Quantum Spectrum of a Hamiltonian with Linear Potential

In summary, the conversation discusses the spectrum of a Hamiltonian with a linear potential, where the variable x and parameter z are involved. It is mentioned that the eigenvalues are numerical in a 1-D infinite potential well with Airy function solutions, as seen in a recent simulation. However, finding the numerical solution is more challenging if the walls are finite. A possible derivation can be found in David Miller's Quantum book.
  • #1
Andre' Quanta
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Suppose to have a hamiltonian with a linear potenzial like z x, where x is the variable and z a parameter. Which is the spectrum of the Hamiltonian of this sistem?
 
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  • #2
Depends... if the particle is to be bounded in a 1-D infinite potential well then a linearly-varying potential gives rise to Airy function solutions and hence the eigenvalues are inherently numerical (I remember this distinctly from a recent Mathematica simulation of a stationary electric field perturbing an electron confined to a 1-D box). You can find a derivation in David Miller's Quantum book. If the walls are finite then the numerical solution is more difficult to find.
 

1. What is quantization in a linear potential?

Quantization in a linear potential refers to the process of breaking down the energy levels of a particle in a linear potential into discrete values or quanta. This is a fundamental concept in quantum mechanics and is used to describe the behavior of particles in a linear potential.

2. How does quantization in a linear potential affect the behavior of particles?

Quantization in a linear potential affects the behavior of particles by limiting the energy values that the particle can have. This results in the particle exhibiting wave-like behavior, such as having discrete energy levels and exhibiting interference patterns.

3. What is the significance of quantization in a linear potential?

The significance of quantization in a linear potential lies in its ability to accurately describe the behavior of particles at the atomic and subatomic levels. It is a fundamental concept in quantum mechanics and has been confirmed by numerous experiments.

4. Can quantization in a linear potential be observed in everyday life?

No, quantization in a linear potential is not observable in everyday life as it only applies to particles at the atomic and subatomic levels. However, its effects can be seen in various technologies, such as transistors and lasers, which rely on the principles of quantization in a linear potential.

5. How is quantization in a linear potential related to the Heisenberg uncertainty principle?

Quantization in a linear potential is related to the Heisenberg uncertainty principle as the uncertainty principle states that the position and momentum of a particle cannot be simultaneously known with absolute certainty. This is due to the quantization of energy levels in a linear potential, which results in an inherent uncertainty in the position and momentum of particles.

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