. This is because the potential energy function for an harmonic oscillator is symmetric about the equilibrium point, meaning it has the same value for positive and negative displacements. Therefore, the system can have equal amounts of energy in both positive and negative directions.

5. How is the energy of an harmonic oscillator related to its frequency and amplitude?

The energy of an harmonic oscillator is directly proportional to its frequency and amplitude. This means that as the frequency or amplitude increases, the energy of the system also increases. This relationship is described by the equation E = (n + 1/2)hf, where E is the energy, n is the energy state, h is Planck's constant, and f is the frequency of the oscillations.

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