Quantum SHM with Finite Potential and Hermite Polynomials

In summary, a special case of quantum SHM involves a particle oscillating around an equilibrium point with a fixed frequency and amplitude, governed by the laws of quantum mechanics. It differs from classical SHM in that it takes into account the wave-like nature of particles and the uncertainty principle. It has various applications in fields such as nanotechnology and quantum computing, and the energy of a particle in quantum SHM is quantized due to the discrete values of frequency. However, this type of motion only occurs in microscopic systems and is negligible in macroscopic systems where classical mechanics is sufficient.
  • #1
Larry89
7
0
Let it be V=(1/2)m(w^2)(x^2) for -L<x<L and V=A finite for x elsewhere.

is it obvious to use the wavefunction for the SHO that has infinite limits for the (-L,L) region and the usual decay of tunneling for the parts outside x</L/, and then play with the boundary conditions to determine the constants?

thanks for any discussion.

PS: I am particularly interested in the case that the particle energy is less than A but any further ideas are more than welcomed.
 
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  • #2
You should use Hermite polynomials (H_n) with arbitrary non-integer n's
 

Related to Quantum SHM with Finite Potential and Hermite Polynomials

1. What is a special case of quantum SHM?

A special case of quantum SHM, or quantum simple harmonic motion, is a system in which a particle oscillates back and forth around an equilibrium point with a fixed frequency and amplitude. This type of motion is governed by the laws of quantum mechanics, which describe the behavior of particles at the atomic and subatomic level.

2. How does quantum SHM differ from classical SHM?

In classical SHM, the motion of a particle is described using classical mechanics, which follows Newton's laws of motion. In quantum SHM, the behavior of the particle is described using quantum mechanics, which takes into account the wave-like nature of particles and the uncertainty principle.

3. What are the applications of quantum SHM?

Quantum SHM has many applications in various fields, such as nanotechnology, quantum computing, and quantum information processing. It is also used in the study of atomic and molecular systems, and in understanding the behavior of particles in quantum systems.

4. How is the energy of a particle in quantum SHM quantized?

In quantum SHM, the energy of a particle is quantized, meaning it can only take on certain discrete values. This is because the particle's energy is directly related to its frequency of oscillation, and in quantum mechanics, the frequency can only take on certain values.

5. Can quantum SHM occur in macroscopic systems?

No, quantum SHM only occurs in microscopic systems, such as atoms and molecules. This is because at the macroscopic level, the effects of quantum mechanics are negligible and classical mechanics is sufficient to describe the behavior of particles.

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