# Raising and lowering operators

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• John Finn
In summary, the method of using "raising" and "lowering" operators to solve second-order differential equations of the Schrödinger type has been well known for many years. This technique involves factorizing the equations into products of such operators, and can be applied to any potential. A useful resource for understanding this method is the paper by L. Infeld and T. E. Hull, "The Factorization Method", Rev. Mod. Phys. 23, 21 (1951). For a more accessible explanation, the section in R. Shankar's textbook "Principles of Quantum Mechanics (second edition)" titled "The free particle in spherical coordinates" on page 346 is recommended.
John Finn
Suppose you take a Schroedinger-like equation $$-\psi''+F(x)\psi=0$$. (E.g. F(x)=V(x)-E, and not worried about factors of 2 etc.) This is positive definite if $$\int \left( \psi'^2+F(x)\psi^2 \right)dx>0$$. Is so, you can write this as the product $$(d/dx+g(x))(-d/dx+g(x))\psi=0$$, i.e. as $$a^{\dagger}a=0$$ for $$a=-d/dx+g(x)$$ [its adjoint is $$a^{\dagger}=d/dx+g(x)$$] if $$g(x)$$ satisfies a Riccati equation, $$dg/dx+g^2=F$$. So raising and lowering operators hold for any potential.

Is this true? Is it useful? Is it well known?

MENTOR NOTE: Post edited changing single $to double$ for latex/mathjax expansion.

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That you can solve second-order differential equations (of the Schrödinger type) using "raising" and "lowering" operators has been well known for many years now, and the general method for factorizing the equations into products of such operators can be found in this nice paper:
L. Infeld and T. E. Hull, "The Factorization Method", Rev. Mod. Phys. 23, 21 (1951)

If you prefer more accessible explanation of the method before jumping straight into this paper, I suggest that you check out the section in R. Shankar's textbook "Principles of Quantum Mechanics (second edition)" titled "The free particle in spherical coordinates" on page 346. The general techniques and many specific examples (as regards to your question about the validity for "any potentials") can be found in the paper by Infeld and Hull linked above.

## 1. What are raising and lowering operators?

Raising and lowering operators are mathematical operators that are used in quantum mechanics to manipulate the energy levels of a quantum system. They are represented by symbols such as a and a and are used to raise or lower the energy of a quantum state by one unit.

## 2. How do raising and lowering operators work?

Raising and lowering operators work by acting on a quantum state, represented by a wavefunction, and changing its energy level. The raising operator a increases the energy of the state by one unit, while the lowering operator a decreases the energy by one unit. These operators follow specific mathematical rules and can be used to find the energy spectrum of a quantum system.

## 3. What is the significance of raising and lowering operators in quantum mechanics?

Raising and lowering operators are fundamental tools in quantum mechanics. They allow us to understand and manipulate the energy levels of a quantum system, which is crucial in studying the behavior of particles at the atomic and subatomic level. These operators also play a key role in the mathematical formulation of quantum mechanics, making them essential for any quantum physicist.

## 4. Can raising and lowering operators be used for any quantum system?

Yes, raising and lowering operators can be used for any quantum system. They are a fundamental part of the mathematical framework of quantum mechanics and can be applied to any system, regardless of its complexity. However, the specific form of these operators may vary depending on the system and its energy levels.

## 5. Are there any real-life applications of raising and lowering operators?

Yes, there are several real-life applications of raising and lowering operators. These operators are used in various fields such as quantum chemistry, solid-state physics, and quantum computing. They are also used in technologies such as lasers and transistors, which rely on the manipulation of energy levels at the atomic level. Additionally, the principles of raising and lowering operators are the basis for many advanced technologies currently in development, such as quantum cryptography and quantum teleportation.

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