# Quantum Mechanics - Lowering Operator

• izzmach
In summary, the conversation discusses how to show that a lowering operator, A, is a derivative with respect to the raising operator, A†. The approach involves defining a function in terms of A† and using commutation relations to solve for the next step. The process includes working out commutation relations for (A†)^2 and (A†)^3, and then proving a general formula using induction.
izzmach

## Homework Statement

let A be a lowering operator.

## Homework Equations

Show that A is a derivative respects to raising operator, A†,

A=d/dA†

## The Attempt at a Solution

I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can someone please explain briefly what i should do in the next step?

##A_+|n> = |n-1> \, , \, A_-|n+1> = |n> ## so
$$A_+\,A_-\, |n> = |n> \,\Rightarrow\,\left(A_+A_- + [A_+,A_-]\right)|n> = |n>$$

theodoros.mihos said:
##A_+|n> = |n-1> \, , \, A_-|n+1> = |n> ## so
$$A_+\,A_-\, |n> = |n> \,\Rightarrow\,\left(A_+A_- + [A_+,A_-]\right)|n> = |n>$$

sorry, i don't quite understand. Mind to explain it?

## |n> = \Psi_n ## my English is very poor for this theme. ## \frac{d}{dA_+}\left(A_+A_-+1\right) ## is that you need.

izzmach said:
I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can someone please explain briefly what i should do in the next step?
You didn't write out the commutation relations in your initial post. I.e., ##~[A, A^\dagger] = \dots ~?##

Start by working out ##[A, (A^\dagger)^2]##, and ##[A, (A^\dagger)^3]## using the Leibniz product rule for commutators. This should help you see the pattern. Then try to prove ##[A, (A^\dagger)^k] = k (A^\dagger)^{k-1}## by induction on ##k##.

## 1. What is a lowering operator in quantum mechanics?

A lowering operator is a mathematical operator used in quantum mechanics to decrease the energy of a quantum state by one unit. It is represented by the symbol "a" and is the adjoint of the raising operator. In other words, the lowering operator "a" and the raising operator "a†" are inverses of each other.

## 2. How does a lowering operator work?

A lowering operator acts on a quantum state by decreasing its energy by one unit. It does this by "annihilating" a particle in the state, causing it to transition to a lower energy level. This process is also known as "lowering" or "lowering the energy" of the system.

## 3. What is the relationship between a lowering operator and a quantum harmonic oscillator?

A quantum harmonic oscillator is a physical system that can be described by a set of quantum states and their corresponding energies. The lowering operator "a" acts on these states to lower their energy levels, making it an essential tool in the study of the quantum harmonic oscillator.

## 4. What are the properties of a lowering operator?

Some of the key properties of a lowering operator include its commutation relationship with the raising operator, which is given by [a, a†] = 1. It also has a zero eigenvalue, meaning that it will always result in a state with zero energy when acting on the ground state of a system.

## 5. How is the lowering operator used in quantum mechanics calculations?

The lowering operator is used in various quantum mechanics calculations, such as finding the energy levels and wave functions of a system. It is also used in the creation and annihilation of particles within a system. Additionally, it is an essential tool in solving the Schrödinger equation and understanding the behavior of quantum systems.

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