Consistency of solutions using operators (Energy ladder)

[mohammed.omar]

Hi,

I've been reading about the use of the lowering and raising operators to solve quantum mechanical problems and I have this question in mind.

In the book "Introduction to Quantum Mechanics", by David Griffiths, page 35, he notes out that there should be only one energy ladder, so what we obtain as energies is the only possible solutions. This pulls me to the following questions.

1. How did we make sure the ladder we obtained is the only energy ladder?
2. If this is true, this means that using the raising operator will get me the subsequent higher energy level. Is this true? How could I make sure there are no solutions between the points I obtained on the ladder?

 

[eljose79]

Hi there,

Great question! The concept of energy ladders and raising/lowering operators is a fundamental part of quantum mechanics. To answer your first question, the energy ladder we obtain through the use of raising and lowering operators is the only possible solution because it is based on the fundamental principles of quantum mechanics, such as the Schrodinger equation and the postulates of quantum mechanics. These principles have been extensively tested and have been shown to accurately describe the behavior of particles at the quantum level. Therefore, the energy ladder we obtain is the only possible solution that is consistent with these principles.

As for your second question, it is true that using the raising operator will result in subsequent higher energy levels. This is because the raising operator is defined as the operator that increases the energy of a system by one quantum. In other words, it adds one unit of energy to the system. This is a fundamental property of quantum mechanics and has been experimentally verified through various experiments.

To address your concern about the possibility of solutions between the points on the ladder, it is important to note that the energy levels in quantum mechanics are discrete. This means that there are no energy levels in between the ones we obtain through the use of the raising and lowering operators. This is also supported by experimental evidence, as we have never observed any energy levels in between the ones predicted by quantum mechanics.

I hope this helps clarify your questions. If you have any further questions or concerns, please don't hesitate to ask. Quantum mechanics can be a complex and challenging subject, but it is also fascinating and has been proven to accurately describe the behavior of particles at the quantum level. Keep exploring and learning, and you will continue to deepen your understanding of this fascinating field.