Driven quantum mechanical harmonic oscillator

Your Name]In summary, the speaker has calculated the quantum mechanical harmonic oscillator with a driving dipole force and used the Floquet-Formalism to find the mean expectation value of the Hamiltonian over a full period. They have obtained a result with a singularity at \omega_0 = \omega and are questioning the possibility of a negative energy expectation value around this point. However, it is not physically possible and they may need to review their calculations for potential errors.
  • #1
tommy01
40
0
Hi.

I just calculated the quantum mechanical harmonic oscillator with a driving dipole force [tex]V(x,t) = - x S \sin(\omega t + \phi)[/tex]

I used the Floquet-Formalism. Then I calculated the mean expectation value, in a Floquet-State, of the Hamiltonian over a full Period T indirectly by using the Hellmann-Feynman Theorem.

What I've got reads:

[tex]
\overline{H_\alpha} = \hbar \omega_0 \left( \alpha + \frac{1}{2} \right) - \frac{S^2}{4m} \frac{(\omega_0^2 + \omega^2)}{(\omega_0^2-\omega^2)^2}
[/tex]

So the first Term is the Energyeigenvalue of the "stationary" harmonic oscillator and the other term accounts for the perturbation mentioned above, which is always positive and therefore lowers the energy with a singularity at [tex]\omega_0 = \omega[/tex].

Now to my question. Is it physically possible to get a negative energy expectation value around [tex]\omega_0[/tex]?

Thanks and greetings.
 
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  • #2


Hello,

Thank you for sharing your calculations and results. It is always exciting to see new developments in quantum mechanics.

To answer your question, it is not physically possible to have a negative energy expectation value around \omega_0. This would violate the fundamental principles of quantum mechanics, such as the uncertainty principle and the conservation of energy. The energy of a system must always be positive or zero.

It is possible that there may be errors in your calculations or assumptions that led to this result. I suggest double-checking your calculations and consulting with other experts in the field to see if they can offer any insights.

Thank you for your contribution to the scientific community. Keep up the good work!
 
  • #3


I appreciate your work in calculating the quantum mechanical harmonic oscillator with a driving dipole force using the Floquet-Formalism and the Hellmann-Feynman Theorem. Your results show that the mean expectation value of the Hamiltonian over a full Period T includes both the energy eigenvalue of the "stationary" harmonic oscillator and the perturbation caused by the driving dipole force. This perturbation can lower the energy with a singularity at the resonance frequency, \omega_0 = \omega.

To answer your question, it is not physically possible to have a negative energy expectation value in a quantum mechanical system. Energy is a fundamental quantity that cannot be negative. However, your results show that the perturbation can decrease the energy from its original value, which is an important observation in understanding the behavior of the driven quantum mechanical harmonic oscillator. Further studies and experiments can help explore the implications of this phenomenon and its potential applications in quantum mechanics. Keep up the good work.
 

1. What is a driven quantum mechanical harmonic oscillator?

A driven quantum mechanical harmonic oscillator is a physical system that follows the principles of quantum mechanics and exhibits harmonic motion under the influence of an external driving force. It is a model commonly used in physics to understand and study the behavior of systems such as atoms, molecules, and solid-state materials.

2. How does a driven quantum mechanical harmonic oscillator differ from a classical oscillator?

A classical oscillator follows the laws of classical mechanics, which describe the motions of macroscopic objects, while a driven quantum mechanical harmonic oscillator follows the laws of quantum mechanics, which govern the behavior of microscopic particles. In the classical case, the motion of the oscillator is continuous, while in the quantum case, the motion is quantized and can only occur in discrete energy levels.

3. What is the role of the driving force in a driven quantum mechanical harmonic oscillator?

The driving force provides energy to the system, causing it to oscillate at a specific frequency. The amplitude and phase of the driving force can also affect the behavior of the oscillator, such as changing the frequency or causing the system to exhibit chaotic behavior.

4. How is the energy of a driven quantum mechanical harmonic oscillator calculated?

The energy of a driven quantum mechanical harmonic oscillator is calculated using the Schrödinger equation, which describes the time evolution of the quantum state of a system. The energy of the oscillator is given by the sum of its kinetic energy and potential energy, which depend on the mass of the particle, the frequency of oscillation, and the amplitude of the driving force.

5. What are some real-world applications of a driven quantum mechanical harmonic oscillator?

Driven quantum mechanical harmonic oscillators have many practical applications, such as in the construction of lasers, atomic clocks, and other precision instruments. They are also used to study the behavior of condensed matter systems, such as superconductors and semiconductors, and in the development of quantum computing technologies.

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